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Hopf bifurcations in magnetoconvection in the presence of sidewalls Zangeneh, Hamid 1993

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HOPF BIFURCATIONS IN MAGNETOCONVECTIONIN THE PRESENCE OF SIDEWALLSByHamid R. Z. ZangenehB. Sc.(Mathematics) Northea.stern University, Boston, Massachussets, 1977.M. Sc.(Mathematics) Northeastern University, Boston, Massachussets, 1980.A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESMATHEMATICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIA1993©Hamid R. Z. Zangeneh, 1993In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.MathematicsThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1W5Date:(9q3AbstractWe study multiple Hopf bifurcations that occur in a model of a layer of a viscous, electri-cally conducting fluid that is heated from below in the presence of a. magnetic field. Weassume that the fluid flow is two-dimensional, and consider the effects of sidewalls withstress-free boundary conditions. Our model partial differential equations together withthe boundary conditions have two reflection symmetries. We use center manifold theoryto reduce the partial differential equations to a two-parameter family of four-dimensionalordinary differential equations. We show that two different normal forms are appropri-ate, depending on the sizes of certain magnetoconvection parameters for large aspectratios. AVe denote the two normal forms by "Case I" and "Case II". In both cases weprove the primary Hopf bifurcation of standing wave (SW) solutions, and we prove theexistence of secondary Hopf bifurcations of invariant tori from the SW solutions. Weprove that the tori persist in 'wedges' in the parametric plane. In Case II we show thatthere are also secondary Bogdanov-Takens bifurcation points. Using this, we show thereare additional secondary and tertiary bifurcations of periodic solutions and invariant tori,and also argue that generically, there exist transversal homoclinic and heteroclinc points,and consequently open regions of parameter space that correspond to chaos of chaoticregions, and show the existence of quasiperiodic saddle-node bifurcations of invarianttori. Also, we show that in this case the system is a small perturbation of a system withthe symmetries of the square, as the aspect ratio approaches infinity.IITable of ContentsAbstractList of Tables^ viList of Figures viiiAcknowledgement1 Introduction 11.1 Basic concepts 31.2 Oscillatory convection in fluids and Hopf bifurcations with symmetry 71.3 Overview of the thesis ^ 92 Oscillatory instabilities of magnetoconvection equations 122.1 Magnetoconvection equations ^ 122.2 Symmetry ^ 162.3 Linear stability analysis ^ 172.4 The adjoint problem 292.5 Asymptotic results as m^oo ^ 312.5.1^Case I (fixed ç and Q) 312.5.2^Case II (decreasing^increasing Q as m^cc) ^ 343 Center manifold and normal form reductions 373.1^Abstract formulation ^ 373.2 The resealed problem  ^433.3 Center manifold reduction  ^463.4 Normal forms ^  513.5 Large aspect ratios  ^564 Evaluation of center manifold coefficients^ 594.1 Center manifold coefficients  ^594.2 Asymptotic results  ^694.2.1 Case I  ^694.2.2 Case II  ^724.3 Numerical results  ^745 Existence of invariant tori^ 785.1 The truncated normal form  795.2 Bifurcating periodic orbits  ^835.3 Bifurcation of invariant tori  ^905.4 Persistence of invariant tori  ^996 Secondary Bogdanov-Takens bifurcations^ 1096.1 Preliminary coordinate transformations  1106.2 The reduced system ^  1136.3 Bogdanov-Takens bifurcations in the truncated system ^ 1236.4 Approximate D4 symmetry^  1397 Conclusion^ 142Bibliography^ 150ivA Normal form coefficients^ 153A.1 Computation of normal form coefficients ^  153A.2 Limiting values of normal form coefficients  171A.2.1 Case I ^  172A.2.2 Case II  174B Numerical values of normal form coefficients^ 179List of Tables4.1 Normal form4.2 Normal form4.3 Normal form4.4 Normal form4.5 Normal form(continued)4.6 Normal form(continued)4.7 Normal form(continued)coefficients (Case I) for a =1,( = .1, Q = 10072^75coefficients (Case I) for a =1,e = .1, Q = 10072 (continued). 75coefficients (Case I) for a =1,( = .1, Q = 10072 (continued) 76coefficients (Case II) for a =1,e = .1,Q = 10072, k =1. . . 76coefficients (Case II) for a = 1,e = .1,a = 10072, k = 176coefficients (Case II) for a = 1, = .1, Q = 10072, k = 177coefficients (Case II) for a- = 1, = .1, Q = 10072, k = 1776.1 The values of v;,A,^aiR for a = 1,e =^= 107, and differentvalues of Q ^  1206.2 Values of aj,,(3j in (6.57), for a = 1,e = .1,m = 107, and different valuesof 0 130B.1 Normal form coefficients (Case I) for a = 1,e = .1, Q = 71.2^180B.2 Normal form coefficients (Case I) for a =1,( = .1, Q =712(continued). . 180B.3 Normal form coefficients (Case I) for a =1,( = .1, Q = 72 (continued).^180B.4 Normal form coefficients (Case I) for a =1,( = .1, Q = 10172  ^181B.5 Normal form coefficients (Case I) for a = 1, = .1, Q = 10472 (continued). 181B.6 Normal form coefficients (Case I) for a =^= .1, Q = 10172 (continued). 181B.7 Normal form coefficients (Case I) for a =1,( = .1, Q = 10672  ^182viB.8^Normal form coefficients (Case I) for a = 1,^=B.9^Normal form coefficients (Case I) for a = 1,( =.1, Q = 10672 (continued). 182.1, Q = 10672 (continued). 182B.10 Normal form coefficients (Case I) for a = 1,( = .01, Q = 712 ^ 183B.11 Normal form coefficients (Case I) for a = 1,( = .01. Q = 10072 ^ 183B.12 Normal form coefficients (Case I) for a = 1,( = .01, Q = 1017r2. ^ 183B.13 Normal form coefficients (Case I) for a = 1,( = .5, Q = 10072^ 184B.14 Normal form coefficients (Case I) for a = 1,( = .5, Q = 10472^ 18413.15 Normal form coefficients (Case I) for a = 1,( = .5, Q = 106r2^ 184B.16 Normal form coefficients (Case I) for a = 10-6,( = .01, Q = 4000072. 185B.17 Normal form coefficients (Case I) for a = 10-6, = .01, Q = 10672. 185B.18 Normal form coefficients (Case I) for a = 10-6, = .01, Q = 10872. 185B.19 Normal form coefficients (Case II) for a =^= .01, Q = 7T2, k = 1.^. 186B.20 Normal form coefficients (Case II) for a = 1,( = .01, Q.^= 72, k = 1(continued) ^ 186B.21 Normal form coefficients (Case II) for a = = .01,Q = 72, k = 1(continued) 186B.22 Normal form coefficients (Case II) for a = =^.01, Q = 71-2, k = 1(continued)^ 187viiList of Figures2.1 Level curves of the stream function ^ 222.2 Graphs of critical Rayleigh numbers 1?„.,(L) and R1(L) ^ 232.3 Graphs of critical Rayleigh numbers Rm(L) ^ 245.1 Bifurcation set for (5.4) ^ 845.2 Parameter values for which bifurcating invariant tori exist ^ 995.3 Curves in the (itiR,P2R) parameter plane corresponding to A1 and A9. . . 1005.4 Parameter values for which normally hyperbolic invariant tori exist^. . . 1045.5 Schematic bifurcation set for (5.1) ^ 1065.6 Bifurcation diagrams for (5.1) 1075.7 (a) Typical region (inside shaded circle) in (R, L) parameter plane forwhich results of this chapter apply to the magnetoconvection equations;(b) Magnification of the region in Figure 5.7(a) ^ 1086.1 The curves F1,^r4 for a- = 1,^= .1, e2 = 10072,k = 1, m = 107 ^ 1216.2 The curves F1, ..., F4 for a = 1,^= .1,^= 30072,k -= 1,m^107 ^ 1226.3 Parameters as in Figure 6.1. The circular shaded region shows the param-eter values that correspond to our bifurcation analysis. 1276.4 Parameters as in Figure 6.2. The circular and square shaded regions showthe parameter values that correspond to our bifurcation analysis. ^ 1286.5 Magnification of the square region in Figure 6.4 ^ 1336.6 Phase portraits for regions of Figure 6.5 ^ 1346.7 Bifurcation diagrams for Figure 6.5 135viii6.8 Magnification of the circular regions in Figures 6.3 and 6.4 ^ 1366.9 Phase portraits for regions of Figure 6.8 ^  1376.10 Bifurcation diagrams for Figure 6.8  1387.1 Schematic bifurcation sets for magnetoconvection equations in Case II . . 1437.2 Bifurcation diagrams for magnetoconvection equations in Case II. . . . 146ixAcknowledgementFirst of all, I would like to thank my parents, who were my first teachers. They fosteredin me the love of knowledge and wisdom. They also taught me to always evoke reasoningand never to accept anything without questioning, and to value education and truth overany material achievement and success. To my brother Bijan, who helped me to choosemathematics over the other fields of study, while I had a chance to study in other fields.He helped and inspired me throughout my studies.A very special thanks to my supervisor, Wayne Nagata, who taught me what I knowabout different aspects of the theory of differential equations and bifurcation theoryduring his very well-organized courses in differential equations and dynamical systems.He introduced my thesis topic to me, and during my research he provided me withconstructive and insightful comments, from which I learned a lot. He carefully read mythesis several times, and proof read my thesis very thoroughly. Overall without his helpand inspiration this work could not have been done.I would like to thank Dr. Colin Sparrow, who gave me a lot of inspiration and createda lot of interest in me for chaotic dynamical systems during my short stay in Cambridge.I would like to thank Dr. Robert Israel for his assistance in using the MG Graphicspackage for the graphs in this thesis. He answered my questions without any hesitation.I would like to express my great appreciation to the entire Mathematics Departmentat U.B.C. for its invaluable and extensive support during my stay here and especially tothose who taught the courses I took.I would like to dedicate this thesis to my wife Afshan and my two wonderful children,Alireza and Maryam, and express my deep gratitude for their patience, support andunderstanding during my studies at U.B.C.; without their support this thesis would nothave been possible.xiChapter 1IntroductionTo improve his living conditions and satisfy his curiosity, mankind needs to find, under-stand and study laws governing nature. In particular, differential equations modellingnatural phenomena have been shown to be extremely useful. The study of differentialequations themselves has produced an extensive theory, which in turn motivated moreabstract mathematical theories such as the theory of Lie groups, differential geometryand functional analysis.For one with little background in differential equations, the subject might be seenas a collection of tricks and hints for finding solutions. But with a little acquaintancewith the theory it becomes clear that, apart from linear equations, it is rarely possible tointegrate systems of differential equations and find the solutions explicitly, while theoremson existence and uniqueness of solutions do not convey much information about thebehavior of solutions. This shows the importance of the ideas and methods used in thequalitative study of solutions, or dynamical systems.The theory of dynamical systems has a rather short history. It can be considered tohave been originated by Poincare, who in the last decade of the nineteenth century revolu-tionized the study of nonlinear systems of differential equations, by combining techniquesof geometry and topology with analytic methods to study qualitative properties of solu-tions. Around this time, Liapunov also made important contributions to the qualitativestudy of differential equations. The work of Poincare and Liapunov was continued andfurthered by Birkhoff in the first part of this century. Birkhoff realized the importance1Chapter I. Introduction^ 2of the study of maps and emphasized discrete dynamics, since the qualitative study ofdifferential equations can often be reduced to the study of the iterates of an associatedmap (the Poincare map). Also, many problems and phenomena in the qualitative studyof differential equations can be seen in their simplest form in the study of discrete dynam-ical systems. After Birkhoff, the study of dynamical systems was relatively inactive inthe West. However, Soviet mathematicians such as Andronov and Pontriagin continuedto study differential equations from the qualitative point of view [10].In the early nineteen-sixties there began a great resurgence of interest in dynamicalsystems, mainly due to influence of Smale, Peixoto and Moser and in West, and Kol-mogorov, Anosov and Arnold in the Soviet Union. In his important survey article, Smale[43] reviewed the concepts of dynamical systems developed by many mathematicians(such as Anosov, Peixoto and Smale himself) during this period, and outlined a programthat was followed by many mathematicians, and which led to a good understanding ofa class of dynamical systems known as Axiom A or hyperbolic systems. His study ofVan der Pol differential equations motivated Smale to construct a two-dimensional map,with chaotic dynamics, which is now known as the Smale horseshoe. This example, stud-ied with the help of differential topological techniques and symbolic dynamics, led tothe study of chaotic dynamics in many other systems. In other significant mathemati-cal work, Kolmogrov, Arnold and Moser used hard analysis to develop their celebratedK.A.M. theory on the persistence of certain solutions (invariant tori) under perturbationsof integrable Hamiltonian systems. In addition, scientists studying nonlinear models ofnatural phenomena came to realize the power and beauty of the geometric and quali-tative techniques developed during this period, and at the same time raised interestingproblems of their own, which provided new sources of motivation for the theory beyondthe traditional questions arising from mechanics. Lorenz [28], a meteorologist, presentedan analysis of system of three quadratic ordinary differential equations which eventuallyChapter 1. Introduction^ 3created great interest in chaotic dynamical systems for mathematicians as well as scien-tists from other disciplines. The advancement of computer graphics has also contributedto great interest in dynamical systems among non-mathematicians. For more informationin the history of dynamical systems see [1, 16, 10, 43, 2].1.1 Basic conceptsA first step in the qualitative study of a system of differential equations is to study thedynamics of the system close to its fixed points or periodic orbits, since these representstationary or repeating behavior. Since the theory of linear equations is well-developed,one can consider the linearization of the system about its fixed points or periodic orbits[16]. If the linearized system is hyperbolic (i.e., all the eigenvalues of the linearizedsystem have non-zero real parts), then one can apply the Hartman-Grobman theorem[16] to show that the nonlinear system is topologically equivalent to the linearized systemin a small neighborhood of the fixed point or periodic orbit. However if the linearizedsystem is non-hyperbolic (i.e., the linearized system has at least one eigenvalue with zeroreal part), then the linearized system does not not give enough information about thenonlinear dynamics. In this case one uses center manifold theory [20, 5, 16] to establishthe existence of a locally invariant (center) manifold of solutions for the original nonlinearsystem and then study the dynamics close to the fixed points or periodic orbits restrictedto the center manifold. If the rest of eigenvalues of the linearized system have negativereal parts, then the center manifold is exponentially attracting, and the product of thedynamics restricted to the center manifold with a linear exponential decay is locallytopologically equivalent to the dynamics of the original system [5]. The center manifoldcan be approximated by its Taylor series to finite order, and this approximation is usuallysufficient to determine the dynamics on the center manifold. To be more precise, considerChapter I. Introduction^ 4a system of ordinary differential equations= Bx + f(x,y),= Cy + g(x, y),where (x, y) E Rn x Rni, and B and C are 71 X n and in x m matrices whose eigenvalueshave zero real parts and negative real parts, respectively. We assume that the nonlinearfunctions f and g vanish along with their first derivatives, at the origin. Then the centermanifold theorem implies that there is a locally invariant center manifold, which can berepresented by a local graph= {(x , y) : y = h(x), h(0) = Dh(0) = 0) ,where h : U^Ft' defined in some neighborhood U C R."2 of origin. The dynamics of(1.1) at the origin is locally topologically equivalent to= Bx + f (x, h(x)),^ (1.2)= Cy.Thus the local study of (1.1) is reduced to the study the n-dimensional system= Bx + f (x,h(x)).^ (1.3)The center manifold function h(x) satisfiesDh(x)[Bx + f (x, h(x))] — C h(x) — g(x , h(x)) = 0,^(1.4)which just expresses the local invariance of the center manifold. Using the center manifoldreduction, the dimensions of the problem can be reduced considerably. The relation (1.4)can be solved approximately for h(x) by expanding in a Taylor series, collecting terms oflike powers, and then solving term by term for each Taylor series coefficient of h(x). Forproofs of the above statements, see [5, §9.2].Chapter I. Introduction^ 5We illustrate these ideas with a simple example [16]. Consider the two-dimensionalsystem= xy,^ (1.5)= —y + ax2,and observe that one of the eigenvalues of the linearization about the origin of (1.5) iszero, while the other eigenvalue is negative. By the center manifold theorem, there existsa differentiable one-dimensional center manifold y = h(x) such that h(0) = 0, le (0) = 0.By substituting the Taylor seriesh(x) = ax2 + bx3 + ,into (1.4), i.e.,(x)[xh(x)] + h(x) — ax2 = 0,we obtain h(x) = as2 + 0(x4). Thus the reduced system representing the dynamics onthe center manifold is= xh(x) = ax3 + 0(x5).^ (1.6)It is easy to see that the fixed point x 0 in (1.6) is asymptotically stable if a < 0 andunstable if a > 0. Therefore (x, y) = (0,0) in the system (1.5) is asymptotically stable ifa < 0 and unstable if a > 0.If after a center manifold reduction the reduced system has dimension greater thanone, the system can be simplified further by using the method of Poincare-Birkhoff normalforms. The basic idea in normal form reduction is to construct appropriate near-identitynonlinear coordinate transformations which annihilate certain nonlinear terms in theTaylor expansion of the system. The method of normal forms is of fundamental impor-tance in local theory of differential equations. For a discussion of normal form theory,see [1, 16].Chapter I. Introduction^ 6The methods of center manifold theory and Poincare-Birkhoff normal forms are im-portant not only in the study of a single system of differential equations, but also inbifurcation theory, where one attempts to analyze a parametrized family of systems ofdifferential equations. One concentrates on bifurcation points, i.e., those parameters forwhich the systeni is structurally unstable (a dynamical system is structurally stable ifunder any small perturbation the perturbed system is still topologically equivalent tothe original system). Thus arbitrarily small perturbations of parameters from a bifurca-tion point will produce topologically inequivalent dynamics. One attempts to find andclassify all the topologically inequivalent dynamics possible when parameters are variedin a neighborhood of the bifurcation point. If the analysis is local in a neighborhoodof a fixed point or periodic orbit, then one can use center manifold theory and normalforms to simplify the analysis. An illustrative example is that of Hopf bifurcation in aone-parameter family. This bifurcation is associated with pure imaginary eigenvalues forthe linearization, and periodic solutions for the nonlinear system. See [16] for more infor-mation. If two or more parameters are varied, then more degenerate bifurcation pointscan be found, and this typically enables one to describe a wide range of behaviors usinglocal analysis. This area of research has been very active in recent years, and there arestill many open questions. See [16, Chapter 7] for a survey of two-parameter bifurcations.There exists a parallel theory for discrete dynamical systems, i.e., qualitative study ofiterated maps.Mathematical models of many physical problems have some sort of symmetry. Thesymmetry can be intrinsic to the physical system, or come from the idealization of anapproximate symmetry. Symmetry leads to more degenerate behavior, yet at the sametime the presence of symmetry can simplify the analysis. The books of Golubitsky et al.[13, 15] give a. systematic treatment of bifurcation with symmetry from the group theorypoint of view (see also the references therein).Chapter I. Introduction^ 7One can use dynamical system methods to study certain partial differential equations.For example, to study local bifurcations in a system of parabolic partial differentialequations, one usually find the parameter values for which the linearized system has zeroor pure imaginary eigenvalues about its steady state solution. Then by considering thesystem as an evolution equation in a Hilbert space, or more generally, a Banach space,one can then apply center manifold theory for infinite-dimensional systems. One thenobtains a finite-dimensional system of ordinary differential equations, and then one canstudy the bifurcation of these reduced equations. If the system of partial differentialequations have some symmetry, then the center manifold reduction can be done so thatthe reduced system of ordinary differential equations has the corresponding symmetry.1.2 Oscillatory convection in fluids and Hopf bifurcations with symmetryIn this thesis, we study the nonlinear dynamics of a model of a horizontal layer of aviscous, electrically conducting fluid that is heated from below in the presence of avertical magnetic field. Such situations arise in astrophysics, geophysics and in laboratoryexperiments. We consider two-dimensional motion near the onset of oscillatory (timeperiodic) convection. Unlike previous studies of magnetoconvection, we consider theeffect of sidewalls, especially distant ones. The magnetoconvection model consists ofa system of partial differential equations, together with boundary conditions. Severalparameters occur naturally in the model, and these represent physical quantities.In models of two-dimensional convection (e.g., magnetoconvection, convection in bi-nary fluid mixtures) the symmetry group 0(2) of rotations and reflection of the circleis often present. This symmetry is due to equivariance of the model partial differentialequations (e.g., Navier-Stokes equations) under spatial translations and reflections, andChapter 1. Introduction^ 8the use of periodic boundary conditions. The theory of Hopf bifurcation in the pres-ence of 0(2) symmetry (e.g. [14, 29]) has been successful in accounting for a variety ofphenomena observed in experiments, especially on binary fluid mixtures. As the fluidlayer is heated from below with increasing intensity, in these experiments the motionlessconduction state loses stability to oscillatory modes and appears to undergo Hopf bifur-cations as time-dependent convection onsets. Spatio-temporal patterns such as standingwaves and travelling waves have been observed. The corresponding experiments in mag-netoconvection are more difficult, and we know of no experiments corresponding to thephysical situation we study in this thesis. However, see [39, 40] for descriptions of relatedexperiments in magnetoconvection.In the dynamical system analysis of this phenomenon, the normal form describing thisbifurcation has 0(2) symmetry (e.g., [15]). In this bifurcation, two branches of symmetry-breaking solutions, denoted by standing waves (SW) (a family of solutions with reflectionsymmetries) and travelling waves (TW) (solutions with spatio-temporal symmetries) arecreated [29]. While it can be hoped that the idealizations that are responsible for the0(2) symmetry (infinite layer, periodic boundary conditions) will not qualitatively affectthe dynamics much, it is interesting and useful to consider the effect of breaking thesymmetry of the system, especially if the corresponding idealization is not satisfied bythe real system. For example, periodic boundary conditions are an approximation tomore realistic models with only reflection symmetry, due to presence of distant sidewallswhich break the continuous translation symmetry of 0(2).To consider the effects of sidewalls, one attempts to reduce the model to a simplerone that captures the dynamics of the original system, at least under certain restrictions.There are several ways to achieve this. One traditional approach has been to use theformal method of multiple scales which results in a simplified partial differential equationfor a slowly varying envelope function [24, 9]. However, there is no rigorous explanationChapter 1. Introduction^ 9for validity of this formal method, and in fact [33] showed that the bifurcation resultsof [9] using this method are only valid for very small range of parameters. To accountfor the effect of sidewalls this model can be considered as symmetry breaking pertur-bation to the idealized system. Another commonly used method is a formal Galerkinreduction, by systematically using only finitely many modes of a Fourier expansion ofthe solutions. This method usually is justified by physical intuition but often can bemade rigorous mathematically. To reduce the problem to a finite dimensional system ofordinary differential equations in a rigorous way, the original system of partial differentialequations is considered as an evolution equation in a Hilbert space (e.g. [36, 37, 38]).At critical parameter values, the linearized partial differential equation has only finitelymany eigenvalues with zero real parts. If the rest of eigenvalues at these parameter valueshave negative real parts, then using center manifold theory [20, 47], the existence of anattracting center manifold can be proved, and the evolution equation restricted to thecenter manifold leads to a finite-dimensional system of ordinary differential equations.The reduced system carries the symmetry of original system, and it can be considered asa system with broken 0(2) symmetry (e.g. [32]). The sidewalls destroy the translationalsymmetry S0(2), but keep a reflection symmetry Z2. Such an approach has been takenby several authors, and in particular has been used to describe the effects of distantsidewalls on the onset of steady convection in the Rayleigh-Benard problem [33]. For0(2)-equivarient Hopf bifurcations, the effects of various different symmetry breakingperturbations have been considered in [29, 34, 6].1.3 Overview of the thesisWe consider our magnetoconvection problem in a rectangular regionQL = {(x,y) : —L < x < L, 0< y < 1},Chapter 1. Introduction^ 10with aspect ratio 2L, and we use boundary conditions which are extensions of thestress-free boundary conditions commonly used in the regular Rayleigh-Benard convec-tion problem [48]. With these boundary conditions the magnetoconvection problem hasZ2 e Z2 symmetry. A parameter R (Rayleigh number) gives a measure of the inten-sity of heating from below. Using standard methods (e.g. [46]), we express the systemas an evolution equation in a Hilbert space. We prove that the spectrum of the lin-earization K(R, L) about the trivial solution of the evolution equation, consists entirelyof isolated eigenvalues with finite multiplicities. Along a particular family of curvesR Rfiz(L), 771 7-= 1, 2, 3, ... (curves of values of the Rayleigh number R as a function ofhalf the aspect ratio L), the linearized operator has pure imaginary eigenvalues. Two con-secutive curves Rni(L), Rm+i(L) intersect at a single point defining a particular value ofL Lm. At such a point of intersection K will have a double Hopf point, as two differentspatial modes simultaneously become unstable. We prove that for large enough imposedmagnetic fields, at (R, L) (Rm(L„,),Lm) all the rest of the eigenvalues of K have neg-ative real parts (Chapter 2). Then using the center manifold theory for parabolic partialdifferential equations [20] we find a reduced parametrized family of four-dimensional ordi-nary differential equations which represents the dynamics on an exponentially attracting,locally invariant center manifold. By using normal form theory, we simplify the reducedequation further. We show that for large L, depending on the size of other parametersin magnetoconvection problem, two different normal forms, which we denote by Case Iand Case II, will be appropriate. Case II corresponds to convection with very strongmagnetic fields, in fluid with a very small ratio of magnetic diffusivity to thermal diffu-sivity. Both the normal forms have a double Hopf point near 1 : 1 resonance (Chapter3). After a long calculation, we find explicit expressions for the normal form coefficientsand their asymptotic behavior for large L, in both Cases I and II. We then evaluatethese coefficients numerically for some parameter values (Chapter 4, Appendices A andChapter I. Introduction^ 11B). In both Cases I and II we prove the existence of primary Hopf bifurcations of twofamilies of standing wave solutions, which we denote by SW0 and SW„. Also, we provethe existence of a secondary Hopf bifurcation of invariant tori from the SW solutions,and the persistence of the tori in open regions ('wedges') in parameter space (Chapter 5).For large aspect ratios in Case II, we also find more complicated dynamics. We prove theexistence of secondary Bogdanov-Takens bifurcations points at a particular parametervalues, and the existence of such bifurcation points implies more complicated dynamicsand leads to further bifurcations of invariant tori, existence of transversal homoclinic andheteroclinic points, quasiperiodic saddle-node bifurcations of invariant tori, and conse-quently the existence of open regions in parameter space for which the dynamics of thesystem is chaotic. Also, we show in this case that the system is a small perturbation ofa system with D4 symmetry, in the limit as L approaches infinity.Chapter 2Oscillatory instabilities of magnetoconvection equationsIn this chapter, we describe the physical basis of our problem, and perform some pre-liminary analysis. In the first section, we present the partial differential equations andboundary conditions that describe magnetoconvection in a two-dimensional layer. Thenin §2.2 we discuss the symmetry which the system enjoys. In §2.3 we discuss the lin-earized stability analysis of the trivial, motionless solution of the magnetoconvectionequations and find that there are an infinite number of values of the aspect ratio 2Lin ofthe layer, in = 1, 2, ..., such that the linearized equation has pure imaginary eigenvaluesand both "even" and "odd" eigenfunctions (oscillatory instabilities). In §2.4 we considerthe adjoint problem to the linearized eigenvalue problem, and compute its eigenfunctions.Finally, in §2.5 we study some of the asymptotic behavior of the linearized system forlarge aspect ratios.2.1 Magnetoconvection equationsIn this section, we consider the partial differential equations that describe the stateof an electrically conducting fluid, in the presence of an externally imposed verticalmagnetic field. The electrical conductivity of the fluid and the presence of magneticfields contribute to effects of two kinds. Due to the motion of the electrically conductingfluid across magnetic lines of force, electric currents are generated and the associatedmagnetic fields contribute to changes in the existing fields. In addition, fluid elementscarrying currents transverse to magnetic lines of force contribute to additional forces12Chapter 2. Oscillatory instabilities of magnetoconvection equations^13acting on the fluid elements. The equations describing this situation are (Chandrasekhar[3]):^at^(1/p0)[vAu — pgey — VP + (1/R0)(V x B) x B1 — (u V) u,OTKAT — u • VT,OtOB71AB + V x(ux B) ,OtV • u = 0,V • B = 0,where A is the Laplacian operator, V is the the gradient operator, u is the fluid velocity,p is the density, T is the temperature, B is the magnetic field, P is the pressure, ey isthe unit vector in vertical direction, g is the acceleration due to gravity, Po is the densityat some reference temperature To , v is the viscosity, K is the coefficient of thermomet-ric conductivity, po the magnetic permeability, and 7/ is the magnetic resistivity. Theparameters g, ito ii are all assumed to be positive constants.The first of the above equations is the equation of motion, and can be derived fromthe conservation of momentum, while the fourth equation is the equation of continuity,and can be derived from the conservation of the mass. We have used the Boussinesqapproximation, which treats the density p as a constant po except where it appears inthe external force in the momentum balance. The second equation is the equation ofheat conduction, and is obtained from the conservation of energy. The third and lastequations, which express the interaction between the fluid motion and the magneticfields, can be derived from Maxwell's equations. We assume that the density obeys anOberbeck-Boussinesq equation of statep = pop. — a(T — TO],^ (2.2)where a is the coefficient of thermal expansion, assumed to be a positive constant.Ou(2.1)Chapter 2. Oscillatory instabilities of inagnotoconvection equations^14We simplify this problem by assuming that u and b are constant in the z direction,and we write u = (u, v) , where u is the horizontal component of the fluid velocity andv is the vertical component. Similarly, we write the magnetic field as B = (B, By). Weassume that the fluid is confined between the two horizontal planes y = 0 and y = h (> 0)and that the temperatures on these two planes are maintained constant at T = To ony = 0, and at T = Ti on y = h, with To >In the presence of a uniform, vertical magnetic field, the system (2.1)—(2.2) has thetrivial motionless solutionu(0) = (0, 0),13(°) = (0, Bo),^ (2.3)T"^To — (To — (yA),P"^Po — PAY + a(To —^(Y2/2/1)].Now we consider finite amplitude perturbations from the motionless solution defined byBTP===B" + b,T" + 0,P(()) + \.(2.4)By resealing the variables as= h , y =^t = (1/2/K)f, u = (K/h)ft,X = (Povh1112), 0 = (To — T1)0, b = Bob,and then "dropping the bars", we obtain-57 = o- [Au — VN' + ROey + (Q (V x b) x (ell + b)] — (u • V) u,Chapter 2. Oscillatory instabilities of inagnetoconvection equations^15DOatabatV • u+ v u•VO,(Ab + V x [u x (ey + b)],=0,(2.5)V • b = O.The parameters appearing in the convection equation (2.5) are all positive, and aredefined byya(To — T1)113R =^(Rayleigh number),4h2Q =( Chandrasekhar number ),Po Po lio va = v^( Prandtl number ) ,=^k (magnetic Prandtl number).Note that the Rayleigh number R is proportional to the temperature difference betweenthe lower (warmer) and upper (cooler) boundaries, and Q increases with the strength ofthe imposed magnetic field.Our system of equations is accompanied by boundary conditions. The simplest bound-ary conditions to work with analytically are the extensions to magnetohydrodynamics ofthe "stress-free" boundary conditions used for ordinary Rayleigh-Benard convection [3].We assume that the fluid is confined to the rectangular regionL^( , y ) : 0 < y < 1,^L < x < LI ,^(2.6)the temperature is kept constant at the upper and lower boundaries y = 0, 1, and thesidewalls at x = L, —L are insulated. The total magnetic flux through the region remainsconstant, and the normal velocity, together with the tangential components of both theviscous and magnetic stresses vanishes on all boundaries. ThusOn^ Ob,v = 0 = br =^=0. oim y = 0,1,Oy Dy(2.7)Chapter 2. Oscillatory instabilities of magnetoconvection equations^16^017^09^x^-g-37 = ba.^3bY =0Ox^' on = L, —L.These boundary conditions do not correspond to a physical situation that is easily pro-duced in a laboratory, but they are commonly used for computational convenience sincethe eigenfunctions of the linear problem yield sines and cosines. Gibson [12] shows thatthe criteria for the onset of instability are not substantially altered when more realisticboundary conditions are adopted.2.2 SymmetrySymmetry can play an important role in the bifurcations of systems of differential equa-tions, and has received much attention in recent years (see, e.g., Golubitsky et al.[13, 14, 15]). System (2.5)-(2.7) possesses a Z2 e Z2 symmetry. To explain this fact, wedefine the action J on the dependent variables corresponding to the reflection x —xthrough the vertical midline x = 0 of the layer:JO(t,x,^=^—x,^if 0 = n or bx,^(2.8)^JO(t, x, y) = cb(t, —x, y)^if ç = v, v, 8 or by.There is an additional symmetry, due to our use of the Boussinesq approximation, withrespect to the reflection y —> 1 — y about the horizontal midline = 1/2 of the layer. Wedefine the action /3 on the dependent variables corresponding to this symmetry by.3 cb(t. , , y) = 0(t, x , 1 — y)^if 0= u, by or N.^(2.9),30(t,^—0(t,^—^if o^v, 0 or b., .The transformations .1, 13 generate a group of symmetries for equations (2.5) and (2.7)that is isomorphic to the group Z2 e Z2. \\Te will exploit this symmetry in our treatmentof the onset of convection.Chapter 2. Oscillatory instabilities of magnetoconvection equations^172.3 Linear stability analysisTo analyze local bifurcations from the steady state solution of the system (2.5)—(2.7)one first considers the linearized equation as an approximation to the original system.We expand an arbitrary disturbance in terms of some suitable set of normal modes, andexamine the stability of the system with respect to each of these modes. In our casewe use Fourier modes which satisfy the boundary conditions. Then we seek solutions4:1) = (u, v,O,bx,by) in the formt) = (i)(x, Y) (2.10)We will justify this formal stability analysis in Chapter 3. Using equations (2.5), (2.7),and (2.10) and then linearizing, it is easy to see that a satisfies the following eigenvalueproblem with boundary conditions (2.7):o-AU + a-CQ^—^— a —=ay^Ox^0:rAVa^-1-aRO—a 04),0YAt) + (2.11)cz\b + Oft„. 0y„ ,Dü(AbyoOa^0i, = 0.+ 0,y0i)x^oby0.ar^ayWe will find two sets of solutions.i) Even solutions ( m, even ):= c1 sin(m7312L) cos(n7y),Chapter 2. Oscillatory instabilities of magnetoconvection equations^18= c2 cos(m742L) sin( wry) ,c3 cos(m/xx/2L) sin(n7y),c4 sin(m71-42L) sin(wry),= c5 cos(771742/) cos(n7y),= c6 cos(mmr/2L) cos(1/7y).(2.12)ii)Odd solutions ( m, odd ):Ii = c1 cos(mrx 2L) cos(wry) ,= e9 sin(in,742L) sin(n7y),= C3 sin(/n:7142.E) sin(n7y),^ (2.13)= C4 cos(nrn-42L) sin(wry),=C5 sin(m742L) cos(wry),= c6 sin(trimr/2L) cos(n7y).Substituting (2.12) or (2.13) into (2.11), we find in both cases that eigenvalues a mustsatisfy the cubic equationa3^ 2+ (a + ( +1)P„,„awhere++[711272aRP',. ;(a( + a + ( ) + an2(Q72( 2.14)4PL21 a,„„ ',T12720.R(an2(Q72pmn^+ a(P ^= °.4L2=^2 ((in/ 2L)2 + i2). (2.15)To simplify (2.14), we putQ n 13;1;111= 9 71 971-^-R = (2.16)71- 2^9a^=^s P111111Chapter 2. Oscillatory instabilities of magnetoconvection equations^19so then s must satisfyS3 + (0- +^1)S2 s [a + (qn, „ —^ ) + ((a + 1)]+0-((1 —^n^g„) = o.^(2.17)Since we are interested in Hopf bifurcation from the trivial solution, we look for pureimaginary roots. Equation (2.17) will have pure imaginary roots s = ±ico„,„, and (2.11)will have pure imaginary eigenvalues = ±iP,,,nconi„, if^[(q.,^+ 11,1•77. =^= (a + ()+ ^awith99^(7(qMn (1 — () WT:717)^+(- +^(a +1)0-W7217,2(1 — () (a +We require that co,2 > 0, which is satisfied if ( < 1 and((1 + a)(111111 > go = a(1 — ()(2.18)(2.19)(2.20)9PT.,„ q0Q >  7 .1) 9 n- 7i-Remark 2.1 Equation (2.17) has a zero eigenvalue if= 1 + q,,,„which implies that (2.11) has a zero eigenvalue for parameters satisfying4L2p72 (1);;?H, + 71272Q)97^9-171-(see Figure 2.2). However, if Q is sufficiently large, the imaginary eigenvalues occur ata lower value of the Rayleigh number, thus the onset of instability is through oscillatorymodes.17 =-  R(L) =Chapter 2. Oscillatory instabilities of magnetoconVOCti011 equations^20Now by substituting a = ir.P71.„co„ into (2.12) and (2.13) we find eigenfunctions of(2.11) for both cases (i) and (ii). In case (i) we denote the eigenfunction as the eveneigenfunction (DE and in case (ii) we denote the eigenfunction as the odd eigenfunctionV. We have2L .sm(rn,71-42L) cos(n7y)712,P71171(1^iWm)27127iii]?,, n^L/711u )I/ 71^cos(m7rx/2L) cos(n7ry)\ Pnin(( +2L— cos(m742L) cos(n7y)insin(m742L) sin(n7y)1PM 11 ( 1 + 1:())717. )2n2 Lirin^n^iWI17 11 sin(7n742L) cos(n7y)Pm. n^iW7/1 /1 )for in even, and° It4)77E111 =cos(rnmv/2L)sin(n7y)1 cos(rnirxi2L) sin(n7y)sin(rn71.12L) sin(n7y)sin(rnmri2L) sin(n7y)cos(m7.42L) sin(n7y)(2.21)(2.22)for in odd, where (I) =^v, 0, ba., by)T . We will not include the pressure term x, since itcan be recovered using the velocity terms (u, v).Now from (2.16) and (2.18) the critical Rayleigh numbers 1?,„.1(L), for which (2.11)has pure imaginary eigenvalues a = ±iPm„,..o, ^aren271-2(Q^(C^1)./,„ 1 4L2P„R„(L) =^+^[ 721112 •a +1 (2.23)To simplify the above expression, we temporarily fix in, and let X = 4L2/7112, P„ (x) =Rmn(L). Then the critical Rayleigh number I? for each n is given byChapter 2.^Oscillatory instabilities of magnetoconvection equations 21f?„(x) = n2Ax(n2^14) + Bx(1/xwhere+ ()(QA^ B72^± 1)(a= =andn2)3,± 074(2.24)(2.25)a + 1It is clear that f/n(x) is minimized when n = 1, andurn^(x) = oo,^lim T?i(x) = oo.x—).00Since(A + B)x: —3Bx — 2BPi = x3has only one positive root x = x*, and since R'I'(x) = 6L3(x + 1)/x4 > 0^for x > 0,R1 will have its minimum at x = f. Since x = (2L/m)2, we have infinitely manycurves Rmi (L) = P1(4L2 /in2) depending on in, and each will have its miniinum point at= in V7r*/2, with minimum value R„,i(L.7) = I? (x).The number in in Rnd(L) corresponds to the number of rolls in the region 12L, thatwe expect to bifurcate from the steady-state solution at the critical Rayleigh numberR = R1(L) of the original convection problem. We refer to these solutions as "even"when in is "even" and as "odd" when in is odd (Figure 2.1).We define L„, by the value of L where the curves of the critical Rayleigh numberscorresponding to odd and even solutions intersect, i.e., L = L„, is the unique solution ofRmi(L) = Rm.+1,i(L). It is clear that L„, lies between L7.„ and /441, i.e.,where A =ITi..7" satisfiesinA^(in + 1)A2 < Lm <^9 (2.26)2^3 ^a(T2+ — —1 0.^(2.27)M A4^+ 1)(1 + 072x-40+Chapter 2. Oscillatory instabilities of magnetoconvection equations^22(a)(b)Figure 2.1: Level curves of the stream function: (a) the even mode 771 = 2; (b) the oddmode in = 3.Chapter 2. Oscillatory instabilities of magnetoconvection equations^2370000600005000040000300002000010000m=1 m=2 m=3 Til= 4 Tr1=5■^1^1^1^\1 I I1^1^I^I1 I I I1^I^I^I1 I I II^I^I^II 1 I 1I^IR„,(L): Hopf bifurcationR,°„(L): Steady state bifurcationFigure 2.2: Graphs of critical Rayleigh numbers Rm(L) and R(L)vs. L for in = 1,...,5,with a = 1, ( = .1, Q = 10072. Solid curves represent the graphs of R1(L) anddashed curves show the graphs of R.,9,(L). For a given in, the graphs of R1(L) andR1(L) intersect at L = (.1112)m and R 56000.R'4000-3000-2000-77i = 1 77 = 2^m = 3Chapter 2. Oscillatou instabilities of magnetoconvection equations^241000-0.5^1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^LFigure 2.3: Graphs of critical Rayleigh numbers R,„(L) vs. L, for .m = 1,...,15 witha = 1,^= .1, Q = 10072.Chapter 2. Oscillatory instabilities of inagnetoconvection equations^25We observe that (2.27) implies that A is adecreasing function of Q, and 0 < A <(see Figure (2.2)).To simplify our notation further, we put(2.28)and for each fixed m, we putPl = Pmi, P9 = PI71+1,11 W = W7711^=^+ j • (2.29)Then if in is odd, the critical eigenfunctions become2L„„ cos(m7.42L„„)cos(7.0711sin(m7x/2L„„.) sin (71-y)( 1 +1 iw ) sin(rnirx/2Lm.) sin(7y)2L,„.7nipi(c ^cos(rn742L7n)sin(rY)7/7T^((^) sin(m742L71,)cos(7Y)2L„,+^sin( (in + 1) mr/2L7n) cos(iry)I/1^1COs(( + 1)7x/2/,,n)sin(Ty)1P2(1 ^1w2) cos( (in + 1)-A-42L71) sin(Ty)271-L„,± P2( ^1c02) siu( (rn + 742L,) sin (7ry)71^ cos( (in + 1)742L,„) cos(y)P2(( iw2)(Di = sena =and(711-1-1,1(2.30)(2.31)Remark 2.2 Without loss of generality we assume in is odd, since when in is even onlythe roles of (Di and 4)9 are interchanged.Now we can prove the following proposition:Chapter 2. Oscillatory instabilities of magnetoconvection equations^26Proposition 2.1 Fix a > 0, 0 <^< 1, Q > 0. and fix a positive integer m. IfQ is sufficiently large, then when L = L,„ and R = R„, the linearized system (2.11)has eigenvatues j = 1,2, and the rest of eigenvalues have negative real parts,uniformly bounded away from the imaginary axis.Proof: For fixed 777,, by (2.14), when L = L,„ and R = Rfi, the eigenvalues a satisfya3 + Ca2 + Da + E = 0,^ (2.32)whereC^(^)+ + 1 . Pn,m2 71-- a Rfi,= 13(a( + a + () + an2(Q72 ‘1P,„„1,= m272R,„ + P?:?3??? = 0,n ) 7T P-^-2( - 2- 11111 4Landp ^2 {(7/2/2.Lid2 + 722] , 171, ii = 1,2, ....By the Hurwitz criteria for stability [17] roots of equation (2.32) will have negative realparts if C, CD — E and E are all positive. Clearly C > 0 and is uniformly bounded awayfrom zero for all in and n. Also it is clear that CD — E and E are increasing in n, andtherefore will be minimized at 71 = 1, so we consider their values when n = 1. Using(2.18) we obtainCD — E a((a^()7r2(2 (17/2Pi1,i — in2Pad) IT?2(a +1)(a + ()(( + 1) [ill 2 P —al 2 (2.33)Substituting the definitions of Prni and Pn,i into (2.33) after some simplification we findthat CD — E is positive ifry<T2^77/277/2(7722 + 7772)^377/27722E(777,) = (7/22 — 7772) ^, + 1   (2.34)[72(a + 1)(( + 1)^(2L7)6^(2.Lth)4Chapter 2. Oscillatou instabilities of inagnetoconvection equations^27is positive. Now using (2.23) and Rth(L77,) = Rth+ (Lth), after some simplification we getaCQ =^ (2.35)30-1+ 1)27112^7112(711 + 1)2[11/2^(ñ1t + 1)1 72(o- + 1)(( + 1) + 1^(2L,)4 (2L,72)6We substitute (2.35) into (2.34) to get3 [(IT/ + 1)2 — ni,2]E(in)^f/-12 (7T/2 - in2)(2L771)4(Th ± 1)2 [7/72 ± (IT/ + 1)2] -^+ 717.2)(2.36)(2.L.,))6From (2.36) it is evident that E(in) = 0 when in = Th. or in = fiz + 1 and we have twopairs of pure imaginary eigenvalues, while the third eigenvalues — (a + + 1) are real andnegative. For 771 > ñi + 1 or in < ili,13(m) > 0, and therefore CD — E are positive forall values of in and n. To find a uniform bound on .E(in) we notice that for all in 0 inand in 0iii + 1, we haveE(in) > min^— 1), E(iTi. + 2)1 > 0,^(2.37)where in is fixed.Now we need to show that E is also positive uniformly bounded away from zero, forsufficiently large Q. For 71 = 1 we can rewrite E in the formE = a( [P2n1 Pr1(72(2 Rip) 4- 72Rm] •^ (2.38)E is a cubic function in P„,.1, by finding its mininnun it is easy to show thatE > 2fl — (4/3) (Rh) — 72Q)3/2^(2.39)therefore E> 0 if— 72Q)3/2 < (3/d/4)72R,,„.^(2.40)For a given a, (, Q and ili , one can always check whether (2.40) is satisfied. However, Rrhdepends on Q, so we proceed to show that (2.40) is always satisfied for sufficiently largeChapter 2.^Oscillatory instabilities of magnetoconvection equations^28Q. It is clear that (2.40) is satisfied and E will be positive uniformly bounded away fromzero if97r-Q — Ra, > 0.^ (2.41)^We note that R,^= Rin.±i(Lat), and therefore9 /-1^1-39^7r^— =^{gth+1,1 — rth+1,1 2L,,T1^2 (2.42)[1 + (fit -I- 1 ) ]}On the other hand, using (2.35), after some calculation we have7F4aCQ2P3<^—372P? (2.43)(a +1)(( +1)and it follows that2L„,^2(a + 1) (<" + 1))2 .(2.44)+ 1 a(q/-0 + 1 ,1Then using (2.18) we have+ a + 0(.bh+1,1^3(a +^ )(172Q +>^[(1 (7+11^ 0-2(1 +^(1 +^(0- +(2.45)(72(qr-i) + 1,1It is clear that the expression inside the square brackets on the right hand side of (2.45)is increasing with respect to qi•i+1,1, and it is easy to check that it is positive for3(1 +02(1 +a) (1,ii+1,1> qi =^ (2.46)a((1 — ()i.e.,97T->qi.^ (2.47)1;?1-1.- 1 ,1From (2.26) and (2.27), it follows that for fixed a, and fit the right hand side of (2.44)is 0(Q2/3) as Q^oc, hence (2.44) is satisfied for all sufficiently large Q. Q.E.D.Remark 2.3 For specific choices of parameter values, we can check (2.40) and (2.41)numerically (see Chapter 4).(4,1, 4).2)L^(24) u 1 ft 2 + 1 2 + 0 1 0 2 + k!. 6 :12. + b6y2) dxdy, (2.48)-L1 j•LChapter 2. Oscillatory instabilities of magnetoconvection equations^29We briefly summarize the results of this chapter so far. We have considered thelinearized stability of the magnetoconvection problem for fixed values of a, (,Q, where> 0, 0 < < 1 and Q > Q0, as we increase the Rayleigh number R through thepositive numbers. For given L > 0, when R < R.11 (L) for all in, all the eigenvalues ofthe linearized eigenvalue problem (2.11) have negative real parts, and by the principle oflinearized stability, the steady state solution (2.3) is asymptotically stable. If L =then when R =^= I?,„(1,n) the linearized eigenvalue problem will have two pairsof imaginary eigenvalues, while the other eigenvalues have negative real parts, and weexpect a double Hopf bifurcation with reflection symmetry.2.4 The adjoint problemIn this section we consider the adjoint problem to the linearized system (2.11), andcalculate its eigenfunctions when L = Lm and R /?. This information will be usefulfor our nonlinear analysis in the next chapter.We define the adjoint of linearized eigenvalue problem with respect to the inner prod-uctwhere(1)-1 =^Oi ,^, bly)T ,j = 1, 2.and the overbars denote complex conjugation. Integration by parts yields the adjointeigenvalue problem to (2.11),0-k* )(11,-0,:raw OM abe,*ay^ar )a (Ai,* pl + r0Y^= ov-,29* + arn, = dr,^(2.49)2L„,^ cos(m71-42L)cos(7ry)insin(in7rx/2L71„) sin(71-y)aR„,sin(rn742L,)sin(7Y)P K —^cos(mmil2L,„) sin(7y)IP K^. )^ sin(m7.17/2L) cos(7Y) j\ I — zwith*^(T)*01 — "--"m P1(1 —20-(Q7L„„Chapter 2. Oscillatory instabilities of magnetoconvection equations^30(Ay; — 0-(Qaolyt* =(Aby* a(Qau*= aby*,axDu* Dv*= 0,ax ay^abx*^aby*^ = 0,ax^aywith boundary conditionsau*Ob.*=^=9* = b; = ---1 =0, on y = 0,1,^(2.50)Oy ayDv*^DO*^Ob*u =^=-- = b* = 0; = 0, on x = L, — L.xax^axUsing Fourier series we find eigenfunctions V = (it, 1,* , 9, b., y) corresponding to theeigenvalues ±iPiwi, j = 1,2, of (2.49)-(2.50)(2.51)2L„,^ sin( (in -I- 1)71-37/2E,„) cos(y)(m, + 1)cos( (in^1)742L1„)sin(Ty)aRn, cos( (171^1)7r312L,„ ) sin ( -,Ty)P9(1 —--a(Q271-E,„(InandGI) :71E+1 = C n ? I ,^(2.52).^ sin( (711 + 1)71-312L,„)sin(ny)1)P2K 1W2)—71-0-(Q— lw9) cos((ni^1)742L,„) cos(7y)P9(( Chapter 2.^Oscillatory instabilities of magnetoconvection equationswhere31Cm = 2^Rin4L.,„0-(Q]-1[4421Pi72m2^ 2 +in2pi ((^)^Pi? (1^ico-1Crn+14L,2,, P2^4L2mo-(Q aR„„. (2.53)+^9[7r2e/7/ -I- 1)2^(777,^1)2P9K —^P.5- (1 — iC4/9The normalization constants Cm,Cm4.1 are chosen so that(4) .7 ,(DZ) r, = (5,k,^j, k = 1,2. (2.54)Using (2.19) and (2.33) and after some simplification we get,in271-2(1^iwi)(( + iwl)8L131wi(wi — i(5)+ 1)272(1 ± iw9)(( + 1W2)(2.55)8g,P)c.o9(w9 — i6)where (5 = 1+ a + ( and the overbar denotes complex conjugation.2.5 Asymptotic results as in --+ 00In this section we discuss some of the asymptotic behavior of the critical aspect ratiosand related quantities as in^DO.2.5.1 Case I (fixed ( and Q)We are interested in behaviour for large aspect ratios. Since (2.26) implies that L m =0(m) as in oa, for fixed a, ( and (2 we consider limiting behaviour as in -÷ oo. Wecall these "Case I" limits.Proposition 2.2 L,„ = 0(m) as 'in --4 x. More precisely, we haveA^,^A(1 + 3A2)^.^ _ A + in-^ ^ in -2 + 0(m-3),IP 2^24(A2+ 1)where A = VT* satisfies (2.27).(2.56)Chapter 2. Oscillatory instabilities of magnetoconvection equations^32Proof: By (2.26) it is clear thatlim 2L,1/777, = A.^ (2.57)711-4 00expanding2L„,/777. = A + i1m1 + (2111 ^0(7n-3).and substituting this expression into the expressionR71(Lin) = Rm+1,1(L7n)that defines Lrn, after some calculation we solve for el and E9 to get (2.56). Q.E.D.From Proposition 2.2, it follows immediately thathiin PI =^= P,111^)00^I71-+00(2.58)whereP = 72 (—Al2 + 1) , (2.59)and(4.) =^11M C,V1 =^11111 W2, (2.60)--+ 00^171 -+where7120-(Q(1 — ()w = —(2 + (2.61)P2(1 + 0)Remark 2.4 There is a limiting relation between the eigenvalues and eigenfunctionsand normalization constant in this problem and the infinite layer (L = oc)problem withperiodic boundary conditions ([35,1). If(^(1)711 = (^I)^—^9 (1)1 — (I)91  ^9 21and if(I)* — (I);(p.,n _ (I)7 + (I); , (Km =^—^2 2iChapter 2. Oscillatory instabilities of magnetoconvection equations^33then for fixed x, y, we haveliiii (1)91"(x, y) = (I)1(x, y)^lint (J).72". (x , y)^(1)2(x , y) ,lint^, y) = (1)1(x , y),L„,—roolim (I);"1 (x , y) = (1);(x , y),L„,—+oowhere (1)1,(1)2,(1)1,(1).2* are eigenfunctions of the linearized equation, and (DI ,(1); are adjointeigenfunctions for the infinite layer with periodic boundary conditions. We also haveurn Pi4'1 = Pw, and^lint r*,ni = 7-(°),L",-->00andurn C„, = C.L„,—+cowhere iPco is the imaginary eigenvalue, r0 is the scaled critical Rayleigh number and Cis the normalization constant used in the infinite layer case.Let c = m-1. Then we also have9=^P — — E^0(E2),97r-P9^P +A2c + 0(c2),(w2^/-2■=^+^if 0(E2),(2.62)(2.63)(2.64)+pwA271.2(c_02 _L=\1E 0(E2 ), (2.65)WPWA2=^.R0 + (2.66)where7r2(Q^((Ro = (a + ()[+ 1)P2 A2P(2.67)a +1 + aProposition 2.3 For fixed L = L„, there will be no other instability unless we increaseR above R,„ at least by 0(c2).Chapter 2. Oscillatory instabilities of inagnetoconvection equations^34Proof: To prove this we need to look at the size of Rffl.4_9(L) — R„,(L„,), since clearlyRm+k(LT„) — 117-„(L) > R7,2H-2(Liii) — R,„(L„,), for all k > 2. Using equation (2.23) wehave—4772 — 4R„,+2(L„,) — R„^a T2 r^1,(L7„) =   (A + B)"j"1 L1n2(in + 2)2]+ Br(m + 2)4 — in4 3 ((in + 2)2 — in2)]+I. 16L4^4L2^(2.68„,= 4B(in ± 1)  13 [(in + 2)2 — (in + 1)2] 2L„, (in + 2)2 1^2L,,(2.69)[(in + 2)2 — (m + 1)2] [m2 + (in + 1)2 + (in + 2)2] }+(2L„,)3,„.,,2 1 on + 1)2 ± (in + 2)2 14B(nt +1)(2iii + 3) {=^ 3 + "( 7-(2L7,)(in + 2)2^(2L„1)2= 24PB e2 + 0(E3)A2712= 0(2),^ (2.70)since Lim and A satisfies(2.35) and (2.27) respectively, where A and B are as in equation(2.25). Also since B^0 this difference is not zero. Q.E.D.2.5.2 Case II (decreasing^increasing Q as in^co)In subsequent chapters, we will also consider behaviour for large aspect ratios, small (and large Q. Suppose we fix real positive numbers a, ç and Q, and consider thesequences given by=^(2„;, =^ (2.71)for fit = 1, 2, 3, .... For each = (711, Q = Q„, we obtain a sequence of critical half aspectratios L,11, and corresponding critical Rayleigh numbers R,t, in = 1, 2, .... We considerlimiting behaviour when 'in = in and in —+ co. We call these "Case II" limits. Taking thelimits in Case II change the values of A, P, I?(. of !i2.5.1. We denote their correspondingChapter 2. Oscillatory instabilities of inagnetoconvection equations^35values in Case II by 5■, P^respectively; their corresponding equations are replacedby2^30^--T171.2(71 + 1),P =0.“27.1.2co2/52(0- + )'rap21 1-71.20.(2Po I 72^ (7 +11The asymptotic expansions of the terms2L rll ,P1 ,^ 5 ‘545)9as M.^oo change, but the leading terms are of the same order in E = m-1. Correspond-ing to equations (2.64)-(2.66) we have2LZ771272;\  ek, 12 + 0 (fie +^ (2.77)A3^P2 =^2712 Ai ch12 + 0 (Ek. +(2.78)^=^+ 0-C(21-0-' + (45k03)72] Eki2 4_ 0((k +^(2.79)2(a +1)P3)cDP3=^+ a(Q[^+^03)72] Eki2 + 0 (Ekpo+o(E4./22()7 + 1)cDP3where0.1.2("2—k5= ^ .12(o- + 1)PThe appearance of R.,„+2(Lin) — Rrn(Lin) also changes, but by (2.69) it is of the sameo-(1Q(2.72)712(U+1)'(2.73)p2i(2.74)(2.75)0" •ek12^0(6.1.7^ (2.76)(2.80)(2.81)order as in Case I and equation (2.70) will be replaced by24PP .R2(L:)^R'" (L ^22) = ^ 62^0(E2+k/2 + (3),411^?II - (2.82)Chapter 2. Oscillatory instabilities of inagnetoconvection equations^36whereB = + 0(E4/2)These results will be useful when we calculate the coefficients of the reduced equation onthe center manifold and the determine the dynamics of the related equations in Case II.Chapter 3Center manifold and normal form reductionsIn this chapter we begin our nonlinear analysis of the magnetoconvection equations. In§3.1 we give an abstract formulation of the magnetoconvection equations as an evolutionequation in a Hilbert space, and then prove the existence of a nonlinear analytic semiflowin that Hilbert space. Then in §3.2 we rescale variables so that L is explicitly introducedas a parameter into the evolution equations, while the domain becomes fixed. In §3.3 weestablish the existence of a locally invariant attracting center manifold WC for the evolu-tion equation, which we will then use to study the dynamics of magnetoconvection. Werepresent the flow on TV' as a two-parameter family of four-dimensional ordinary differ-ential equations with Z2 e Z2 symmetry. In §3.4 we use the symmetry and near-identitycoordinate transformations to simplify the family of ordinary differential equations byputting them into normal form. Finally, in §3.5 we discuss the choice of normal formswhen the aspect ratios of the fluid layer are large. We use different normal forms for theCase I and Case II limiting situations introduced in Chapter 2.3.1 Abstract formulationIn this section we reformulate the magnetoconvection equations (2.3), (2.7) as an equationin a Hilbert space, which generates an analytic semiflow. For fixed L, let= (XI Y) 0 < y < 1, — L < < Ll ,37Chapter 3. Center manifold and normal form reductions^ 38be the rectangular region as in equation (2.6), and let FL denote its boundary. LetL2(f2L) be the space of all square-integrable functions on f/L, with norm110111,2( fl )[^1/2I 0(x5 Y)12 (137(1Y] (3.1)and inner productpi pL(0^L2 (f L) =^00:^(S 'il)(13: dY0 - L(3.2)where the overbar denotes complex conjugation. When k > 0 is an integer, the Sobolevspace 1/171'2(c4) of order k is defined byIV" (Cid =^: Dç E L2(C21) Va, kJ < k} ,^(3.3)whereaa +^D =^ = {a, (1'9} ,^1(11 = al + co.c1xa aya2and the derivatives D°0 are taken in the weak (distributional) sense. The space Mik,2(12L),with the normIt ^=^wk,2010 = E HD"0112,2(ad I 1/2^(3.4)and scalar product= E (Dao,D.0)/,-20-20,^ (3.5)lnl<k^is an Hilbert space [21^0, tvo,2(Q1 ) = L2( Q L ).. When k =Let L2(C2L) = L2(L) x L2(L) and Wk'2(C2L) = wk,2(2L ) x Tk,2(Q L ), with thenorms and inner products inherited from the product structures. In order to introducesuitable function spaces for the magnetoconvection equations, we first consider the fol-lowing boundary value problems:(1) Given f E L2(C2L), find u = (u v) and \' satisfying—Au+ Vx = f in C2L,Chapter 3. Center manifold and normal form reductions^ 39V • u = 0 in QL,^ (3.6)u n = 0 on FL (i.e. u = 0 on x= +L, v = 0 on y = 0,1) ,V x (u x 11) = Don FL (i.e. Ov/03: = 0 on x = ±L, Oulay = 0 on y = 0,1) ,where n = (nx,ny) is the unit outward normal vector on FL. (This is the Stokes problemwith "stress-free" boundary conditions.) Define the following Hilbert spacesH1L =^L2(C2L) : V u 0 in QL, and u • n 0 on FL} ,^(3.7)VjL = { u E \V"2(L) : V • u = 0 in C2L, and u n = 0 on FL} ,^(3.8)u E W22(c4) : V • u = 0 in C2L, and u•n=0,V x (u x n) = 0 on FL} .^(3.9)Then one can define an unbounded self-adjoint operator A1 in H1L with domain DIL,such that Aiu = —ILAu for all u E DIL, where II is the orthogonal projection of L2(C1L)onto HiL. The problem (3.6) is equivalent toA1u=ITf, u E^ (3.10)since the pressure term^E W1'2('4) can be recovered from u and f. The inverseoperator 4I-1 is a compact operator in H11 [46, pp. 104-103 and Remark 2.4 on pp.110-114(2) Given f E L2(C2L), find 0 satisfying Laplace's equation with mixed boundaryconditions:—AO ==00f in C2L,0 on y = 0,1,o on 17= +L.Chapter 3. Center manifold and normal form reductions^ 40DefineH2L = L2(CIL),^ (3.14)V2L, = {0 E IT71'2(121j : 0 = 0 on y = 0,1} ,^ (3.15)D2,t,^{0 E 1172,2 (.-NL ) : 0 = 0 on y = 0,1, 00/ax = 0 on x ±L} .^(3.16)Then one can define an unbounded self-adjoint operator A2 in H2L, with domain D2L,such that A20 = —A0 for all 0 E D2L. The problem (3.11)-(3.13) is equivalent to-420 == f,^0 E (3.17)and AV is a compact operator in H2L [46].(3) Given f = ( fs, fy) E L2(c1) with fo„ f = 0, find b = (b„, by) satisfying a mixedproblem:—zb = f in QL,^ (3.18)^b„. = 0 on FL, (3.19)^Dby/On = 0 on FL,^ (3.20)The boundary value problem (3.18)-(3.20) is just a pair of &coupled Dirichlet and Neu-mann problems. If V • f = 0 we note that the solution automatically satisfies V • b = 0.The Dirichlet and Neumann problems are classical, and solutions are given in almostevery book on elliptic partial differential equations (e.g. [46]). Let us define the followingHilbert spacesH3L^{b E L2(S2L) : f by = 0},^ (3.21)V3L^{b E W1'2(21) :^by = 0, and b.,. = 0 on FL}.^(3.22)oLD3L^b G w2,2( -4),./ by = 0, and b„. =0, Dby/On = 0 on FL, . (3.23)Chapter 3. Center manifold and normal form reductions^ 41There is an unbounded self-adjoint operator .43 on H3L, with domain D3L, so that A3b—Ab for all b = (bs, by) E D3L, and 43-1 is a compact operator in Hu.Each of the above boundary value problems has been considered as an abstract equa-tion of the formAjcb = f, i = l,2,3in a Hilbert space HL, i = 1,2,3. The operators Ai are positive, linear unboundedoperators in HL with domains DL and their inverses 4i-1 are as self-adjoint compactoperators in HL. For fixed parameters a,(, let (I) = (u, 0, b)T, define the Hilbert spaceXL = H11 x H2L X 113L,with inner product as in equation (2.33), and define the unbounded linear operator A inXL byA(I) =(o.4iu420, (A3b), (I) E D(A) = DI L x D21 X D3L.^(3.24)Then the operator A is a strictly positive self-adjoint operator in XL with domain D(A).The inverse operator A-1 is self-adjoint and compact, and the spectrum of A consistsentirely of isolated eigenvalues with finite multiplicities, so its eigenfunctions are densein XL [46]. The eigenfunctions have the form( [A,1 cos(inmil2L) + .13;n71sin((m+ 1 )7s/2L)] cos( wiry)[A„ sin(in7/7/2L) + B,2 cos((in + 1)742L)]sin(n7y)=^[A3 sin(in7.112L) + B„ cos((ut + 1) 71-37/ 2L)] sin(n7y)^(3.25)[AL cos(in7rx/2L) + B„ sin( (in + 1)742L)] sin(n7y)[4.,5 sin(wirx/2L) + BL, cos((m, + 1)7//2L)] cos(n7y)where in is an odd positive integer and 11 is any positive integer. The coefficientsand A2„,„,^and B,2,„„, AL and A,5„„, 13,4„„ and B,5,„ are related in such a way that theChapter 3. Center manifold and normal form reductions^ 42divergence of the first two and the last two components are zero, i.e.,ifl^ 171A2 ^A 4inn^2Ln "1"'^n 2Ln in—B^m +172„„21,71^BL" B4  171 + 1 B5 .2Ln i""The operator A is sectorial in XL [20], and for each -y E R we may define the fractionalpowers A7, with domain X7., = D(A7) in XL. The space D(A), endowed with the scalarproduct((J)1 , 4)2 D(and the graph norm= ( A7 (D1 A7 (1)2 ) L11 / 21(1) D(.4.7 )^4), (I))D(A ) f( 3 .26)(3.27)is a Hilbert space, and A7 is an isomorphism from D(A7) onto XL. For -y = 1, werecover D(A) . Since A is a sectorial operator with compact resolvent, the embeddingsD(A) C X C XL are continuous for 0 < -y < 1 and are compact for 0 < -y < 1[20, Theorem 1.4.8]. Now for fixed a,C,Q, we putB(R)(1) = (all[R0e11 + (Q(V x b) x ey], v, V x [u x eypT,^(3.28)A1(4),(1)1) = (II[a(Q(V x b) x b' — (u^—u VO', V x (u x b'))T, (3.29)where (13 = (u, 0, b)T, .213' = (u', 0', b')T. We now write the magnetoconvection equations(2.5) and (2.7) as an evolution equation in XL,(RI) = —.44) + B(10(J) + N(4),^ (3.30)where N(4)) =^(I)) We observe that B(R) :^Xi, is a bounded linearoperator and 111 : X-1", x XI —4 XL is bounded bilinear operator for 1/2 < -y < 1[20, Theorem 1.6.1 and the arguments on pp. 79-81]. and so B(1?) + N(.) is analyticfrom X2 into XL when -y > 1/2. By [20, Theorem 3.4.4 and Corollary 3.4.6] we have thefollowing fact:Chapter 3. Center manifold and normal form reductions^ 43Proposition 3.1 If 1/2 < 7 < 1, then the evolution equation (3.30) generates a localserniflow in X2 that depends analytically on t > 0, on R E R and on the initial conditionCO) EWe note that if V • b^0 when t^0, then the solution (I)^(u, 0, b)T satisfiesV • b --= 0 automatically for t > 0.Now let^K(R) = —A + B(R), D(K(R)) D(A),^(3.31)denote the linearization of the vector field of (3.30) about (I) = 0. By the above argumentsand [46, p.54], it is clear that for each fixed R the spectrum of K(R) consists entirelyof isolated eigenvalues with finite multiplicities. To find them, we solve the eigenvalueproblemK(R) (1)^tv(1),^E D(A),and we seek eigenfunctions in the form of (3.23). This is system (2.11) that we havealready considered in §2.3, and we found all eigenfunctions in the form of (3.25). Sincethese eigenfunctions are dense in D(A), our formal calculations in §2.3 are justified.3.2 The rescaled problemSince we would like to study the dynamics of rnagnetoconvection when L is near one ofthe Lim which were found in Chapter 2, we introduce L as parameter explicitly to theequations. For this, we use the resealing=^u = LU, bx^Lb.^ (3.32)Then system (2.5), after "dropping the hats", becomesauat —a [Alu —^+ ROey + (Q (vi x^x^+131)1 — (u • V) u,Chapter 3. Center manifold and normal form reductions^ 44aoatabatv • u= Alt9 + v — u• VO,= (Alb + x [u x (e11 +b)],=0,(3.33)V • b = 0,in the fixed domain= (x, y) : —1 <17 <1, 0< y <1} ,where^= L-202/0x2 02/ay2,^= (L-20/03.0/0y) ,= (L2b„„by), u = (it, v), b = (b„.,by).The boundary conditions (2.7) becomeDby=^v=t9=b„.=^= 0, on y = 0,1,^(3.34)Dvay ay00^Oby=^— —^=^— 0, on 3: = +1.ax^axThe Hi^ corresponds,space XL defined in §3.1 j^nder the resealing, to a Hilbert spacesurviX, and for an inner product in X we may take11^9 _9 + el + L2 btx2^byi -1;y2)^dy,( (I) 4)2 = /2)11^(L-711 11- +0 -1(3.35)where (Di =^i = 1,2, and the overbars denote complex conjugation.The linearized operator IC(R) of the previous section corresponds, under the resealing(3.32), toK(R, L) = —A(L) + B(R, L),^ (3.36)where.A(L)(1) =^—Al 0, —(L\Ib)'^ (3.37)B(R, L)(I) = (o-II1[ROey + (Q(V1 x b) x e 1 ] ^x (u x ey))T,^(3.38)Chapter 3. Center manifold and normal form reductions^ 45and IV is the orthogonal projection that corresponds to n under the resealing (3.32). Forfixed R and L, A(L) is a sectorial operator in X, and B(R, L) :^X is a boundedoperator if 1/2 < y < 1. The eigenfunctions 4)1, (I)9,^(1)2* now take the form—2 cos(m7rx/2) cos(y)712,sin(m7rx/ 2) sin(rry)1 1(1 +^) s n(7n7rx/2) sin(7ry)P 27rrrTi (C + .^ oc s(rn7rx/2) sin(7ry)i)7rPt (( +^s.^ i n(m7r.r/ 2) cos(71y)^j\2 + 1 sin((m,^1)7I-x/ 2) cos(7r1J)cos((rn + 1)7rs/2) sin(7ry)1 P2(1 + 1.w2) cos((m.^1)7rx/2)sin(ry)27rsin((nt + 1)742) sin(7ry)(in + 1)P2(( + iw2)^ P.2 () cos((rn +1)71-42) cos(7ry)—2 cos(m7rx/2) cos(y)711,(DI = C„,\ p((^) sin(m7r.v/2) cos(7r11)sin(rn7rx/ 2) sin(7r1J)P1(1°--R icoi) sin(m7r42) sin(7ry)2a(Q7rP9(( — t.w9) cos(m7-142) sin(7ry)—7ro-(Q(3.39)(3.40)(3.41)Chapter 3. Center manifold and normal form reductions^ 462^ sin((m, 1)742) cos(7ry)(in + 1)cos((in^1)7rx/2) sin(7ry)crR^cos((m + 1)7rx/2) sin(7ry)•P2(1 — 1c07)---a-CQ27r(in + 1)P1(( — t. wi) sin( (m,^1)7rx/ 2) sin(7ry)• cos((m + 1)7rs/2) cos(7ry)P2(C — iw2)and= Cm+1 (3.42)(3.43)with Cm, Cm+1 as in equations (2.55). The evolution equation (3.29) becomesd(I) = —A(L) (1) + B(R, L)(1) + 1\'((1 L),whereN(4), L) = M(,, L) = (f1[cr(QV1 x lat) x b— (u- '71)ub—u- V18, V1 x (u x b))T ,(3.44)and Proposition 3.1 holds for (3.43).3.3 Center manifold reductionThe study of bifurcation and stability in differential equations can often be greatly simpli-fied by the use of center manifold theory. This theory allows one to reduce the dimensionof the state space, while preserving the local behavior of solutions of differential equa-tion. In this section we apply a suitable version of the center manifold theorem to reducethe parametrized family of evolution equations to a parameterized family of ordinarydifferential equations in a four dimensional phase space, the dimension four of the phasespace being determined by the number of eigenvalues with zero real part at the criticalparameter values.When L = L,„, "odd" and "even" curves of the Rayleigh numbers 1?,„(L) and Rm+i(L)intersect at R = R„, and modes corresponding to both odd and even numbers of rollsChapter 3. Center manifold and normal form reductions^ 47are marginally stable. (Recall that we assume in is odd). Therefore for R, L in a smallneighborhood of R,„, Lm, we may see bifurcations of even solutions, odd solutions, orother solutions arising from nonlinear mode interactions.When (R, L) = (R„„ Lm) the center eigenspace Ec, corresponding to the eigenvaluesof K(R„„ Lim) with zero real part, is given byE, = {Z14)1+ Zi4)1+ Z24)2 +Z24)2 : (Z1, Z2) E c2},^(3.45)where (Dj, j = 1,2 are given by (3.39),(3.40). Now we define the projection P of theHilbert space X onto E byP(I) = ((I), (1*1)(I)i + ((I),(1)1)4)i + ((I),(1);)(1)7 + (4),(1)2)(1)9, (3.46)where the overbars denote complex conjugates, the inner product is given by (3.40) withL = Ln.„ and (I);!, j = 1,2, are the eigenfunctions of the a.djoint operator K*(Rm, Lui)for K(Rm, Lin) given by (3.41)-(3.42).Proposition 3.2 When L = L„,„ and R = Rm the space X decomposes into a direct sumX = e Es, where E, = 'R,(P) and Es = Ar(P) are K(R„,,L,„)-invariant subspaces.The spectrum of K(R,„ L) restricted to Ec. is {±iP1w1,±iP9w9} , and the spectrum ofWm, Lm) restricted to Es is contained in the left complex half plane, with real partsuniformly bounded away from the imaginary axis. Thus if E(K) denotes the spectrum ofK(R„„L„), we have R(E(K)) < 0, and F.,(K) n iR is a spectral set.Proof: Since any (I) e D(.4) can be expressed as a convergent series of eigenfunctionsof K = K(Rm, L„,), and since the eigenfunctions are orthogonal, it is clear that theprojection P commutes with operator K (i.e. PA-4) = KP(I) V (I) E D(A(L)) ), thusK leaves the subspaces(P) = span {^(1)1, (I)9.^= Fe,Chapter 3. Center manifold and normal form reductions^ 48.A1(P) = {4)7, (II, (I)2*, (I;}± = Esinvariant with X = E, e E„. We have already proved the second part of the propositionin §2.4. Q.E.D.We now state two results on center manifolds due to Henry [20, Theorem 6.2.1 andCorollary 6.2.2, Theorem 6.2.4Theorem 3.1 . Consider the abstract differential equation(1(J) = —A(D + f(1),^ (3.47)where A is a sectorial operator in a Banach space X, 0 < < 1, U is a neighborhoodof the origin in X7, and f : U X is C1 with f(0) = 0 and f' Lipschitz continuousin U. Assume K = A + 1(0) has R(f,(K)) < 0 with E(K) n iR a spectral set. LetX = E, + E, be the decomposition into K -invariant subspaces with n(E(K}E)) = 0 andRPICIE.)) < 0. Then there exists a C' attracting locally invariant manifold (a centermanifold)1,17^{(I)^+ (1), :^=^((De ) ,^E E, IIH < 6}^(3.48)tangent to E, at origin. The flow in We is represented by the ordinary differential equationd(I)^-dt = P [K (4), + ((De)) + N ((De + 'i'(4))] ,where N(1)) = f(I) — (0)(1) and P is the projection of X onto E along E„.(3.49)If the nonlinear part f in Theorem 3.1 is smooth enough, the center manifold WC issmooth and we can approximate tali (4) by the first finitely many terms in the Taylorseries for T((1),):Theorem 3.2 . Assume the hypothesis of Theorem 3.1. and assume that N : U X isCP, where N() = f(4))— 1(0)(1) . If there is a CP function h with Lipschitzian derivativeChapter 3. Center manifold and normal form reductions^ 49from a neighborhood of the origin in E, into E, with range in D(KIE,), such that—(1 — 13)[K(4), h((1)c)) + N( + h(4),))] = 0 (IR10:1. ,) P [K (4) , 1(4) „)) + N(1I + h('IIc))]^(3.50)as 41),^0 in Ec, thenI (1) - h (4)1 = 0(11411")^(3.51)as 4),^0 in X,, where T ((k) defines the center manifold of Theorem 3.1. If N is CP nearthe origin, there is a unique polynomial function h of order p satisfying the conditionsabove.We have shown that for each fixed L the operator A(L) is sectorial, and we can applyTheorems 3.1 and 3.2 to the systemd(I)7-7t- = —A(L,„)(1) +^+ It, L,n)(1)^N ((I), L„,),dt =0,(3.52)and obtain the existence of a center manifold for R near R7 and fixed L = Lm. Butsince the operator A(L) depends on L, to also treat L varying near Lin we need to applya parameter dependent version of Theorem 3.1 due to Vanderbauwhede and boss [47,section 2, comments on page 136]. Then the center manifold can be represented asWc --= {(1)c,(1)8) E EceE ,s7 (I) = (4, t, v), JJ1cIJ < (51, < 2 1111 < 63} (3.53)where it = v = L — L„, and T is a smooth function from a neighborhood of theorigin in E, X R2 into E,"' = E flX, with T(0,0,0) = 0, t110, 0,0) = 0. Actually Tis CP for any integer p> 0, since N(4). L) in X7 is analytic (although T is not necessarilyanalytic). The flow in WC is represented by the ordinary differential equationd(I),= P - (R,„^L + OK), + ((Der^+ N((1), + (4)ii,vn) •dt (3.54)Chapter 3. Center manifold and normal form reductions^ 50The study of bifurcation and stability of solutions of the evolution equation (3.29) nearthe origin in XI', for R near R„, and L near L„, is now reduced to the study of equation(3.54).The projection P commutes with the action of the group Z2 ED Z9 on X defined byequations (2.8) and (2.9), and hence the reduced equation (3.54) is equivariant withrespect to the action of J and 13 restricted to E. If we identify E, with C2 byE, =^= Zi(1)1+ Z1(1)1 + Z2(1)2 + Z22:^Z2) E C2},then sinceJ(Di= 4>i, J4)1= (-Di J(I) 9 = - (I) 9 , 1129 = — (1) 9/3 (I) = _ (i) ,and/3) = _ (-1) ,^= 1, 2,the action of the group z9 Z2 on C2 will be given byZ^Z2^) j—* ( Z1 1 Z9 -22),^(3.55)(z1,21,z9, 22)Using the definition of P, the reduced equation (3.54) becomes^= (.1-ie(R,„ + t , Ln, + v)(4), + T(4),,,ct, v) + N((I), +^j = 1,2.(3.56)where the overdot denotes differentiation with respect to t, (I) = Z1(11 + 2^+ Z2c1)2224)2 and (Z1, Z9) E C2. By our remarks above, (3.56) is equivariant with respect to theaction of Z2 Z2 given by (3.55). Note that Zj. 3 = 1,2 satisfy the complex conjugatesof equations (3.56).Thus, for L near L„, and ? near R„, the trajectories of the magnetoconvectionequation are attracted to a center manifold, where the time evolution is given by aChapter 3. Center manifold and normal form reductions^ 51parametrized family of systems (3.56) of the general form= F(Z, p, v),^ (3.57)near Z = 0, p = 0, v = 0, where Z = (Z1,21, z9, 22) T E C4, and it, ti are realTparameters. The vector field F = (F1, P1, F9 -F9 ^is CP for any positive integer p in aneighborhood of (Z,^= (0,0,0), andF(0, it, v) = 0,for all (it, v) near (0,0). Also0 0 00 —iPiwi 0 0Ao = DzF(0,0,0) = (3.58)0 0 iP9w2 00 0 0 —iAc.4.)9 /where Pi^P21 W2 are as in equation (2.28). Furthermore, the family (3.57) is equivari-ant under the action of J and /3 described in (3.55), i.e.,F (J Z, p, v) = JF(Z, p, v),^ (3.59)F( /3Z, p, v) = ,3F(Z, p, v),for all (Z, p, v) near (0,0,0). We have reduced our problem locally to that of a doubleHopf bifurcation with Z2 ED Z9 symmetry.3.4 Normal formsNow we put the system (3.57) into Poincare-Birkhoff normal form. This involves sim-plifying (3.57) by removing terms from the Taylor series expansion of F order by order,using near-identity changes of coordinates. We can demand that these coordinate changesChapter 3. Center manifold and normal form reductions^ 52respect the symmetry (3.59). Those terms that cannot be removed at a given order arethose with an extra symmetry under:(Z1,21, z2, 22)^(eiozi, ei021, eioz„, eio22) .We now outline the calculation required to put (3.57) in normal form. First, we notethat the Z2 e Z9 symmetry (3.59) implies that each component of F(., p, v) is odd in thecomponents of Z, thus the lowest order terms in the Taylor series expansion of F aboutZ = 0 are odd in the components of Z, and (3.57) has the form±1 =^+^+ v + ci l z112 + c2Iz2 12 + c7z i' + (784+ c92) + 2 (c3z.? + c10z12 + c11z2 2 + c122? + c132,1)+ o (IV + lit, vIlz) 3 + lit, vI21z 1),^ (3.60)Z2^z9 (iP9W0 (1211 b9/1 C41Z112 C5IZ912 C144 + Cl5Zr+ c162) + 22 (c6zr + c171z9r2 + C18iZ112 + C192.;2 + C202)+ 0(145^lit, v1143 + lit , vi2lZ1),where al, b1, a2, b2, C1,^, C90 are complex numbers that can be determined from themagnetoconvection equations (2.15), but as we will see below we will not need to computethem all. (The equations for 21 and 22 are the complex conjugates of those for Z1 and4).We temporarily fix it = 0, v = 0, and write (3.57) asA Z + 0(1z13).^ (3.61)For any positive integer it, let H„ denote the finite-dimensional vector space of vectorfields P(Z) in C'1, whose components are homogeneous polynomials in the componentsof Z, of degree n. Define the linear map adA0 : H„ H„ byad.40(P) = [P, A0] = DA0P — DPA0,^(3.62)Chapter 3. Center manifold and nonnal form reductions^ 53where •] is the Lie bracket. Let G„ be a complementary subspace to the range adAo(H„)in H„ so that H„ = adA0(H„) Gn. Then by normal form theory (e.g. [16, Theorem3.3.1]) and using the Z2 ED Z2 symmetry, for any odd positive integer k > 3 there is acoordinate transformationZ = W + P ( TV) ,^ (3.63)where the components of P(W) are Z9 El) Z2-equivariant polynomials in the componentsof W, of degree at least 3, such that (3.57) becomesTV = 40W + g(3) (TV) + g(5)(W) + + fj(k)(W) + Rk(W),^(3.64)with g(n) E Gn for 3 < 71, < k,n odd, and Rk^(11171k+2 ) Furthermore (3.64) is stillZ2 Z2-equivariant. To characterize Gn explicitly, we observe that= span {WI join, n79 j kl 7-rt ,W3, kl m W4 j k nifwhereW1 j k 1^=^ti/.17Tir , 0 , 0 , 0)T,^= (o,^1471I47 , 0 , 0)T ,W3,jklm = (o, 0,^, 0)T^IV4,jklin = (0 ,0,0,^ITT)and j, k, 1, in run over all positive integers such that, j + k + 1 + in = n. Applying (3.62)we find thatad40 (TV' Juni) = i[(k — j — 1)P1c.,..)1 + (in — 1)P2w21WLikau^(3.65)ad40 ( W9,jom ) = i[(k — j + 1)P1 col + On — 1) P9 W9i1-179 , j Hi,^(3.66)ad4o(II73 jki,„) = iRk — j)Plz..oi + (in —1 —1)P9c,..)91W3 jki,„^(3.67)ad-4o ( W4,jklm ) =- i[(117 — i)Pib) 1 -1-- (111. — 1 + 1) P9W21117.1j kl m •^(3.68)Since the matrix of adAo(H„) is diagonal in this basis, G„ can be found merely by locat-ing the zero eigenvalues of ad:40(H„). We observe that the conditions on ad40(WiJkim) =0 014711147112001 //0^/00047211v21 2 j0TT/2113/412 jChapter 3. Center manifold and normal form reductions^ 540,i = 3,4 can be derived from those of adA0(Hijo1n) = 0,i = 1,2 simply by in-terchanging (j, k, 13w1) with (1, m, P9w2). Also, the conditions for adA0(W2,fith,, = 0can be derived from ad4o(T171,ju2) = 0 simply by interchanging (j,/, /31co1, P2c.o2) withi(k, m,^, —P9w2), so it is enough only to consider adA0(1471,kt771) = 0. To find thecondition on cubic terms in G3 we must find j, k, 1, in with j + k + 1 + in = 3 such thatPiwi(1 j + k) + P2w2(m, —1) = 0. (3.69)The Z2 ED Z2-equivariance also implies that j + k be odd while in + / should be even.Since ./31chi1 and P2c,i2 are both positive, (3.69) will be satisfied, only ifj = k + 1 and nt =or1 — j + k = n1 and in — I = n9 with n ii9 < 0.The possiblities j k + 1 and in = 1 show that the vectors/ ivilw212 \^/^o^\^/o^MITIT212o o0^0\^/^\^/^\are in G3. To consider the only other possibilities first we notice that—2 < 1 — j <^< 1 + k < 4, —3 < —1 < 119 < 3 ancl als° IiI+1n2I<4.Chapter 3. Center manifold and normal form reductions^ 55But neither ni nor n2 can be odd, since this implies that j + k is even or in + 1 is odd,respectively, which was ruled out by Z9 ED Z2 symmetry. The only remaining possibilityis when1 — j k = 2, and in — 1 = —2,which implies that j = in = 0. For any fixed L„„ however, 2(P1c,..4 — P2w2) is not zero,therefore the coefficients of the vectors/-1711/V1 /0" 0^1 /^00 0 0(3.70)0 0 1,179Wi2 00 0 0^/ 117.- TV?2^Ican be removed by near-identity normal form transformations. However we note that,by doing so, the term (Picot — P2co9)-1 will be appear as a factor in the coefficients ofsome fifth order terms in the transformed equation.Ignoring the possible implications of this last remark fo now, normal form theory andthe results of our calculations imply that for k = 3, (3.64) takes the form^T/I71 = TF1 (iPtcot + C11147112 4- C21W212)^0(1W15)^(3.71)ti72 = W2 (iP2CO2 + 11751 L17212 + c4ITV1 12) + o(Iw15),(the equations for Wi and W2 are the complex conjugates of those for Wi and W2). Nowto restore the parameter dependence, we apply normal form theory to= F(Z, 7/),^= 0,^= 0,^ (3.72)seeking coordinate changes of the formZ = IT' + P(W,^it = ft, V =^ (3.73)(see, for example, [16, p.1451). The calculations then reduce to those outlined above, andwe obtain the following result.9 9= Pw + ^wiPoW92Aco^++(3.75)Chapter 3. Center manifold and normal form reductions^ 56Proposition 3.3 There exists a Z9 e Z9-equivariant coordinate transformation ZW + P(W,p,v) that transforms (3.60) into= 1171^+ alp + blv + C11147112 + C2111/212)• °(1/1,11121W1 ± l i,1111W12^1W15),^ (3.74)1472^1472 (iP2c4,2^a2p b2v +11751W2)2 C4111/112)+ 0(11t, v1211171 + 111, v11W12^MI5)where a1, b1, a2, b9, C1, C9, C4, C5 are the same complex numbers that appear in (3.60).Note that (3.74) is still Z9 (1) Z9-equivariant.3.5 Large aspect ratiosWe would like to study the third order truncation of system (3.74) as a means to studythe system itself, but this is reasonable only when the higher order terms are muchsmaller than third order terms. For fixed a,(-, Q and all L„,, the normal form (3.74) givesa valid description of the local dynamics of the magnetoconvection equation (2.5), forparameters (L, R) in some sufficiently small neighborhood of (L„,, m = 1, 2, .... Weare also interested in behavior when the aspect ratio is large. Therefore for fixed a,<" andQ, we consider the "Case I" limiting behavior of the coefficients in (3.74) as in oo.As m (and L„,) increase to oo, the domain of validity of (3.74) shrinks to zero, so in ourapplication we are mainly interested in large, but finite, 171, and L„,. Since some of higherorder terms in (3.74) are proportional to (Piwi — P9w9)-1, and Picvi — /39w9 is small forlarge Lm, we should study the size of Piwi — N.4,9 further. For fixed a,(,Q from §2.5 wehaveChapter 3. Center manifold and normal form reductions^ 57therefore(272— Aco9 = ^LT„1 + 0(L,-„2),Aw(3.76)as Lm --> oo. We can expect the center manifold reduction, used to obtain (3.74), tobreak down if p, and v are large enough so that modes other than those corresponding to(Di and 4,2 have eigenvalues for the linearization K + p, L„, + that are close to theimaginary axis. Proposition 2.3 suggests that for large in, we should restrict our analysisat least so that lajp+ ()Jul < 6, j = 1,2 where 6 = 0(in-2) as in --> oo. The solutionsof interest from the third order truncation have size IIVI2 < constant • 6, and in this casethe higher order terms in (3.74), even if some of the fifth order terms have coefficientsproportional to (Piwi — P9w9)-1, can be expected to have size only up to 0(in4).Since the principal parts of (3.74) have size up to --, 0(in-3), it is reasonable to neglectthe higher-order terms of (3.74).Now suppose we take limiting values in "Case II". Recall that we fix a but take asequence of small ( and large Q,=^Q = filk/2^ (3.77)where k,(', Q are fixed positive quantities and fit = 1,2, .... For each in we get a sequenceRit), in = 1, 2, .. of critical parameter values giving double Hopf bifurcations. Weconsider our "Case II" limit, when in = 1)1, and 711 oo. We recall that 1:17,1, = 0(m) asbefore, and from (3.76) we havePico' — Aw9 = 0(1117(k+1) + in-2)^ (3.78)as in -4 oo. In Case II, fifth order terms in (3.74) with coefficients proportional to (Piwi —P2w2)-1 will have size up to^0(ink-'). and if k > 1, such terms could be comparablein size to the principal parts and should not be neglected, unless laitt +^J. Toavoid this last restriction, we can use only a near-identity normal form transformationChapter 3. Center manifold and normal form reductions^ 58that do not result in fifth order terms with coefficients proportional to (Piwi — P2w2)-1.As discussed above, this means the cubic terms corresponding to the vectors (3.70) arenot removed. Thus the resulting normal form is=^+ asp + biv + Cilt17112 + C21117212) + C3 ft-7,^+ h.o.t., (3.79)472 =- 1T2 (iP2w2 +^+ b2v + C1IW1I2 + C51W212) + C6147211712 + h.o.t.,where h.o.t. = 0(1/1, P121W1^vliW12 + 111715). Since the matrix adA0(H„) is diagonal,the coefficients a , b , j = 1,2, C1,^, C6 in (3.79) are not affected by the normal formtransformation and are the same complex numbers as in (3.60).We will study (3.79) in Chapter 6, expecting it to give useful information on themagnetoconvection problem if ( < 1 and Q >> 1, for a wider range of parameter values(R, L) than the normal form (3.74).Chapter 4Evaluation of center manifold coefficientsSo far, we have shown that the dynamics of the magnetoconvection system (2.5)-(2.7)can be reduced to the normal forms (3.74) or (3.79). Both are valid, (3.74) is simpler, but(3.79) will give additional information in the Case II limit of decreasing and increasingQ and L. However, to make specific predictions of dynamical behavior, we need moreinformation about the coefficients in the normal forms.In this chapter we evaluate the normal form coefficients of (3.74) and (3.79). In §4.1we give explicit formulae for the coefficients in terms of the parameters of the originalmagnetoconvection system (2.5)-(2.7). Then in §4.2 we study the asymptotic behaviorof these coefficients for large aspect ratios, in both the Case I and II limits.. Finally, in§4.3 we evaluate the coefficients numerically for some specific values of the parametersof the magnetoconvection system.4.1 Center manifold coefficientsIn §3.4, we showed that the coefficients in the normal forms (3.74) and (3.79) are thesame as the corresponding coefficients that appear in the system (3.60) which repre-sents the phase flow in the center manifold The relation between (3.60) and themagnetoconvection system is given by (3.56).First, we determine the terms in (3.60) that depend on the parameters it and v. ByChapter 4. Evaluation of center manifold coefficients^ 60comparing derivatives with respect to it in (3.54) and in (3.56), we havea • = — (K(R„, + it, L„ + v)(1)i,C^i(12,0=(O,o)^j = 1,2Using equation (2.55), after some simplification we get(a + 1)a7r2wi — ia-72(6( w?) 4(2L,,/7/1 )2p12(.01 (,t)? + (52 )(a + 1)a7r2w9 — io-72(6( b.4)a2 = ^4(21,771/7/02Pi2W2(w3 + 62)where6 = 1 + + (.^ (4.4)From equations (4.2) and (4.3) it is clear that aim j = 1,2 are positive. We now findexpressions for bl and b9. Since K(//, v) is analytic in v, and iPiwi are simple eigenvaluesfor K0 = K(Rn„, Lin), there exist analytic functions oi(v), j = 1,2, such that a(v) isan eigenvalue for K(R„„ L„, + v), with ai (0) = iPicoi anddo -bi =^j = 1,2,dv(4.5)By equation (2.14), ai(v) satisfies the cubic equation0 =^+^v ) + Pi(v)(a( + + () + a(-(2rn 2 2 0. Rm+a-(Q72Pi (v)4(L,„ v)2^a(P13(0,where171272o- Rn,41.51(v)(L, + v)2)(4.6).P,^= 72 f[mj2(Lm + v)]2 +^.But from (2.19) and (2.23), we haveaçQ = (w? + (2)f)i (0)2(a + 7120 — ()4.L.'„^(0)3(0- +^+ L44)^(4.7)a 1?,„ = . •) ^— ()al(4.1)(4.2)(4.3)Chapter 4. Evaluation of center manifold coefficients^ 61therefore Si = ai/P(v) satisfies(a + 1) (c.o? + (2)P? (1 — OP?(v)(2)(0-^1)P1(0)2(1 — ()Pi2(140 = s31+(a+(+1)51+[(0-(+6r+0+LPi (0)3(a+()(1 + w?)] s^(b)?aPi (0)3 +^w?) + (7((Lim + v)2Pi3(v)(1 — ()h(si, v),(Lni + 0215(00. —(4.8)Since fi(iwi3O) = 0, and Ofi/Ov(iwi,v)^0, by the implicit function theorem, forsufficiently small v there exists a unique solution of fi(s1,v) = 0,^= C2i(v) =^+(But +^+ 00v12). To find B111, B11 explicitly, we substitute s1 = fli(v) intoequation (4.8), and calculate(a + 1)(a + ()72 [m2(1 — (2) — 2L(1 + w?)}2(c.o? + 62)(1 — ()PiLlin2^+ (2)(a +2P1'1(1 —72(2.2„ —17/2)(o- + ()(1 +4)(0 + w?)2Piwi (1 — OL1, (w? + (52)where P1 is given by (2.28). Similarly, we find^(a + 1)(u + ()7r2 [em+ 1)2(1 — (2) —^+4)]B2/1^2(01 62)(1 OP2L1712(in + 1)2(4 + (2)(a + 1)2P2c020 — 01),72(2L.:2n —^+1)2)(a + ()(1 +4(0 +4) 2P2w2(1 — ()Li2I (w + 62)where P2 is given by (2.28). Since= daj(0) = idP;(0)(v) + Pi (Bin + iBii),.7^di/we have= 1, 2,bin = PiBjll, j = 1,2,^ (4.13)B91B1RB11(4.9)(4.10)(4.11)(4.12)Chapter 4. Evaluation of center manifold coefficients^ 62— 71. 2 171 2C/Jib11 = ^ PiBu,213,7„—72 (711, + 1)2w.)b 1 1 = ^2 L 73„.^+ P9 B9 .Proposition 4.1 For all odd in, we have b111 < 0 and b2R > 0.Proof: To determine the sign of bil? we note from (4.10) thatP? A = [m2(1 — (2) — 2r,2,,(1 + w?)]271-4g, (1 — (2)and biR have the same sign. But by (2.28) we have(4.14)(4.15)1 + w = 1— (2 a(Q72(1 — () (a + 1)Pi2 (4.16)and by (2.35) we haveaCQ ^3m2(in + 1)2 m2(m + 1)2 [m2 +^± 1)2](4.17)72(a + 1)(1 + ()^(2L„,)^(2L„,)6Therefore using (4.16) and (4.17) we getA 2^m6 — (in + 1)27n2 [m2 + (in + 1)2]^32 [m2 — (in + 1)2]  < 0^4.18)^. (=(2L771)6^ (2L)4The proof that b2R > 0 is similar. Q.E.D.We now calculate the coefficients C1, ..., C6 of system (3.79) in terms of the parametersof the original convection problem. To do this, we need to approximate the centermanifold function ti1(4, it, v) at it = 0, ii = 0. We observe that by Theorem 3.2,kli(c1),,, 0, 0) can be approximated by its Taylor series to any finite order. We expandkli(4),, 0, 0) = 421-119000 +^+ Z1Z9T1010 + Z122T1001+ 212To9oo + Z1Z2kI10110 + 2129To10i + 4T0090+ Z9Z2Tooll + 2;2T0002 + 0(143),^ (4.19)Chapter 4. Evaluation of center manifold coefficients^ 63where(Pc = Z14)1 + 2-1 (Di + Z2(1)9 + 224)2.Then using the chain rule and (4.19), we have= iP1w1Z1 (2T9000z1 + 2it I I Iwo + Z9 loth + 22 Km)— iP1c4.)121 (ZiWiloo + 221T0200 + Z9Toli0 + 29Thno)• iP2w2Z9 (z1 i1010 + Tom) + 2Z2T0020 + 29T0011)- iP9(.4.19^(Z1 T 100 1 + 21k11 0101 + Z9 T 00 1 1 + 229T0002)+ o (jz i 3).On the other hand, from (3.54) we have= (I — f.3)[K^(4,0,0)^A.10 ((I)c, (1),)] •(4.20)(4.21)where /(0 = K(Rrn, Lr„), A10(11,19) = M(4)1, (1)2, E„,) for any 01)i, (1)2, T(61),, 0, 0) isgiven by (4.19), andmo (cbc, (De) = Z?Al0(1,(1)1) + 2W0(4)1,(1)1) + Zi27 [A10(4)1, (1)1) + M0(4)1, (DO] +Z1Z2 [1110(4)11 (1)2) + -110(4)2, (1)1 )1 + Z122 [M0(4'1, (1)2)^-110(4)21 (DO]+ z122 pi/0(4)1, 4)2) + A10((I)2, 4) )1 + Z1Z2 ph(4)2, (DI) + m(1(4)1, 432)]z.22/110 (4)2, 4)2) + 4,110(4)2, 4)2) + z24 [mo(4)2, (12) + M0(4'2, 4)2)] .(4.22)Then by identifying the coefficients of quadratic terms we have(K0 — 2iPiw1)T9o00^= _ ppl0(1)1,(1)1), (4.23)(K0 + 2iP1wI)To900^= —(I — P),11(4)1,(1)1). (4.24)(K0 — 2i.P9w9)T0090^= —(/ — Amo(4)2,4)2), (4.25)(Ko 2iP2w2) 0002K01/1100KO W0011- i-P2‘.02)4'1010- iP2W2)T0110• iP2W2)T1001▪ iP2W2)4j0101(K0 +Chapter 4. Evaluation of center manifold coefficients^ 64-(/ - P)11/0(4)2, (1)2),= -(i - P) [ itio ( 4)1 , (Di) + Alo(4)1,4)11,▪ -(I - P)Plo(4)2,(1)2) + .7110 (4)2, (1)2)]-(/ -^[11/0(4)1, 4)2) + Allo(cD2,4)01,= -(1 - 15) [mo(1,4)2)±/[0(4)2,(D1)i,P10(4)1, 4)2) + M0(4)2, 4)1)] 7=^P) P10(4)1, 4)2) + M0(7, (1)1)] •(4.26)(4.27)(4.28)(4.29)(4.30)(4.31)(4.32)Since M0(4)1, 4)9) = M0(4)1, (D2), from the above equations it is clear thatTioio =^T1001 = 011O, Tow = t2000, Toon = C009•Also, we note that Toil°, Tian can be obtained from Tioio under the changes w1and w2^-w9, respectively. To find C1, ..., C6 we substitute (4.19) into (3.56) withp = 0, v = 0, and compare with (3.79). Identifying the coefficients of the cubic termsgivesCi =C2 =C3 =C4 =C5(1110 ( T1 1 00, 'D O + MO ((1) 1, T i m ), (I) 7) +(M0(2000, (I)1) -1- M0(4)1, 4'2000), 4)0 •/11 ((.'...0\t--0011,(1)1)^-A10(4)1,ID0011)^-1/0(1j1001, 4)2),(1)0 ±010( 4)2, Tim) + -Al0 (T1 010, (D2) ± 2110(4)21 tP1010)• (1)1'),(-0^0020, -21^0M (Ti^1 4- A/ ra^D 0020/,-, 21, -21(m0('110110, (I)2) + A10(()2, Tom), (DT)iloo, )2, + L'40,^-• How +^- 1010, -In =1)d^1^(d^fri^\^(fri+ (M0(()1, tP 1010) + 1110(T^(1)1) + 2110((J)1, olio), (I);)(Mo(T owl, 4)2) + Mo(4)2,^+(4.33)(4.34)(4.35)(4.36)(4.37)Chapter 4. Evaluation of center manifold coefficients^ 65(1110 (4'002014)2) +111-0 CC 7 410020)1 (1)2*)C6 = (MO(tI I 10017 C I) 1) + 1110(4)1, 1001) ( 2*)^(4.38)(Aio(T9000• (19) +^T9000), (I);) •After lengthy calculations, which we have checked using the symbolic computationprogram Maple, we find explicit formulae for C1 , ..., C6. The calculations consist of threemajor steps:1) Computation of the terms A10(1,2) Using the results of Step 1 to compute the Taylor series coefficients ijkl of the centermanifolds by solving (4.23)-(4.32).3) Computing C1, ..., C6 according to (4.33)-(4.38).The results are given below; more details of the calculations are given in AppendixA. We have1971-Ci. = (C1/2) f al?"'^1   ^ ]^(4.39)1P(1(1 + iwi) [ 4P1(1 + wf)^2Pi2(1 + iwi)(t + 2iwi)io-(Q(2/,,,,,./in)472w2Pi2(C + iwi)((2 + w?)(wi( + 2ib.)1(2/„./111)2) 1 'C3C2{ ^-^aRln^7 ( (2M + 1.)(e2 + d2) + C- 12 + (112) +. C/2 1^+ 1101) 4(711. + 1)71 ((2rn + 1)ci + c'i)+ 7r ((217-1 + 1)d1 + d'i) + 41)2(11+ 4)]16P2(1 — ic02)^16P2(1 + iw2)+ '2g,P1 [D3 + D.4 + b3 + n4 ±712171• (( + iW1 )P1(TC(27 (E3+ E 4 + E3 +=2L2 Pi r ;_-,^A 1 ,^2(7(..Q [E,73 ± EA(Cni/2)^7;n [D3 + /-41.1 -h in7r(( + Iwi )(4.40)(4.41)9^o-Rin^71"P1(1 + )1) 2 P.1 (1 + W2) (7)2 + 2 u)2)^7r ((2m^1)d2^2)]}4(rn +1)((2m^1)(11 — d'1)16P2(1 +^IChapter 4. Evaluation of center manifold coefficients^ 66,^o-R„, ^[ ^((2777 + 1)(c2 + (12) + cl2 + d'2)C4 = Cm+1/ 2 (4.42)P2(1 + 477/7r ((22:p.21)ci + c'1)^( (2171 _1)d1 d 1) 1 16P1(1 —^161-'1(1iW1)^4P1 (1 + c4.4)1+ 1)72 [DI + D.2 + Di + 2 + a C(271- (El E2^1 +-E 2)]}P2K jW2)C5 = (Cm+1/ 2) aR,n^1P2(1 + iw2) I 4P2(1 + (.03)^211(1 + ic02)(z2 + 2iw2)]   ^7T io-(Q(2L71/(7n +1))472w2 Ph( + iW2) (C2 W)(W2( 2iW2(2tro./(71/ 1))2)2„. ^+ n21C6^21cr(Q ^{-(Cm4-1/2) {^21,(In + 1)P2 72 I 7r(m + 1)(( +1w2)^+ E2]((2m + 1)d1 —P2(1 + iw2) 1_2Pi2(1 +^+21w1)^16P1(1 +1w1)71 ((2m + 1)d2 + d'2)]}47n472where Cm, C„,+1 are given by (2.55), raj =.1.--17- for j = 1,2,72^a(Q ((2777 + 1)c3 + c',3 + 8(ci + c'4)){((27n + 1)ci + ^7r(m + 1)(27n + 1)(c1 4m^327ng,72 72I cr(Q ((27n + 1)c3 + c',3) + 72o-(Q(27n + 1)c3P2^2177,P1(C — iW1)^327ng,,Pj(( — icy')+A-20-(Q R2777. + 1)3c/3 + 8 (c4 + (27n + 1)2c.)]32777,E;iiP1K — iwi)+71" ((2m + 1)ci + cc)^7r(171, + 1) ((2m + 1)ci — cc)8m.^32E,72 o-(Q (—(2m + 1)(3 — C/3 + 8 ( C.1 + Cil ) )=P1^8(m + 1)(( — 1w2)+71 ((2777 + 1)ci + c)^71771(21)1 + 1)(ci — cC)-14(777 + 1)^32g, (77/ + 1)^i72 72o-CQ ((27n + 1)c3 + ci) +^7120-(Q(27n ± 1)c3 {^Pi^2(m + 1)P2(( — 7:c4,2)^32(m + 1)g,P2K — iw2)9D1D2D3D412 8711(( —(4.43)(4.44)(4.45)(4.46)(4.47)Chapter 4.^Evaluation of center manifold coefficients7-20-(Q [(2m^1)3c13 — 8 (c4 + (2111 + 1)2e4)]67(4.48)32(m + 1)L1P2(( — iw2)7r ((2m + 1)c1 +^In^((2m + 1)ci — c11)8(m + 1) 3247 ((2M,^1)C3 + C13) —87(c4^c14) (4.49)8m72 ((2m + 1)ci +E2 (4.50)8m.Pi(C —E3=[(2772^1)C3^8(C4^e4)] (4.51)8(m + 1)E472 [(2m + 1)ci + cid(4.52)8P2(m + 1)(( — 7:w2)'72 cr(C2 ((2m + 1)d3 + d'3 + 8(d4 + d'4))P2 1^81.11K^iW1)7r ((2m + 1)d1 + (IC)^7r(m + 1)(2m + 1) (di +4m^32 mg, (4.53)D272 f 71-2a(Q ((2m + 1)(13 + d'3)^7r2^mo-(Q(2^+ 1)d3P2 2mPi(( + iwt )^32mg1P3^+72.902 [(2m + 1)3d3 + 8 (d4 + (2m + 1)2d4)]32mE,2nP1 (C + iwi )7r ((2171 + 1)d1 +^)^7r (In ± 1) ((2m + 1)d1 — di)(4.54)32g^}f)3o-(Q (—(2m + 1)d3 — d'3 + 8(d4 + d!4))8(rn + 1)(( + 7:W2 )((2m + 1)d1 + d'1)^in(2rn + 1)7(d1 — d'1)(4.55)4(m + 1)^32E(m + 1)D471-20-(Q ((2171 + 1)d:3 + d'3)^72o-(Q(2m + 1)d32(m + 1)P2(( + iw2)^32(m + 1)g,P2((^iw2)Pic)-(Q [(2111 + 1)3(13 — 8 (d4 + (2/7" + 1)2(114)132(111 + 1)g,P2(C + ico2)((21/1^1)(11^11/7 ((2/11^1)d1 — di1)(4.56)8(m + 1) 32g,71.^+ 1)d3 + (I'3) — 87(d4 + (1) (4.57)8rnChapter 4.^Evaluation of center manifold coefficients 68E.=72 ((2111+ 1)di + d'1)(4.58)87nPi (( + iwi)71 ((2117.^1)(13^ + 87(d4 + d)= (4.59)8(in + 1)E4=712 ((2771^1)Cli^(111)(4.60)8P2(m, + 1)(( + iw2)and ni, E , jchange w2C1—w2.=1,...,4 are obtained from Dj,Ej,j = 1,...,4, respectively, byAlso(A1/4) + (a(QA0A3/27113) + (aRA2/4aqi)making the(4.61)'(712A0/471-2) + (a(QA0/713) — (aR/16g,.10C2 = (Cii4Th) - (42/111)1 (4.62)27c1 + A3C3 = (4.63)113C4 = —A4/174, (4.64)C 1 = (B1/4) + (a(QB03/2703) + ((277/, + 1)aRB2/ LIL-,01) (4.65)(02Bo/472) + (0-(QBo/03) — ((2111 + 1)2aR/16g,.01) '/C2 = (2rn + 1(c/1/401) — (B2/0),) (4.66)IC3 =27cii + B3(4.67)03 'c4' = —B4/04. (4.68)where- -40 + iP1w1 + iP9w2,^719 = a.:40 + iP1w1 + iP9o.)9,773 = (A0 + iP1w1 + iP2w9, 714 = ((7/2/,)2 +^+= Bo +^+^1132 = aBo + iPjwj + iP9w9,= (Bo +^+ lAw9, 0.1 = (((2in + 1)7/2L)2 + iP1w1 + iAw2,Ao^71-2(1/4/2 + 4),Bo = 72((217/ + 1/2L1„)2 + 4),Chapter 4. Evaluation of center manifold coefficients^ 69(2m + 1)74Lm(', + 1)'0- C(27 (P2 - P1)1071 + 1)P1P2(( + iwi)(( + iw2)A1 =- (2m + 1)B1,A =^7(2m + 1) [m,P2(1 iw2) +^+1)P1(1 iwi)] 2 4M(771+ 1)PIP2(1+^+ iW2)7r [-7nP2(1 + iw2) + (in 1)P1(1 + iwi)] 41n(rn 1)P1P2(1 iwi)(1 + iw2)72^(( +^— P2(( + 2b)2)] 711(711 + 1)1311).2(( + if-4)1) (C^iC4)2rA3 = (2m+ 1)B3,^7r2 [P1((^iW1)^P2K ib)2)] Azi =^4771( 77 1 + 1)-P1P2K +^+ iw2)B.1 = (2m, + 02-44.and d • (I'. j = 1, ..., 4 are obtained from c• 1 C1j, j = 1,...,4 by making the change W2 "4J4.2 Asymptotic resultsIn this section we give explicit formulae for the limiting values of the coefficientsal, co, bi, b9, C1, •• •, C6^in the normal forms (3.74) and (3.79), as^oo. We consider Cases I and II discussedin §3.5. Details of these calculations are given in Appendix A. Throughout this chapterwe assume-1=4.2.1 Case IIt is easy to see from (4.2) and (4.3) thatinn^= lim 0,2 = a,-B1=B2B3Chapter 4. Evaluation of center manifold coefficients^ 70where(4.69)and A, P,w are given by (2.27), (2.59), (2.60) 6 = 1+ a + (". It is clear from (4.13)—(4.15)that b , j = 1,2, are 0(c) as in oo, but we will show that actually the real parts biRare 0(c2). By equation (4.18) and (4.9) we have—71-6(o- + 1)(o- + ()(1 + ()Awhere A is as in (4.18), after some simplification we get=and similarly we have1271-4(a +Oa^1)((^1)(-2 +0(c3),^(4.71)A5P(w2 + 62)1274(a +^+1)(C^1)E2^O(E3),^(4.72)A5P(w2 + 62)Also, after some calculation we find asymptotic expansions of bi1, j = 1,2 and we get{471-2 [(a + 1)(w2 + (2) — w2(1 — ()] w(1 — ()A3272 (A2 —2)^+ (1 + w2)((6 w2) } c + 0(c2), (4.73)w(1 — () (62 + w2) A3{47r2 [(a + 1)(w2 + (2) — w2(1 — ()] w(1 — ()A32 2 (A2 — 2) (0- +0(1 +2)((a+(-02)} e+ o(c2), (4.74)_ (6-2 + w2)Remark 4.1 We notice that in the asymptotic expansions of b1 and a2„b9 the termsat order 0(c) are equal while the terms at order 0(c2) are negatives of each other. Thisis due to the fact that the same relation holds for the asymptotic elpansions of P1,wi andP2, w2. We will find similar behavior for the center mamfold coefficients.(a + 1)o-ir2w — ia 72 (6( + (.4.)2)a =^4A2p2w(w2 ± 62)bin -^nip 12 (4 + (52)^1^ (4.70)bilb21Chapter 4. Evaluation of center manifold coefficients^ 71For the normal form coefficients C1, ..., C6 we havelim Cr = lirn c5 = A + B,n1— O0^711—).00u rn C2 = liM C4 = A,whereA + BA =711—Y 00liM C3771-+ 007r2o-R0(( + iw)7/1—) 00_^lirn C6 = C,In -+00i^1^72(4.75)4A2P2c..o(co — i6)io-(QA271-4 (1 + iw)I_^4P(1 + w2)^2P2(1 + iw)(a7 + 2iwzddii+ 4P3((2 + w2)(w( + 21:wA2)(w — i6) '—72(( + iw)(1 + iw)^o-/?0^+ d21)[7(c21 + 4AP(1 + icii)4A2Pw(co — i6)^P(1 + iw)^2{1^1o-Q(2A2c1,11^271-20-(Qc„^a(0-1141A2+ 4.+ 4P(1 + w2)i^((2 + w2)^P(ç — iw)^( + iw7rd1 1 (3)2 — 1)^72cr(Q(A(1:31 — 2d4 1 )^70-('Qdir^} (4.76)+^+ ,4A^P(( + iw)^4P(( + iw)2—72(C + 7:(4)) (1 + kJ)^aR0 71 24A2Pw(co — i6)^P(1 + iw) [2P2(1 + iw)(w + 2iw)„ 2 71-2a(Q)(d3i + 2d1)4AP(1+ iw)^7a21/P(( + iw) 0-(QAT-cli^7rdi (3A2 — 1)+o-(Qd'41A2 + (4.77)+ iw^4P(( + ico)2^4A^}where and A, w, P. R0 satisfy equations (2.58), (2.59), (2.66) and (2.67) respectively,^=472/P, andC91 = (4.78)P2(1 + iw)(7; + 21144'272)2C41 (4.79)P2(( + iw)(t-z( + 2/wA2)'Allo(271-d11 (4.80)p((-2^w2)(472 + Q) 'd29 = 1 (4.81)47rP(1 + w2)2Aiunr2(133 =(P(472 + Q)((2 + w2)(4.82)Chapter 4. Evaluation of center manifold coefficients^ 72d'41 = (4.83)2P ( (2 w2 ) •4.2.2 Case IIFor (4.2) and (4.3) we obtain (for any k > 0)lirn a1 = lim a2 = a,^ (4.84)m--+oo^m--+where+ 1)a7r2c.T.) — io-7r2 + co2= ^ (4.85)4A21926)-(c_b2 +(a +1)2) 'and^c.:) are satisfying (2.72), (2.73) and (2.74). Using (4.70) We also obtain4o- (o- + 1) (71-2cD2 + /5and4or(o- + 1) (22 + P) .;x3P(c452 ± (0- +1)2) ) (2 + 0(E2+(k12)).472a-j)^272a (Â2 — 2) (1 + c.D2) ± 0(61+(k/2)),^\3 ^((0" + 1 )2 + C4)2)472aw- + 272a (A2 _^(1 + (5)2) cz, E^0(61+(k/2)),3^((a +1)2+2)3^j(4.87)(4.88)(4.89)as in^cc. However, since by (2.77)—(2.80), Pi — P9 and w1 — w2 are OW, it is easy toshow that 1)11 —^= 0(c2). This will be important in our analysis in Chapter 6.In Appendix B, we show that if k > 2, then C9 C3, C4 and C6 become unbounded asin^cc, and if k = 2, then limits exist butlim C9 0 Um C4, and^lim C3^lirn C6.L,„->oo^L0 -+00^Lni —3 001R^)^62 ± 0 (62+(k/2)),-6^=^A3p(w2 + (a + 1)2) (4.86)Chapter 4. Evaluation of center manifold coefficients^ 73In order to keep the limiting behavior of the normal form coefficients similar to that forCase I, we restrict k < 2. Then we findwhereA +Alirn^Inn C5 = A +13,771-400 77/ 00Ern c2 =^= A,/71-4 00^ 772 00hill C6 = 0)lilll C3m—ooi7r2O-P0^1^1^72+4A2P2(6.) — i(a + 1)) 14.P(1 + cD2)^2/52(1 + ic;))(eb + 2iJ4]o-(Q7r4(1 + i)+83p.^.^8cD3P3(c.;) —1(a + 1)) '= ^7r2o-Poi^7r(e-21 +(121) + mil 1^+^1^1- - -4A2P2(a) — i(o- + 1))^2^4AP(1 + ic;.))^4P(1 + cD2)..17r2(1 ± iCo)i^171-2(7-(Q(2-e'41 — 2(11,11 ± A(131) + 7a(Q(1114i)X'2(ci, — i(o- + 1) [^ii5cD^415c112+ 7(111(3A2 — 1) + A2(7(0'41]4A(4.90)(4.91)and ft =472Pi7r20.110 7r-^7rciii^  ±4A2P2[JJ — i(a + 1)][2P2(1 +ic.-0)( + 2i64^4AP(1 + iCo)^271-(121]9i7T-2(1 + icD)^I 77-2o-(Q)■((i31 + 2d'41)^71-d1 1 (3A2 — 1)4/5A2( — i(o- + 1)) I^if)co^4A+ a(Qd'ili A2^(7(-QA7rd1liJ^j42 (4.92)7rC)1 = /52(1 + icD)(471-2/P +^ (4.93)7241^ (4.94)=^I PC),^ (4.95)Chapter 4. Evaluation of center manifold coefficients^ 74d91 = 1(4.96)(4.97)(4.98)4715(1 + 4.D2)'27i;\U41(QPCD5k22Pc.D•4.3 Numerical resultsBecause of the complexity of the formulas for the normal coefficients Ci , ..., C6, we werenot able to find the signs of the real and imaginary parts of these coefficients analytically.The signs will be important in our bifurcation and stability analysis in Chapters 5 and6. Therefore we have evaluated the coefficients numerically, using Maple to carry outthe numerical computations. The results are summarized in Appendix B, and a repre-sentative selection of them is presented in this section. The symbols oo in the tablescorrespond to the limiting values of the coefficients as in oo, as calculated in §4.2.We note that numerical results appear to verify our analytic asymptotic results on thenormal form coefficients.The values of L„, were calculated by using (2.35). We have also calculated the valuesof C1 and C5 for a relatively large set of parameter values a, ( and Q (see AppendixB). We found that CH? and C511 are negative for all values of a, C, Q and in we used.The numerical computations of C9, (3, C4 and C6 took much longer and more care wasrequired to avoid memory overflow and round-off error. For ( = .1, a = 1 and all thedifferent values of Q and in that we used, we found that. C911, 6'4R are negative, andC11105R— Conalll < 0.^ (4.99)We have also checked this inequality for a wider set of parameter values of a, and Q inthe limit as in --> oo. This will be important for our results in Chapter 5.Chapter 4. Evaluation of center manifold coefficients^ 75In our calculations we observed that the signs of the real and imaginary parts ofC1, C2, C4 and C5 for all finite in and in the limit as in -4 oo are the same, but the signsof C3R, C6R, C31 and C61 changed as in was increased.We have also computed the values of normal form coefficients in Case II with a =1, (- = .1, Q = 10072, k = 1 and for increasing values of in (Tables 4.3-4.6 ). The essentialfeatures are preserved, and our analytic asymptotic results seem verified numerically.However, convergence appears to be slower: C1 and Cs, C2 and C4, C3 and C6 approachtheir limits much slower than their differences approach zero as in^oo. For example,it is easy to verify that both C1 = A + B + 0(E1") and C5 ==^+ + 0(E'12), but- C5 = 0(E).Table 4.1: Normal form coefficients (Case I) for a = 1,( = .1, Q = 10072.in 2L,,./in /?„, 100ai 100021 1.411 2399. .1762- .1372i .2296- .1146i11 1.022 2018 .2270- .1446i .2342- .1405i101 .9835 2013 .2307- .1431i .2315- .1426i1001 .9791 2013 .2310 - .1428 .2311 - .1428i10001 .9787 2013 .2311 - .1428i .2311 - .1428ioo .9787 2013 .2311 - .1428i .2311 - .1428iTable 4.2: Normal form coefficients (Case I) for a = 1, = .1, Q = 1007r2 (continued).in L;72„ bin L. 72,,,b2R L 771 bii Lin b2i1 -5.305 8.642 17.18 24.0411 -6.621 7.037 19.85 20.83101 -6.803 6.850 20.28 20.391001 -6.824 6.829 20.33 20.3410001 -6.827 6.827 20.33 20.3300 -6.827 6.827 20.33 20.33Chapter 4. Evaluation of center manifold coefficients^ 76Table 4.3: Normal form coefficients (Case I) for a = 1,C = .1, Q = 10072 (continued)771 Cl C2 C4 C51 -.0460 + .1125i -.0743 - .0352i -.0686 - .0026i -.0595 + .0931i11 -.0523 + .1045i -.1360 + .0849i -.1360 + .0849i -.0541 + .1020i101 -.0531 + .1034i -.1000 + .1300i -.0859 + .1363i -.0533 + .1031i1001 -.0532+ .1033i -.0936+ .1333i -.0921 + .1339i -.0532 + .1032i10001 -.0532+ .1033i -.0928+ .1336i -.0927+ .1336i -.0532+ .1033i100001 -.0532+ .1033i -.0929+ .1336i -.0928 + .1336i -.0532 + .1033ioo -.0532+ .1033i -.0928+ .1336i -.0928+ .1336i -.0532+ .1033iTable 4.4: Normal form coefficients (Case II) for a =^= .1, (2 = 10072, k = 1.in 2L7,/in R„, 100a1 100021 1.411 2399. .1762- .1372i .2296- .1146i11 1.011 1832 .2362- .1379i .2442- .1315i101 .9692 1775 .2432- .1333i .2441- .1325i1001 .9638 1757 .24470- .1322i .2447- .1321i10001 .9630 1752 .2450- .1320i .2450- .1320i100001 .9629 1750 .2451 - .1319i .2451 - .1319i1000001 .9628 1749 .2452- .1319i .2452- .1319ioo .9628 1749 .2452- .1319i .2452- .1319iTable 4.5: Normal form coefficients (Case II) for a =^= .1,(2 = 10072, k = 1(continued).in ET2,,b1 R E;2, i b2R L7?, b1 1,71,1)211 -5.305 8.642 17.18 24.0411 -6.181 6.583 17.96 18.947101 -6.229 6.274 17.92 18.031001 -6.206 6.210 17.82 17.8310001 -6.193 6.195 17.78 17.78100001 -6.190 6.190 17.77 17.771000001 -6.188 6.188 17.76 17.76oo -6.188 6.188 17.76 17.76Chapter 4. Evaluation of center manifold coefficients^ 77Table 4.6: Normal form coefficients (Case II) for a =^= .1, e2' = 10072, k = 1(continued).in C 1 C2 C4 C51 -.0460 + .1125i -.0743 - .0332i -.0686 - .0026i -.0595 + .0931i11 -.04321 + .0968/ -.1061 + .0066/ .0119 + .0512i -.0444 + .0946i101 -.0412 + .0938i -.1205 + .0747i -.0109 + .1131i -.0413 + .0935i1001 -.0404+ .0930i -.0947 + .1116i -.0517 + .1264/ -.0404 + .0930i10001 -.0402+ .0928i -.0810+ .1196/ -.0668+ .1244/ -.0402+ .0928i100001 -.0401+ .0927i -.0762+ .1215i -.0717+ .1230i -.0401+ .0927i1000001 -.0401+ .0927i -.0746+ .1220/ -.0732+ .1225i -.0401 + .0927i107 + 1 -.0401 + .0927/ -.0785 + .1722/ -.0784 + .1723i -.0401 + .0927i108 + 1 -.0401 + .0927i -.0739 + .1223i -.0738 + .1223i -.0401 + .0927i109 + 1 -.0401 + .0927i -.0739 + .1223i -.0739 + .1223i -.0401 + .0927ioo -.0401 + .0927i -.0739 + .1223i -.0739 + .1223i -.0401 + .0927iTable 4.7: Normal form coefficients (Case II) for a = 1,e = .1, ej = 10072, k = 1(continued).in C3 C61 .1376- .0938i .0309- .0866/11 .0890 - .0031/ -.0405 - .0381/101 .0539 + .0673/ -.0569 + .0305i1001 .0112 + .0820i -.0319 + .0674/10001 -.0042 + .0803/ -.0184 + .0754/100001 -.0092+.07891 -.0137+.0774i1000001 -.0107+.0784i -.0121 + .0780i10i+1 -.0107+.0784i -.0121 + .0780i108 + 1 -.0113 + .0782/ -.0115 + .0782/109 + 1 -.0114 + .07821 -.0114 + .0782/cc -.0114 + .0782/ -.0114 + .0782/Chapter 5Existence of invariant toriIn Chapter 3 we have shown that the dynamics of the magnetoconvection equation in aneighborhood of the origin in the Hilbert space X, for L sufficiently close to L,„ and Rsufficiently close to R„, (for all finite L„„ and in both Case I and II limits as in oo),will be determined by the dynamics of the four-diinensional ordinary differential equation(3.74), which we rewrite here as21. — v + C1Zi2 C91 Z912] h.o.t.Z2 [iP2W2 b21,/ -f-051Z212+C41Z112] h.o.t., (5.1)whereh.o.t. = v121ZI, ii, 1l11Z13 +IZ15), ii = R — v = L — L..and Z = (Z1, Z2) E C2. The normal form coefficients Ci, j = 1, 2, 4, 5, and ci , b , i = 1,2were given and evaluated in Chapter 4. Recall that system (5.1) possesses the Z2 ED Z2symmetry (3.59).In this chapter, we determine the dynamics of (5.1), for Z, t , v near zero. First,in §5.1 we study the "truncated" normal form where we ignore the higher-order terms(h.o.t.). In this case, the equations decouple and the system is reduced to a planer onewhich is straightforward to analyze. We must then determine whether certain structures"persist" when the higher-order terms are restored, and in the rest of this chapter weprove results for (5.1) that, are valid in the generic case, i.e. when the higher-order78Chapter 5. Existence of invariant tori^ 79terms are not necessarily identically zero. In §5.2 we prove the existence of primaryHopf bifurcations of symmetric "standing wave" periodic orbits SW0 and SW, from thetrivial solution, along two curves F1, P9 in the (it, v) parameter plane. In §5.3 we provethe occurrence of bifurcations of invariant tori from the periodic orbits STF0 and SW,,along additional curves A1 and A9 in the parameter plane, which implies the existence ofthese tori for parameter values in a region adjacent to the two curves A1 and A2. Finally,in §5.4 we prove the existence of normally hyperbolic invariant tori for parameter valuesin the interior of a wedge in the (it, v) plane bounded by A1 and A9, but away from theboundaries. Then we combine this result with the result of §5.3 to prove the existence ofnormally hyperbolic invariant tori for parameter values throughout the wedge boundedby A1 and A9.From our analytic results in §4.1, we know that am,0211 and b911 are positive, whilebiR is negative for all odd m. Also, based on our numerical results in §4.3, throughoutthis chapter we assumeHypothesis 5.1 Ci R, C9 n, C4 R5 C5R are all negative.5.1 The truncated normal formiWe write the normal form^ =(3.1) in polar coordinates Z1^= 7,2eo2 and obtain97" I^ri (a^+ C1pi + C91111 + 0(.5), (5.2)19 (amp + b911v + C 4 R721 C 5 nil 0(r5),01 =P1w1 +^+ buv + C1jv + C911-:; 0(r4),02 = P2CA.)2^ '2!i ^b9iv + C414 + C51r + 0(r4).We observe that the first two equations of (5.2) &couple from the last two, up to terms oflower order, but generically the higher-order terms of 0(r5) and 0 (r4) depend nontriviallyChapter 5. Existence of invariant tori^ 80on 01 and 02. As an approximation, we ignore temporarily the higher-order terms andconsider the truncated normal form-= vi (ai^+ bi Rv + C1j1r? + c91)^(5.3)r2 (a2Rit + b2Rv + Cartri + C5A)01 = P1w1 + aiiit + bi/v + Crir? + Cnr,09^P2w9 + a2/P + b91, + C4/4 + C54Later, we will restore the higher order terms and prove results for the original system(5.1), based on the analysis of truncated system (5.3). Since the last two equations of(5.3) are decoupled from the first two, we need only consider the two dimensional system7'1^r1 ((quit b1^+ C1Rlq + C9 nr:0^(5.4)r9 (amp, + bmv+ C4 1v1 C511) .This system has been discussed by Guckenheimer and Holmes [16, §7.5]. To make use oftheir analysis, we simplify (5.4) by using the following scaling:= ri VJCi ,17 = r9005R1, = s ga(C R)t.Then (5.4) becomesdi^ _9^.991?, ( CI n)C2R -971 sfin(CIR)(aireit + biRv) +^+ 12dt IC5nidr2F9 51111(C 1?) (a9Tha + b21/v)  + ^ + syn(C5R)s911(C1n)T391dtBy Hypothesis 5.1, sgn(C1R) = syn(C5R) = —1, and we are reduced to^f 1 =^( -ft + 17‘f +=^-/121? + CF.; +^•(5.5)(5.6)( 1, 2) = (17* 1, F*9) = ( 1 — BC)112 R — Oli n 1111t — B 119R 1 — BCChapter 5. Existence of invariant tori^ 81whereairdt + bug), i 1,2, B = 6'9R/C5R, C = Cin/Cirt•According to the sign of B, C and 1—BC, there are six different cases. The coefficientsB and C are positive in our application, so our study of (5.6) falls into cases /a (if1 — BC > 0) and lb (if 1 — BC < 0) of [16, p.399]. In our application for the parametervalues that we have checked, only the case Ib is possible (see §4.3), but most of ouranalysis will be valid for both I„ and lb cases.For (5.6), the origin (Ti , 12) = (0,0) is always an equilibrium point, and there are upto three other equilibrium points in the first quadrant:( T1 , 2 )^( /111R, O), if /1111 > 0;, f2) = (0, 0190 , if / 19 11 > 0;ifRIR^B It2R 11211^C/111? > 0.1 — BC^1 — BCThese fixed points correspond to the following equilibrium points of (5.4):SIVe) (ri,r2) =^ )al nit + biRv, 0CIRbiaif it > --v;(li (0, 1 a9^+ 1)9R(1.; (PI^r;^0)sw?) :( if p> --=v;T(°) (r1,r9) =wherer7(p,v)111(a9RC9 ll — al 110511 ) + V OM C9/1 — bl 110511)1C1 BC511 — C911a1 11/Mai BC411 — amCIR) + v(bi Ram — bmCu?),C1 110511 — C21?Ci a2( F1*)2E(ii I I?, /121? ) =2C 17117;213F1 9•2(T;(5.9)Chapter 5. Existence of invariant tori^ 82if^v) belongs to a wedge 1,17(°) in the parameter plane defined byp(a9RC9R — RC5R) + v(b91C211 — RC5R) CinC5R — C9RC4Rp(aiRC4R— 0,211C1n) + v(bittain— b2RCIR) CinC5R — C211C41?Thus, the boundaries of WM lie along the lines> 0,>0.Ar •^. p = Ao*v, sgn(v)^scia(A0*^but/a111),AV) :^Av, syn(v)^sfin(A*, + b9n/a20,whereA'(!; =A; =1?bl^C Rb2 (5.7)(5.8)C1Ra2 R — C1 ROi RC21b9 R — C511b RC5 Rai R — C9 Ra0 RSee Figure 5.1.The family of equilibrium points SWe) of (5.4) correspond to a family of periodicorbits of (5.3) which bifurcate from the trivial solution at the origin as the parameterscross the line al Hp b1 pi = 0. A similar correspondence holds for the family S147,°) of(5.4). In §4.2 we will prove the existence of these two families of periodic orbits for thenon-truncated system (5.1). The family of equilibrium points T(°) of (5.4) corresponds toa family of invariant tori for (5.3), for parameter values in the wedge WM. In §5.3 and§5.4 we show that a family of invariant tori for the non-truncated system (5.1) exists forparameter values inside a wedge W in the (it, V) plane, that is approximated byTo study the stability of the equilibrium points TM, we linearize the vector field of(5.3) about the fixed point (Fr, T.,), obtaining the matrix[ 2C1 ()22C4Rri 7 912C9Rri*r.;E ( p,, v ) = ,2C5Rkr.2*/(5.10)Chapter 5. Existence of invariant tori^ 83Since BC 0 1, E has eigenvaluesA1,2 = (Tr(E) VTr(E)2 — LIDet(E)) /2,whereTr(E) = 2((fp2) + VD2), Det(E) = 4(1177;)2(1 — BC).Depending on the sign of Det(E), (771*,i72*) is either a sink or saddle point for (5.6).The corresponding linearized vector field of (5.4) about (71,r;) is given bywhich has eigenvalues'A‘ 1,2 = (-Tr(E) ITT(E)2 — 4Det(E)) /2.Depending on the sign of CiRC51/ — C911a111, (7'7, r;) is either a sink or saddle for (5.4).These fixed points correspond to either normally hyperbolic attracting invariant tori, ornormally hyperbolic invariant tori of saddle type for (5.3). Bifurcation sets for (5.4),corresponding to the two different cases depending on the sign of CIRC5R — C2RC4R, aregiven in Figure 5.1.5.2 Bifurcating periodic orbitsIn this section we return to consider the reduced four dimensional system of ordinary dif-ferential equations (5.1), and prove the existence of bifurcating periodic solutions. Thesesolutions correspond to nonlinear standing waves in the magnetoconvection problem, sowe will denote them by SW solutions. First, we note that the subspacesVo = {(Z1,0) :Z1E CI,^{(0, Z2) : Z2 E Cl,b1 H= -^I./a1 Rb2R= —a 9 RChapter 5. Existence of invariant tori^ 84(a) it^ bI? l)= ---V(b)b211—^1,/a711Figure 5.1: Bifurcation set for (5.4): (a) CiRC5R-C9RC.in > O (14 CinC5R-C2RaiR < 0.Chapter 5. Existence of invariant tori^ 85are invariant manifolds for (5.1) due to the Z9 e z, symmetry (3.59). So we may studysome of the dynamics of (5.1) by restricting it to these subspaces.The system on Vo becomes:= z1(iPi^+ iti + Ci l Z i 12 ) + 0 (1z115 + Iii, vl^+ lit, 1) I 2 IZi I),^(5.11)where 111 = alp biv. Now let Zi = el, so in polar coordinates (5.11) becomes= 7-1(R1 11^C1 114) + 0 (rj + 111,1)14 + lit, vI27-1),^(5.12)= P1w1 + /117 + Cur? + 0(4 +1/1, vir? + it, v12).Since Cm < 0, by [16, Theorem 3.4.2], (5.12) undergoes a supercritical Hopf bifurcationas we increase ill!? through some value near zero, and there exists a family of periodicorbits Z0(t) Zo(ft, v)(t), which we denote by the branch of SWO solutions. The SW0solutions bifurcate from the trivial solution for parameter (it, v) along the curve F1 definedby et/ = ito(v), where110(1) ) = —vbill/am + 0(1v12),and the ST470 solutions satisfy1412 =^(11 1111 + + 0(1/1, 02), Z2 7=--- 0.Cin (5.13)By a similar argument, we restrict (5.1) to the the subspace '17,, and prove the Hopfbifurcation (supercritical, since C511 < 0) of a family of periodic orbits Z1, (t)= Z7,(it,v)(t)from the trivial solution for parameters along the curve r, defined by it = [1„.(v), wherefir ( V) = —0)9 R /09R + 0 (I v12).We call these solutions, satisfying(12 /?/l^b9 ill/Z1 E 0 , 1Z 212 =^+ O(I ji, 1112),C51?(5.14)Chapter 5. Existence of invariant tori^ 86the branch of SW,_ solutions. The SW solutions Zo, Z, are periodic solutions of (5.1)with periods To = = v) respectively, with ro near 27/Picoi, and Tr near27r/P2w2. The solutions Z0(t) and Z,(t) satisfyJZo(t) = Z0(t) V t E R,^ (5.15)/3Z(t) = Z,(t) V t E R.To consider the stabilities of the SW solutions, and secondary bifurcations, we trans-form coordinates near each periodic orbit. For v 0, we rescale variables by puttingZ = II/11 210, Z2 = 10112 U2, P = iv1A,where u1, n2 E C, A E R. therefore we have=^+ u1^12 + cd,11,121 + 0002)tiu1,\ + bisfin(v) +^ (5.16)7i2 = ivIu2^+ a2A + b2sgn(v)In order to study stability and bifurcation of the SWo solutions (u1^0, u2^0) we let= re' and u2 = u, and obtainit = iP2w2u +^[(12A + b9sgn(v) + C5Iu12 + C47-2]+ 0(1v12),^(5.17)Ivir PIRA + sgn(v)bill+ Cilly2 + C9R1u1l + 0(102),+ IvIktidt + sgn(v)bil + C11r2 +^+ 0(102).Since 0 0 0 for all sufficiently small^we can parameterize the flow away from r = 0by 0, obtaining+ 675171212 +^+ 0(1v12).du/ (10 = iAtt +101 4(u, r, A) + 1021;1 (u,r,0, A, v),^(5.18)chide =^r, A) + Ivrk(u, r, 0, A, v),Chapter 5. Existence of invariant ton^ 87where A = P9w2/P1w1, tl(u, r, 8, A, v) and n(u, r, 0, A, v) are periodic in 0 with period27r, andU(u, r, A) = (u/Piwi) [A(a2 — jAaii) + s9u(v)(b2 — jAbil)— iAC1i)r2 + (C5 — jAC2/)1u1217Zett,r, A) = (r/Piwi)[aIRA syn(v)bili + CiRr2 + C21?InI2].The ST/V0 solution (5.13) now corresponds to the branch of 27-periodic solutionsu = uswo (0, A, v)^0, r = rSj 0 (0,^= ro (A) + 0(v),of the rescaled, reparameterized nonautonomous system (5.18), whereaiRA syn(v)biur.PA) =C 1 11Now we define a moving coordinate system byu = v, r = rstv0(0, A,^+ x,where v E C, E R. Then (5.18) becomesdvide = iv^37, A) + 1v121)(v, x, 0, A, v),dxid° =^x, A) +1/212(r,' 37, 0, A, v),where(5.19)V(v,D, x, A) = (v/Piwi) {A(a9 — Aait) + sfpl(v)(b9 — iz.\141)+ 2(C.1 — iL\Cii)vo(A)x + (C4 — lACil)rO(A) + (C5 — iAC21)iv12}+ 0(I11211,1),X(v,v,s, A) = (1/P1c.01){2C111rO(A):r + C2nro(A)iv121+ (:)(13712 + I.riivi2),Chapter 5. Existence of invariant tori^ 88and 1)(v, V, x,0, A, v), (v, V, x,0, A, v) are both periodic in 0 with period 27r. (Note thatV satisfies the complex conjugate of the equation for v.) By Floquet theory, there is a27r-periodic coordinate transformation, linear in the spatial variables, of the form(v, 17, x)T = [I + vP(0, A, v)]ew,^y)T, to E C, y E R,^(5.20)which removes the 0-dependence of the linear terms in w, t , y to all orders in Iv'. Equa-tions (5.19) are then replaced bydw/d0 =^+ 1014)(w,^y, A, v)^Iv121'New,^y, 0, A, v),^(5.21)^dy/d0 = IvIY(w,^y, A, v) ±1v12)1(w, ü, y, 0, A, v),where1/1)(w, tL , y, A, v)^V(w, D , y, A) + 0011w,yew,^y, A, v)^X(w,'tp, y, A) + 0(11111w„ y1),1/ikw, t , y, 0, 6,^= 0(1012 + lilbwl + 11012),5)(w, 7-D, y, 0, 6,^=^(1012 +^+ w12),and 1;V- , are both periodic in 0 with period 27r, and 11) satisfies the complex conjugateof the equation for w.^The trivial solution to a-  0, y^0 of (3.21) now corresponds to the S1470 solutions of(5.1). The linearization about to = 0, 'CO = 0, y = 0, is given by the matrix4=f20(v, A)0(11112)00(102)C-20(v,00021vICHtr(A)0002)+(5.22)where --20(v, A) = i + I vIC20 (A) + 0(102), andC20(A) = (1/P1w1) [A(a2 —^sfin(v)(b2 — iAbit) -I- (C4 — iACH)r(A)].Chapter 5. Existence of invariant tori^ 89Due to the Floquet change of coordinates (5.20), the entries of the matrix .4 are indepen-dent of O. Moreover, its zero entries are clue to the reflection symmetry. The eigenvaluesof A are Floquet exponents for the STI70 solution. Since the entries of the Jacobian ma-trix A are analytic in v and its eigenvalues are simple for small uI, by [23] the Floquetexponents of SW0 solutions depend analytically in v, for small Iv' and have the formv), 27k0(A, v), 27r7y0(A, v),wherev) = C'20(11, A) + 0(11112),21vICIRrd(A) 70(A, v) = ^ + 0(1v12) > —sgn(v)binlain.(5.23)Floquet exponents for the SW, solutions have the form27(A,v), 27/1,(A,v) 271---y",(A,v),where^ti,„(A,v) =^+ IviQ,(A) + 0(11)12),^ (5.24)21v1C5Rq,.(A)^v) = ^ + 0(102),P2w2C27,(A) = (1 /P2w2) [A(al — iAa21) + sgu(v)(1)1 — izA1)21) + (C2 — iLS,C5i)ii(A)] ,a9RA sgn(v)b911r,i(A) =C5 RA > —sgn(v)b911/a9R.If I, is fixed so that(a9Rb1n— al itb9n^( 1)-')9 \li + 0 ^> 0,al Ra9nChapter 5. Existence of invariant tori^ 90which corresponds to V < 0, then the Stilo solutions bifurcate before the SW„ solutionsas it increases. Near the bifurcation of the SW0 solutions, ro(A) is small and therefore,R(Ro(A, v) < 0 for small 14 and all three nontrivial Floquet exponents of the SW0solutions have negative real parts, implying that the solutions are stable. Near thebifurcation of the SW,, solutions where r,1(A) is small, we have (M,-(v) > 0 for small v,implying that the SW,, solutions are .unstable. On the other hand, if v > 0, the roles ofthe SW„ and SWo solutions are interchanged. We summarize the results of this sectionin the following lemma:Lemma 5.1 Assume (tin, i = 1,2, and Cl,?, Cs!? in (5.1) are not equal to zero. Thenthere are primary Hopf bifurcations of symmetric periodic solutions of (5.1) from thetrivial solution, for parameter values (it, ii) belonging to the curves= i.to(v) = —vbin/aill 0(11112), (5.25)F2 it = 11,(i)) = —11b7R1 am+ 0(11112),near the origin in R2. We denote the periodic solutions in the two branches by SW0and SW,, respectively. If ail? > 0, i = 1,2, Cin < 0 and C511 < 0 (in our applicationthese conditions are satisfied) then for fixed sufficiently small v, both solutions bifurcatesupercritically as it increases and the solutions that bifurcate at the lower value of it arestable, while the other solutions are unstable.5.3 Bifurcation of invariant toriIn this section, we show that for parameter values (ft, v) belonging to one of two curvesA1 or A2, one of the SW solutions for (5.1) has pure imaginary Floquet exponents.We then prove that parameter values along these curves correspond to secondary torusbifurcations from one of the SW solutions, which implies the existence of invariant torifor (5.10), for (it, 1)) near A1 or A2.Chapter 5. Existence of invariant tori^ 91We first consider secondary bifurcations from the ST/I70 solutions. We writeko(A,v) = i + iviko(A,v),and notice that R[ico(sgn(v)A;, 0)] = 0, where Ao* is given by (5.7). The implicit functiontheorem now implies the existence of a unique smooth curve^A = A0*(v), 897/(v) = slin(AO + birdain),^(5.26)such that R[A:0(-A-;(v), v)] 0, with 5■(v) = sgn(v)A; + 0(Izi1), v sufficiently close to0 . Thus for parameters A, v along À.1, the SW0 solutions have conjugate pairs of pureimaginary Floquet exponents. In terms of original parameters of (5.1), Ã1 correspondsto the smooth curve^: p = P0*(v) = Iv1;\0*(v) = vA0* +^, sgn(v) = sgn(A0* + bin/am),^(5.27)such that ntko(ito*(v), v)]^0Similarly, for the SITI, solutions we writev) =^v),and note that R[ii-,,(sgii(v)A;, 0)] = 0, where A; is given by (5.8), and the implicit functiontheorem implies the existence of a unique smooth curveA9: A = -k,r(v), sgn(v) = syn(A; + b9 09R),^(5.28)such that R[k.„.(",;(v), v)]^0, with ;\;(v) = sgn(v)A; 0(14. In terms of the parame-ters of (5.1), A9 corresponds to the smooth curve-^A2 : p = ftv) = IvIA;(v) = iiA + 0(Iv12), sgn(v) = sgn(A; + b211/a211),^(5.29)such that nrkir(it;(v), v)]^0. We summarize the above arguments in the followinglemma:Chapter 5. Existence of invariant tori^ 92Lemma 5.2 There exist smooth curves A1, "2 in the (ii, v) parameter plane, along whichthe SW0 and STIC solutions of (5.1) have conjugate pairs of pure imaginary Floquetexponents.To consider the behavior of the system near the SW0 solutions for parameter valuesnear A1 we letA = .A0*(v) S, syn(v) = syn(A0* biRlain)in (5.21) to obtaindwdedydeK,(6, v)to + jvIVV(w, , y , 6) +^, , y , , 6, v),=^v):t/^1111Y(w, 113, y,^+ Iv12SAw,^y, 0^,(5.30)wherek (6, v) = !Via (8, V) + i[A, + 1V113 (6 , 14], (5.31)(5.32)a(0, v)^-. . 0,C11a91 — C411a1 II ^Oa/06(0, 0) = (5.33)CI BPI wi^,/3(0,0) = (1/Piwi) [Q0 + Ao*(a2/ — Artii)],^(5.34)2Ci R(To*)2 (5.35)7(0, 0) == rO(PAO*IC4::1 (5.36)ro= —(b2/ — Abu) -I- (au — ACH)(e6)2,^(5.37)Qo (2p1',.//,/v) [c4^iAc:11] + tpvlilwur^W (w , ID, y, 6) =^ : [C5 — jAC21], (5.38)+ iSiilliiwi),^(5.39)C+2/Off((111:1jUI!21V + 161111)1312^^Y(w, 'ffi, y, 6) ,^+ 0(11112 + Iellwl +1(5114),^(5.40)PiLoi);‘)(/v, 11). Y , 0 , 6, v) = 0 (iYi2 + 11111 id + lw12),^ (5.41)(5.42)5)(w,17', y, 0 , 5, 0 = ° (iYi2 + Wiwi + I wi2 ),Chapter 5. Existence of invariant tori^ 93and Vi1, 5) are both periodic in 0 with period 27r.Since a(0, v) = 0 and -y(0, v) < 0 for jvj sufficiently small, we can we apply aninvariant manifold theorem for periodically forced systems [18, Theorem VII.2.1] to obtainan attracting center manifold, which can be represented as a smooth functiony = h(w,defined in a neighborhood of to = 0, 6 = 0, withOh^Oh,^Ohh(0, 0 , 0 , 0, v) = 0, —(0,0,0,0, v)^—(0,0, 0,0, v)^—(0,0, 0 , 0, v) = 0,Owfor all 0 E S1 and all sufficiently small H. To find h, we use the fact that the centermanifold is invariant, which implies that h = h(w, ft), 0, 6, v) satisfiesOwOh {IvIN(6, v)w +^,^h, 6) + Iv12)!1). (w , t, h, 0,6, v)}{wo, otp +^11,6) + 1v12);11- (w t, Ii , 0,6, v)}ti)Ivi-y(5, v)h +^, h, 6) + IvI2S)(w, , tD, 11, 0,6, v),^(5.43)Substituting the Taylor series expansiony = h(w, zi), 0, 6, v) = all (0, v)w2 + a12(0. vga..712 + 0,13(0 , v)1D2 + O(11013 + 161111/1)into (5.43), we identify the coefficients of w2, '1.2 and jw12 in both sides of equation (5.43)and obtain^aii(0,v) = 0(10, ai3 = 0(ju1), a12(0, xi) = —C2R/(2Ciro-) + 0(10),^(5.44)We then obtain the following periodically forced equation in the complex plane whichrepresents the flow restricted to the (attracting) invariant center manifold:dw—(10 = (z(6, v)w + IOW ( w, t , 6) +^w, ti), 0,6, v),^(5.45)Chapter 5. Existence of invariant tori^ 94where(W, 6) = "11112 [EC5 - jAC2n^jAC1/1]^0(12V15 + 16114)P1W11;%,- 4(W, IMO, 6,^= 0(1W12),and 1;1,- 1 is periodic in 0 with period 27r We write (5.45) in polar coordinates w = pei"cb,and then apply the theory of normal forms for periodically forced systems [1, section 26]:if qlv10(0, V) is not an integer for q = 1, 2, 3,4 or 5 (this is satisfied for all sufficientlysmall IIJI) then we may change coordinates to put (5.45) into the normal formwheredpdOdi.1)dO= IiiIai (v)6 P + vIA1 (v) P3 +^, 0 ,=^+ Iv1130(v) +^01 046 + 1v1131(14 P2 +^-8(1), , 0 , 6, v)(5.46)Oa^C1/10,91? - C4 04—06(0,v) = ^i3(ii) =^013/ 3(0 , v) , /31(v) = —06 (0, v),C511C111 C211C 4 I? C 1 11,^+ 0 (1111) ,Ai (V)C RC51 C2Ra I^P2W2(C111C21 C211C11) 81(11) =^ ± 0^,C111 P1W1C 1 rtA(P,O,S,v) = (AP5 + i61P3 +1612 9),8(P,11,,6,v) = 0(P4 + l6IP2 + 1612),and A, B are periodic in both 0 and V.), with periods 27r. In our application (11(0 < 0 andA1 (v) > 0 for all sufficiently small V . We now state and prove a result on the existenceof bifurcating tori for parameter values near the curve Al.Lemma 5.3 Suppose A, B in (5.46) are CI functions with A1 (v)^0 and o'i(v)^0.Then there exist vo > 0, 60 > 0 such that for all^0 < Iv' < vo, sgn(v) =^bin/Gill)a l (V) 11Chapter 5. Existence of invariant tori^ 95and all^0 < 1(51 < 60, s gn((5)^—syn(cri(0)/A1(0)),the system (5.46) possesses a C1 invariant torus p = p* (0, 0, (5, v). Moreover, the torusis attracting for (5.46) if cri (0)Ai (0) >0 and is of saddle type if ai (0)Ai (0) <0.Proof: The proof of existence of p* follows from [5, Lemma 12.6.1], all the conditions ofthe lemma being obviously satisfied for sufficiently small Iv' and 161. Now we follow [5,Lemma 12.5.2], adopting its notation, to find an estimate on 60. Since (5.46) is not in aform that we can directly apply this lemma., we first use the standard resealing(.111((liv);5) 1/2P = ^( 1+ 1(511/2/i) (5.47)Then (5.46) becomes^= —2ai(v)1v1(5/3 + (531v170,0,3,v,(5),^(5.48)= A + 1v1 /30^+ (51v1 (0, 0,^v, (5),e = 1,where 7Z, are periodic in 1/., and 0 with period 27r and are bounded, together with theirderivatives, as 161^0 and 1v1 --+ 0. We note that kP has the form^( )] ^0(1(511/2).19T(0,, v, (5) = [31(0^(001 ^+Now we let^, (0,9)T,^v, 6)7',and write (5.48) in the form= A(;\)j) + R(O, /3, 50,^ (5.49)= w(50 +w(A) = ( A + ivii30(v) 61vIT(0, 0,^v, 6)()(1/), 9, [5, v, 6) =1Chapter 5. Existence of invariant tori^ 96where A(A) = 2vai(v)(5 andAlthough [5, Lemma 12.5.2) does not actually apply to (5.49), mainly due to thedependence of A(A) on A, we can use the same proof with minor changes. We areinterested in integral manifolds in the form j f (O, A), with f being periodic in eachcomponent of O. Following [5], for each integer p > 1 we putFp =^E CP(R2 , R) : f(0) is 27-periodic in each components ofWith the CP-topology, Fp is a Banach space. For any f E Fp, we now consider= A(A)i)+Q(o,A,f),=^+ P(O, A, f).whereQ(O,A, f) =^f(9),), and P(O,A,f) = e(O, f(0),;\).For a fixed 0" E R2, let O*(t, 0-, A, f) denote the solution of= w(s) +^f)with initial condition O*(0, 0, A, f) = 0. Now assuming that A(A) > 0, we defineK(u, A) = 0^for n < 0,= —c-1(5')u for U > 0,and form the mapping(5.50)T(A, f)(0) = —^K(u,A)Q(O*(u, 0, A, f), A, j)du.^(5.51)andOP ( , A, f)ao = 0 (v11.513/2) • (61112 + Ifl ).7(A, f) = sup6Chapter 5. Existence of invariant tori^ 97We note thatIK(u, A)I <e' ^V u E R,wherea(v) = 12 (r i (v ) v(5 1 = 0 011 1 16D,Thus, if we choose a bounded neighborhood V of 0 in F1 and choose Iv1,1(51 sufficientlysmall so that 7(A, f) < (4), then by proof of of [5, Lemma 12.5.2] we have^T (A, •) : V^F1.In particular we may takeV = ff E^:^< 11.Also by of [5, Lemma 12.5.2], (5.48) has an integral manifold of the form ji = j(O, A) ifand only if T has a fixed point in F1. We prove the existence of such a fixed point, byproving that T is a contraction mapping. By the mean value theorem we haveIT(, 1)(c7)) — T(A‘, f)()1 < sup ID fT(5k, f)((i))11.7 - f11,where the supremum is taken over all 0 E 112 , f E F1,1 f < 1. ButD fT(‘, f)() = —f K(1.1,5) [D(7°*("),, f)^ac2 (O. (11), f) Oi)*(u) ^ ^0'0^Of duwhere ê*(u) =^(7), A‘, f). In our problem0(20* (11),A, f)Q•^(I)* (11)^f) = o or/ 214, = 006131214f^ OOand< Cci(54)", where -5'(;\, f) = 000161312), Chapter 5. Existence of invariant tori^ 98and therefore1-Dif (T(5%, f)(0)1 < IvI3/2k L°O e-fa(j)jud",for some real constant k. Since a(;) = 0(1v1161), we have 1Df(T(A, f)()1 = 0(1611/2)uniformly for all ch E R2, f e F1 with fI < 1. Similarly, ID(T(5%, f)()1 = 0(1611/2),hence there is a positive constant C such thatf)—N, f)1 C61/21i - ffor all sufficiently small 1v1 and 1(51, and for all 1/11,1111 < 1. Also we observe that= 0(1S1112),^1-Dc-b(T(1, f )()I = 0(151112),and thereforeIT(5‘,/)11 < C1611/2for all sufficiently small 1v1 and 161. Thus for all sufficiently small 1v1 and 161,^•) is acontraction mapping on the closed unit ball in F1. Q.E.D.Remark 5.1 a. With more lengthy estimates, one can prove that T(5%,•) maps the unit ballin Fp into itself for p > 2 by showing that f)lp = 0(161112) and therefore showingthat the bifurcating tori is C. However, the region of parameter values for which CP toriexist may shrink as p increases (see [5, p.492]).b: In the above proof, based on our numerical results in §4.3, we have assumed thatA(5t) > 0. However the proof easily can be modified if A(;) < 0. In that case we simplyredefineK(u, in the obvious way, and the rest of the proof will be the same.We observe that v0,60 in Lemma 5.3 can be chosen independently of each other.In terms of the system (5.1), this implies that the bifurcating invariant tori exist forparameters (it, ti) in a thin wedge-shaped region of width 0(1v1) adjacent to the curveAl. In a similar manner, we can show that invariant tori will bifurcate from the SW,solutions along the curve "9. See Figure 5.2.Chapter 5. Existence of invariant tori^ 99Figure 5.2: Parameter values (shaded regions) for which bifurcating invariant tori existfor (5.1), when CI RC5R — C9Ra41? < 0.5.4 Persistence of invariant toriIn this section we prove the existence of invariant tori, when the higher-order terms arerestored in the full system (5.2) or (5.1), for parameters (it, v) in the wedge-shaped regionbounded by the curves A1 and A2, but sufficiently far from the boundaries. Combiningthis with the results of the previous section, on the bifurcation of tori along the curvesA1 and A2, we thus prove the existence of invariant tori for parameters (it, ti) throughoutthe interior of the wedge bounded by the above curves.It is convenient to use the parameters pin =^i =1,2 introduced in (5.6).In terms of these parameters, the curves A1, A9 (see Figure 5.3) become1‘*1 1121? = 110*(P1R)= C1i111+0(/11n12),A; /1211 = it; (RI It) = MOB + OOP in12),Chapter 5. Existence of invariant tori^ 100P2RFigure 5.3: Curves in the (fil1?,/19W parameter plane corresponding to A1 and A2.where itiR > 0 andC4R^C9 ftC =^B =^.U5RBased on our numerical results of Chapter 4, we assume that CiRC5R- C2RC4R < 0,and thus 0 < 1/B < C. We introduce the scaling= El Jo = EA, r=^= 1,2.for c > 0, then after dropping the hats, (5.2) becomes= crl +^+ c2 R + e20'2 (A + Cie'? + C511T:) 0(c2),(5.52)Ol^ au(b9R— biRA)^b11(o911 — (11R/\)-=^+  ,^ + Cur? + C911-3 + OW),Rb9n — amblR a9Rbi R — a nom= p2w9^e a2/ (b9/? — RA)^b91(0.911 — (IlnA)C4/ 11 + C5/11 ± O(E2).—^ambin— al numChapter 5. Existence of invariant tori^ 101Now we use the coordinate transformation defined by=11(A)^= 7,9.(A)^61/2p2,where and A satisfiesr7(A) =^1 — BA V CIR(BC — 1) =-r(A) = j  A — C V C5R(BC — 1)C5R AC2R C9RC4R C1RC5R'ACiR C4R•C91?CI11 C11105RC511 — AC911, AC11,— C4 R > 0.Note that this is equivalent to 1/B < A < C. We then find that p = (P1, p2), 0 = (01, 02)satisfy an equation of the form= EE(A)p + €312.T(^p, A, c), (5.53)= w(A, e)^E31 201(p A, E) + €202(6,1), A, E),whereE(A) =[^2C2 eV )01'.' ( A)2c1R(ei(A))2(5.54)2C 4 ul'IPOT; ( A)^2C 5 R(IVA))2W 1 + OWc)(Pi=)(5.55)'P2W2 + 0 (f)and T, 02 are 27-periodic in both components of O. Furthermore, for all sufficiently smalla > 0, co > 0 and for fixed q, 0 < q < 1/2, the functions .7., 01 and 02 are continuouslydifferentiable on(01, 02,m ,p9, c) E R2 x Q(a,Eo),whereCl(a, €0) = (P11 P2, A, e) :1(91,p2)1 < a, 1/B + Eq < A < C —^0< e < co}Chapter 5. Existence of invariant tori^ 102Now we use a linear coordinate transformation of the form p = S(A)ji, where S(A) is anonsingular 2 x 2 matrix, that diagonalizes E(A), and we get^P = A(;\■)p^ (5.56)0.0) + oco , ,3,50where A = (A, €),( al (A)^0 )A(A) = E^ 5 al (A) <0 < Et2(A),0^a2(A)and, i5, 5t)^c312S(A)-1.F(0, S(A)/5, ), c),0(0, P,;\)^E3/201(S(A)/),A,E)+ E202(S(A)j),A,c).We observe that (5.56) has the same form as (5.49) (the dimension of /3 is different), andwe can use the same method used in Lemma 5.3 to prove the existence of invariant tori.We construct the 2 x 2 matrixK(v)^diag(e-Eal (4", 0) for it < 0diag(0,(-'2(A)") for u > 0,then form the mappingT(, f)() =^10C K(11, A)(2(0* (a, c , A, f),^f)dit.^(5.57)- 00for f E F1,q; E R2, where Q and 0* have the same meaning as in the proof of Lemma5.3. In the present case, we have-y(A, f) = sup0OP (0,^f)00=^( €3 / 2 ) • ( E 1 / 2 + 1.111), Chapter 5. Existence of invariant tori^ 103where P has the same meaning as in the proof of Lemma 5.3. To obtain useful estimatesfor T we estimate the eigenvalues of al (A), co(A) of E(A). Near a boundary of the wedge,one of the eigenvalues approaches zero, so first we put A = C — Eq, and after somesimplification getcri(A) = eq /2+ 0(1€129,a2(A) = —2 + 0(16129,^ (5.58)If we let A = 1/B+ Eq, we get the similar result, and thus for all A, C+ < A < 1/B — eq,there are positive constants 0, k such thatU Efor all sufficiently small e. Since 0 < q < 1/2, we choose a bounded neighborhood V of 0in F1 and choose lAt, Id sufficiently small so that7 (;\,^< kel+q,then by proof of Lemma 12.5.2 of [5] we again haveTCA',•) :^F1,We prove the existence of invariant tori, by proving that T is a. contraction mapping onthe close unit ball in F1. In a. way similar to how we obtained the analogous estimatesin the proof of Lemma 5.3, we obtainIDAT (A , f) l i = o (E 1/2 -" )^ITC\ ,^, = o (E1/ 2-0for all f EF with fI < 1, and for all A with C^< A < 1/B — Eq. Thus for allsufficiently small E, T is a. contraction mapping on the closed unit ball in F1, and we haveChapter 5. Existence of invariant tori^ 104'LFigure 5.4: Parameter values (shaded region) for which normally hyperbolic invarianttori exist for (5 1) with C C- 1^5R - C21,C411 < 0.Lemma 5.4 For any fixed q , 0 < q < 1/2, system (5.4) possesses Cl invariant tori for11B + eg < < C —^0 < < 112,for all c sufficiently small. The tori are normally hyperbolic, and have the same stabilitytype as the invariant tori T" of the truncated system (5.3).In terms of the parameters of (3.1), Lemma 3.4 implies that the invariant tori for (5.1)exist in a region bounded by curves that are tangent to A1 and A2 at the origin (seeFigure 5.4).Since Lemma 5.3 already implies that tori exist in thin wedge-shaped regions of width0(10) adjacent to A1 and A9 (see Figure 5.2), the parameter regions corresponding tothe two lemmas overlap in a neighborhood of the origin:Theorem 5.1 System (5.1) possesses invariant tori (denoted T1) for the parameter val-ues (it, v) throughout the interior of the wedge of the parameter plane bounded by thecurves A1 and A9, and the tori have the same stability type as the invariant tori T" ofthe truncated system (5.3).Chapter 5. Existence of invariant tori^ 105We summarize the results of this chapter in Figure 5.5, showing the schematic bifur-cation set for (5.1). Corresponding to one-parameter paths with fixed v and increasingp, are diagrams shown in Figure 5.6. We note for p sufficiently large there exists thephenomenon of bistability: both SW0 and SW, solutions are asymptotically stable, andthe behavior of a typical solution (transient) as t oo is determined by its initial condi-tion. The boundary between the basins of attraction for the two SW solutions containsthe invariant torus T1 and its stable manifold.We briefly mention some implications of the result of this chapter for the originalmagnetoconvection problem. For a fixed a,(, (2 and odd integer in, for L < Lin, andsufficiently close to L„, (v < 0 and sufficiently close to 0: see Figure 5.6(a)), as weincrease the Rayleigh number R through Rm(L), stable standing wave solutions SW0(corresponding to odd mode solutions) bifurcate from the motionless solution and an oddnumber of time-periodic rolls will be observed in the fluid. As we increase the Rayleighnumber further through R7+1 (L), a branch of unstable SW, solutions (corresponding toeven mode solutions) will bifurcate from the motionless solution. Increasing the Rayleighnumbers still further, the competition between odd and even modes produces a branch ofinvariant tori T1 (typically corresponding to quasiperiodic or weakly resonant solutions)which bifurcate from the branch of even mode SW, standing wave solutions, and coexistswith the branches of stable SW0 and SW, solutions. For L = L„, (v = 0) and L >L„, (v > 0), we have similar interpretations of the bifurcation structure. See Figure 5.7.Stable SII0solutionsChapter 5. Existence of invariant tori^ 106A2Stable SW0, SW„,^Stable SW0 solutions Unstable Ti^Stable SW„. solutions^Unstable SW, solutions solutions^Unstable SW0 solutionsF1Stable SIT17, solutionsF2Figure 5.5: Schematic bifurcation set for (5.1), with CI110511 — C9RC4R < 0. (Comparewith Figure 5.1(b)).Chapter 5. Existence of invariant tori^ 107(a)(b)(c )Figure 5.6: Bifurcation diagrams for (5.1), for fixed v and increasing p: (a) v < 0;(b) v = 0; (c) v> 0.Chapter 5. Existence of invariant tori^ 108(a)(b)Figure 5.7: (a) Typical region (inside shaded circle) in (R, L) parameter plane for whichresults of this chapter apply to the magnetoconvection equations; (b) Magnification ofthe region in (a), showing parameter values (darker shaded region) corresponding toinvariant tori and bistability of standing wave solutions.Chapter 6Secondary Bogdanov-Takens bifurcationsThe results of the previous chapter give rigorous results on the dynamics of small-amplitude oscillatory magnetoconvection, when L is sufficiently close to Lin and R issufficiently close to R„-„, so that lajRp binvi < Piwi — P9w91, j = 1, 2. However,as we discussed in §3.5, for small large Q and large L, the quantity JPiwi — P2w2I isextremely small, so the rigorous results can be expected to be valid only for a small rangeof parameter values. To get a more complete picture of the dynamics of our problem fora wider range of parameters values in the Case II limit, in this chapter we will analyzethe alternate normal form (3.79), which we rewrite here as:4 =^[iPi^+ ci1 t bi v + CilZ112 + C21Z212] + C3Zi^+ h.o.t.^(6.1)4 = z2[iP2w2 + a2it + b2v + C4IZil2 + C5IZ212] + C6Z24 h.o.t.,where h.o.t. are 0(lp, v121ZI + IP, v11Z13 +1Z15) and ai , b1, a2, b9, C1, •.., C6 are the samecomplex numbers as in (3.73).In §6.1 we consider some coordinate transformations of (3.79) which make the analysisof the normal form easier. Then in §6.2 we consider a three dimensional reduced systemobtained from the third order truncation of the normal form (6.1) and find bifurcationparameters p, v for which the linearized vector field about its nontrivial fixed points havedouble zero eigenvalues. These nontrivial fixed points for the reduced system correspond109Chapter 6. Secondary Bogdanov-Takens bifurcations^ 110to the periodic orbits SW0 and SW, of the full system. In §6.3 we consider the Bogdanov-Takens singularities which correspond to the double zero eigenvalue, and their unfoldingsfor the reduced system. In this way we predict the existence of secondary and tertiarypitchfork, Hopf and global bifurcations from the SW solutions. In §6.4 we consider thereduced system as a small perturbation of a similar reduced system coming from a Hopfbifurcation with D4 symmetry that wa.s considered before by Swift [41].6.1 Preliminary coordinate transformationsTo study the bifurcation of solutions in (6.1) we would like to decouple the average phasefrom the radial direction. The most straightforward way to accomplish this (up to finiteorder) is to write the complex amplitudes in terms of polar coordinates Z1 = ri el , Z2 =r2e2e2 as we did in Chapter 5. Then the average phase (01 + 09)12 does not appear in theequations for 7'1,79, or (01 — 09) up to cubic order. But coupling of the phase difference(01 — 02) with 7-1 and 7-9 in the system makes calculations somewhat awkward. We willinstead use a different coordinate transformation which is easier to work with. Likethe polar coordinates, the phase angle of the S' symmetry in the normal form will bedecoupled from the other variables up to finite order in this coordinate system.We use the coordinate transformation [41]U iI7 =2Z1Z2, TV =1 Z1 12 -1 Z2 1214 — 4 IZ? - Z11 (6.2)Although this coordinate transformation is singular at origin, we will use it away fromthe origin. System (6.1) in this new coordinate system becomes:= U[61?it -1-611 r(B41? B5I? B61?)I2} - VP + ?WI +1)1111^(6.3)-V 1[B - B2I B;31]/2 -^[B ± B21 + B01]/2+ UTII/34n — B5R± B:ig]12 ±U(U, V, IV, Ili, it, v),Chapter 6. Secondary Bogclanov-Takens bifurcations^ 111= U[„,1:1 -1-, alp biv r(Bii — B21 -I-, B31)/2} VVeRit+Vr(BzIR B51? B611}/2 U1TT[B11 + B21 — B61]12V117[B411 — B511 B311]12 V(U,V, W, T, p, v),W[ãmp+^+ T(C111 C511)] r[aRit + Ril]+ UV B61i-U2 (Bin — B211 B3RV2 ± V2 (B111 — B211 B311)12+ W2 (C111 — C51?) + W(U, V, TiV, it, v),tif =^+ P2w9 +where r = .Vu2 + 172 + 1172, andU, V,^= O(7-3 + I p, 1/.2 + p, v 12r)are 2r-periodic in tJI , and= C1— C1, B9 = C5 — C2, B3 = C3 — C6,B4 = C1 + C4, B5 = C5 + C2, B6 = C3 + C61= a — co,^b = b1 — b2,^= P1w —=^a2,^= + b2.Due to the symmetry (3.55) of system (3.79), (6.3) is equivariant under (U, V, W,(—U, —V, W, IF), therefore(—U, V, W,^v) = —U (U, ^, ji , v),^(6.4)V (—U, —V, W, , p, v) = —V (U, V, IV, , p, v),W (—U, —17, W, 'IP, p. v) = W (U, V, W, , p, v),and this implies that the set U = V = 0 is an invariant subset for (6.3). This invariantsubset corresponds to the invariant subspaces Vo (if IV > 0) and^(if IV < 0 ) inChapter 6. Secondary Bogdanov-Takens biThrcations^ 112the (Z1, Z2) coordinates. NW can find some information on the dynamics of (6.3) byrestricting to invariant subset U = V = 0. Then117 = W[aRit + bRv + 'WI (Cm + C511)] + 1WP111t + Lllvi^(6.5)+ II72(CI11 — C511) W(0, 0, TV, T),= Picui + P2w2 + 0(IW)•Since kif > 0 for WI sufficiently small, we can reparametrize (6.5) by and obtaindW= [Piw + P24-1 {WV" Rit +^+ W (Ci^C511) IWI(C111 C511)]+ 11471[a11dt + 6riv] +^kli), (6.6)where WI T, t , v) is periodic in "P with period 27r and is 0(114713 + t , v111412 -I-Ip, vJ2 IWO. Now the Hopf bifurcation theorem implies that (6.6) will have two differentperiodic orbitsWo(ft,v)+ 0(lit, v12),^(6.7)1/i7,(T,/t,v) = 1/17,(t1,v) + 0(1 ii, (6.8)whereal + v)^ain't + nv > 0,C1Rb9ilv1.1) 0.9 nit^b9^> 0.C5R(6.9)Solutions 1470(4J, ft, 14 and^p,^are the periodic orbits SW0 and ST/17, of Chapter5, which bifurcate from the trivial solution along the curvesF1: ^= -0111/0,0v+ 0(102),^(6.10)F2 :^= —(1)211 / a2R)1-1 + 0(1142),respectively.Chapter 6. Secondary Bogdanov-Takens bifurcations^ 113To study secondary bifurcations from the SW0 solutions we use a moving coordinatetransformation= U, V= V, W= W —^v).After dropping the hats, (6.3) becomes^U[a2Rit + bye/ (C4 R C 6 OW01 - [(2 + alit+ biv]^(6.11)— V Wo [Bi + C611 — VW[Bu C61] UW[B411 + C6R] + (U, V, W, , ft, v)=^+ a1t + blv + W0 (B1, — C61)] + V[amit + b2rty + Wo(C4R — C6R)]+UW[Bli — C61] + I/1T/[13411 — Cm] + 1)(U, V, W, P,1i , v)—2W[aiRit + Hp] + UTT[B6d + U2 (B1 R - B2R + B30/2B30/2 + 2W2 C111 Vi)(0, V, W, T, v),+1/2(B111 - B2 RPlW 1 + P2W 9 ± 0(r),where the higher-order terms Cf., :1),.);1) satisfy the symmetry condition (6.2), and are 27-periodic in T.6.2 The reduced systemIf the higher-order terms in (6.11) are ignored then the first three equations decouplefrom the fourth equation, although in general T will appear in the periodic coefficientsof higher-order terms. As an approximation we consider the following reduced system,obtained from a truncation of (6.11) that ignores the higher-order terms -/,-/,1) and 1;1):U^U[a2111u + b2 Iry + (Cm + Q0110] — VP + jt + v}^(6.12)— V Wo[B + C61J — V TIT[B + C11] + UW[B4R + C6r],= UP + äjp + blv+1170(Bil—C61)]+v[andt+b2Rv+wo(c4R—c6R))Chapter 6. Secondary Bogdanov-Takens bill ircations^ 114+UIV[/311 — (761] + VITIB411 — C611],= —2W[ai11it + biRv] + UV[B61] + U2 (B111 — B211+ B311)/2-1-172(B1R - B211 ± B311)/2 2W2C111•The origin in (6.12) now corresponds to the SW0 solution.Remark 6.1 By using a similar moving coordinate transformationU, V = V, Ili = IV — W , p, v),we could consider the dynamics of (6.3) near the SW, solution. Due to the symmetry(6.4) of (6.11), the analysis about the SW,, solution is quite similar to that about theSW0 solution, therefore we only present our analysis of (6.12).The eigenvalues of the linearization of vector field to (6.12) about the origin (i.e.,about the SW0 solution) areTr0,,(1t, v)^Tr20(p, v) — 4D et0,,(p, v)(6.13)2—2(ainp, b^n+ < whereTr0,7r(p, v)^2[a9R11 b911v^W0C411],Det0,,,(p, v) = —10612In + Nit + b ay + B RT'Vo]9+^+ 11111 + 6111+13111V0] ,(6.14)2(6.15)and amp+ b1 11v > 0. We notice that one of the eigenvalues of the linearized vector field,which corresponds to the W direction, is always negative .Chapter 6. Secondary Bogdanov-Takens bifurcationsRemark 6.2 By considering the truncation of (6.3) about the SW, solutionscorresponding expressions for eigenvaluesTr,,o(tt, v)^\ITr 7,2 03(i1, v) — 4D et,,o(ii, v)v)2A13r (it, v) = —2(a9Riu bmv) < 0,115we find the(6.16)wherev) = 2[ai^Rv — Wir Cud/^ (6.17)D et, ,o(it, v) = — I C 312 tiC2 [6, nit + bRy + B2 RTICJ2^(6.18)+ [Co +^biv + B2/1N2and a2Rii b9Rv > 0.We expect pitchfork bifurcations of fixed points of the three-dimensional system (6.12)along one-parameter paths transversal to the curveF3 : Deto,„(it, v) = 0.^ (6.19)Such bifurcations correspond to secondary pitchfork bifurcations of periodic solutionsfrom the periodic SW0 solutions in the four-dimensional normal form. Along one-parameter paths transversal toF4 : Tro,,(tt, v) = 0, Deto,„(iu, v) > 0,^(6.20)we expect Hopf bifurcations of periodic orbits of (6.12) from the origin. These correspondto secondary bifurcations of invariant tori from the SIV0 solutions.To find multiple bifurcation points where F3 and r4 intersect, we look at the signof Deto,„ when Tro,„ = 0. The line it = vAo*, where N; satisfies (3.7), corresponds toTr0,7, = 0. If we put IT70 = Ii'ITV6, we have11a9R — Rb9R,1170* = — sgn ( v)^ 5 sgn(v)^syn(A0*^bill/ain)•u Ram — na R(6.21)Chapter 6. Secondary Bogdanov-Takens bifurcations^ 116Remark 6.3 Based on our numerical results in Chapter 4, in our application sgn(v) =syn(A0* + bin! = 1 when in is odd. To simplify our notation, we continue our analysisassuming that syn(v) = 1; however if syn(v) = —1 we can treat the problem similarly.Therefore for syn(v) = 1 we have1,17* =0and thenb Ram — aiJb9R C Ram — C4 Rai rt' (6.22)Detoor G to* (v),^=^612w0.2 + {co + [à1A* +1)1 +Bi 1147,;`]v} .^(6.23)Then Deto,,(it(v), v) = 0 ifv =v • ^  > 0, j = 1, 2.[±1061 — B1j]1170* — 1)1 — el/A6'In the above equation, and in what will follow throughout1 or 2, with the "plus" sign in front of IC6I correspondingsign corresponds to j = 2. Using (6.22) and (5.7) and after(6.24) in terms of normal form coefficients:al Rain)(.2)( CI Ra2117^ - . (6.25)+IC611^/ r,R. 9/? — Rion] + Ra(Bi b) — R(a‘Li la —Let us denote= v;Ao*, j = 1, 2,^ (6.26)and note thataDet0, , ( 11.7OV =-2 1T0* ( +1 c61 )^o, (6.27)which implies that any point of intersection of the curves T ro,, (ji , v) = 0 and Deto,,(u, v)0 is transversal. To find a condition on the number of solutions for v in (6.25), we consider—M = laiRa(Bib) — b,o(B,n)-cidavtEd 2 (6.28)2(6.24)this chapter, j can be eitherto j = 1, while the "minus"some simplification we write= VChapter 6. Secondary Bogdanov-Takens bifurcations^ 1179+ IC6121^h bLai R- - -R-1 RI-(c/102 [iC6121;2R — [(1316)]2 +^[Ic6124 - [-s(Bia)]2]^(6.29)—2ambiR[IC612e1R6R — cs'(131-0(Bi-e71)] — (CiR)2[Wk2—2Cida(â-g)[ai0(B:76) — biO(Biet)]•There will be one solution /4 if and only if M > 0. On the other hand Deto,,,(14(v), = 0will have two solutions v = //` if— [1061 + B11]IV0* —^— a/A0* > 0,^(6.30)or equivalently, ifIC61 <^— O(B15) — CI(a) (6.31)al rtbR — 1)1 ORNow we consider the curve Det0,7, = 0 in general, after substituting for 1/170 from (6.9)into (6.15). We haveDet0,,(it,v) = D1t2 + Fv2 + 2E itv + G1i. + Hi' +^(6.32)whereD =E =FG =H =(al R/C102 [1B112 — Ic612] + 11/12 - 2(aut/Ci R)R(Bi^(6.33)(i( r,,IRbi 2 R)[1-8112 — IC611 + RA)LAI?—(biR/CiR)(B1a)) — (al 11/C11R(B1 6)),(biR/Cilt)2 {iBi 12 - 106121 ± 1612 - 2(biRICin)R(B16),—2C4(aiRICiR)Bil — ad,—1)1].The graph of Det0 = 0 is a conic section in It and v, the type of this conic sectionbeing determined by the sign of E2 — DF; it will be an hyperbola, parabola or ellipse ifChapter 6. Secondary Boglanov-Takens bifurcations^ 118E2 — DF is positive, zero or negative respectively. After some simplification we get(E2 — DF)(C1n)2 =or in another form10612 {ai2R.ILI2^b?R1et12 — 2a4RbiniR(etb)]- 2— [ains(Bib) — bida(B4) — C4R(etb)]2[(tiR(Bib) — biO(Bilft) — O(âb)1+ ic6i2 [a R 1 61 —^2(C R)2 (E2 — D F)^+ [1061(1, R — aibin)]2.(6.34)(6.35)^This implies that in the region in (it,^plane for which there is only one positive rootfor (6.23), the graph of Deto, = 0 will be a hyperbola.Since we are interested in the behavior of the system for large in, let us denote c = m-1as in Chapters 2 and 4. We recall some of our asymptotic results on the normal formcoefficients. For fixed a, ( and Q (i.e., Case I) we have, as 771 -4 00(2722^P1W1^E +(EP2W2 =  ^),^ (6.36)= .4+B+0(c),^C5=A+B+0(e),^(6.37)^C2 = + OW,^C4 = A + 0(e),C3^C OW,^C6 = C 0(0,= E2b + o(E3),^= 0(e),e2bn/2+ 0(e3), NI? = —e2b11/2 + 0(e3),B + OW,B + OW,where Pi, , j = 1,2 and 1),w are as in Chapter 2, and Ci (j^1, ..., 6), A, B,C,a2, b1, b9 are as in Chapter 4, and a, b, are as in equation (6.3). Now we consider thebiR =B1 =B2 =Chapter 6. Secondary Bogdanov-Takens bifurcations^ 119c-dependence of the terms involved in Deto,, and Tro,,. From (6.36) we observe that theslopes of the curves F1, r9 and r4 near origin are 0(e2). This implies that in Case I for afixed a, ( and Q and for sufficiently large m, Deto,,(p0*(v),v) cannot be zero for small pand v. For example, for in = 107,Q = 10072, = 1, and ( = .1, there is one intersectionpoint, but with the value of v 9519. In fact we haveDeto,,0(m),=^0(E3).(6.38)However, if we consider a. decreasing sequence in ( and an increasing sequence in Qas in our Case II for large aspect ratios that was discussed in Chapters 3 and 4, we willget different results. Recall that for fixed (' and Q, we put= Q = ck/2Q, (6.39)where 0 < k < 2 . Under the scaling (6.39) we have checked numerically the conditionson the number of intersection points of the curves Tro,, = 0 and Deto,„ = 0, for differentparameters of the original magnetoconvection problem. We found that it is possible tohave both one or two intersection points. When the graph of F3 is an hyperbola thereis a very small region in parameter values for which there are two intersection points,however the value of v at the second intersection point is large. For fixed a and asthe value of Q decreases, the value of v seems to increase without bound, but as thevalue of Q increases, the values of yr and 14 become smaller and the intersection pointsappear to approach the origin. We did not find parameter values for which there are nointersection points (see Table 6.1).We have sketched the scaled graphs of F1, r9, F3, El for a = 1, = .1, and for= 10072 and 30072, which correspond to cases with one or two intersection points,respectively, in Figures 6.1 and 6.2. The directions of the axes in all the figures in thischapter are as in Figure 6.1.Chapter 6. Secondary Bogdanov-Takens bifurcations^ 120Table 6.1: Values of //I, 4^/dam for a = 1,‘" = .1,771 = 107, and different values ofQ.14 10-4 in2A; 10-4m2(-biR/ aiR)10-472 4.75 - .247965 .247964.0172 .4814 - .2481 .24807T2 .0546 - .2718 .258010072 .804 x 10-3 - 3.9458 1.088720072 .417 x 10-3 - 6.330 1.808220.0479372 .3901 x 10-3 - 6.733 1.945220.0479472 .3901 x 10-3 76667 6.733 1.94530072 .319 x 10-3 .407 x 10-2 8.238 2.47740072 .259 x 10-3 .143 x 10-2 9.997 3.11350072 .241 x 10-3 .127 x 10-2 11.682 3.727100072 .173 x 10-3 .551 x 10-3 19.616 6.57710572 .248 x 10-1 .312 x 10-4 .111 x 104 .312 x 10310872 .101 x 10-5 .104 x 10-5 .475 x 106 .100 x 106To consider the asymptotic behaviour of the location of the intersection points, weput62 + (63 ) = b R(2 A R^R) 62 o(E3),aRBRWO* ^+ 0 (E3) = -3RE 2 + 0(E3),Dil(6.40)where Ao*,1470* are as in (5.7) and (6.21). Recall that we have shown that in Case II, wehavecz) = (272 ek-Fi^0(ek+2) = ek+iwo^0(ek+2).Au)(6.41)For 1 < k < 2 we haveDeto,,(v,e) = e1l/21-1(712N + [-BIII7(;" + b1]21 + 0(€2k+2 e5),^(6.42)and if k = 1 we haveDeto(v,e) =^9 - I tri; + MI/ 1 2Chapter 6. Secondary Bogdanov-Takens bifurcations^ 121Figure 6.1: The curves r1,...,r4 for a = 1, = .1, = 10072,k = 1,m = 107, using thescaling p = 200€2it, v = .01V. Note that vr = .000804. Dotted lines show the parts ofcurves Tro,, = 0, Tr,,0 = 0 for which their corresponding values for Deto,, and Det,,c,are negative.Chapter 6. Secondary Bogdanov-Takens bifurcations^ 122Figure 6.2: The curves F1, r, for a = 1, = = 30072,k = 1,m = 107, using thescaling it = 400E2 v = .01/i. Note that vi* = .000319, v.; = .00407. Dashed lines showthe parts of graph Tro,, = 0, Tr,,0 = 0 for which their corresponding value for Deto,,and Det,,0 are negative.(+IC I — B1)1176* — b1—Bllwob R ICI± 0(c).^ (6.45)Chapter 6. Secondary Bogdanov-Takens bifurcations^ 123-Elci2(ovIn2 + 0(c5).^ (6.43)The implicit function theorem implies that D eto,,^v, c) along the line Tro,„(it, v, c) = 0is zero ifV = (E)^± 0 (E),3 (6.44)wherewo6.3 Bogdanov-Takens bifurcations in the truncated systemIn the previous section we showed that the linearization of vector field of (6.12) about(U, V, T47)^(0, 0, 0) (the SW0 solutions ) has double zero eigenvalues when (it, v) =v;), i.e., where the curves r3 (Deto,, = 0) and F4 (Tro,, = 0) intersect. Similarresults hold for the SW, solution. The parameter values (pi, j ) correspond to Bogdanov-Takens singularities in (6.12). In this section we unfold the singularities, and analyze thenon-linear dynamics of (6.16) for (it, v) near (it;, v;).We first use the coordinate/ x \transformation0^1^0 \ / U \= 2j— ^—1j^s, (6.46)where\ Z 0^0^1 \^T'17=^(amp + bmv)^W0[C411 — Col],^(6.47)A2j^(;)+ ahtt+b,v +Wo[Bil — Cod. (6.48)Chapter 6. Secondary Bogda,nov-Takens bifurcations^ 124Under this transformation (6.12) becomes1 0 1 0 / x / xo(x,y,w)1.7 —Detoor0Tro,,00—2itin Z jY°(X,Y,I17)zo(X,Y,w,)(6.49)wherex0(x, 37, z)^Z ^j-Y +^- C611 + .X.[BLIR CM]A2jy°(x-,Y,z,)^z (—Aux +11[B4R ±C6R) ± j(B11 C61] ZN2j-X[A2j(C61 + B11) ± A 1i(C611 — B4a)ll ,Z°(X,Y, Z) = X (  (-AUX + 37)B6/  )+ (-AO: + 37)21rLSr,A2j^:IR - B2R + B3R1I2A2j± X2[B11? - B21? - B3R1I2 + 2C1RZ2 .When Tro,,r = 0, Deto,, = 0, the linear part of the vector field for (6.49) has doublezero eigenvalue, while the third eigenvalue —2As! where= a nit;^b1 11v ,^ (6.50)is negative. By the center manifold theorem, there exists an attracting center manifoldrepresented by a smooth surface Z = h(X,Y,p, v) for X, Y sufficiently small and p, vclose to p, v. Moreover, the reflection symmetry (6.4) implies that h can be chosen sothat11( —X, IT, p, v) = h(X, 17, v).This center manifold can be represented by its Taylor series to any finite order, and tothe lowest order it will be in the formh(X,Y, [1'1,14) = euX2 + c9XY + c3.11/2 + 0(IX, Y14).^(6.51)Chapter 6. Secondary Bogdanov-Takens bifurcations^ 125To calculate eu, e2, e3.; we substitute h = h(X, Y,^v;) given by (6.51) intoeu =[B1R — B2R - B3R]/2(6.53)2As1eu + B61^Au (B111 — B211 B3R) (6.54)ezi = .\;!^26.2jA; 2A2.A'!23 3B1R B2R B3R ^i = (6.55)3 2AL49A';Then the dynamics of (6.49) restricted to the attracting center manifold when t =1/ is represented by3=^g1x3 92ix2y g3 jxy2 g4 Y3,^ (6.56)= g5i X3 + 96i X217 + j X y 2 + gsi 37 3where we have ignored the higher order terms. Using normal form theory, we can removesix of the eight nonlinear terms at cubic order. After transformation, (6.56) in normalform can be taken as= Y,^ (6.57)= crix3^j 9wherej = 95j,^ (6.58)(2euX + e2117)(Y + X°(X, Y, h)) + (e2iX + 2e3iY)Y°(X, Y, h) =—2A; (euX2 + e2jXY + e3jY2) + Z° (X, Y, h,^ (6.52)Then by equating coefficients of powers of X and Y in both sides of (6.52), we get—(,13/A23) [B61] + (Au / A2)2[B — B211 B311]/2 13; = YGj+391 j.Chapter 6. Secondary Bogdanov-Takens bifurcations^ 126The dynamics of this normal form has been discussed in [16]. Allowing it, v to vary nearp; and VI, we obtain the unfoldingy,^ (6.59)^=^+ -y2y + aix3 + OjX2Y,where -yi = —Deto,, 'y2 = Tro,,. When (p,v) = (p, v;), the system (6.59) has aBogdanov-Takens singularity at the origin and undergoes a codimension two bifurca-tion in a small neighbourhood of the origin, for it, v close to i.e., (-y1,72) close to(0,0). See Figures 6.3 and 6.4. The shaded circular and square regions in these figurescorrespond to two different case of bifurcations.System (6.59) can further be simplified. Using the scalingsx = - (Vm ) 17, y = (l13/2I O) M t = — (iilaji) f,71 = (Ctj/13.02;-Y17 72 = — ( ICU.i Pi) )"42,^ (6.60)then dropping the "bars", (6.59) becomes= y,^ (6.61)y = 71x + 72y + syn(cti)x3 — x2y,Depending on the sign of^cei = —a1 {2Ai.j[C6R]^A2,j [B11^C61j^(z:i,j/A2,j)1B11 — C611} ,^(6.62)for each j we will have two distinct cases (up to time reversal). In the above expressionfor cri we assumed the nondegeneracy condition= 3g1j g6j^0, (6.63)Chapter 6. Secondary Bogclanov-Takens bifnrcations^ 127Figure 6.3: Parameters as in Figure 6.1. The circular shaded region shows the parametervalues that correspond to our bifurcation analysis.Chapter 6. Secondary Bogdanov-Takens binircation,s^ 128Tr,,0 =Figure 6.4: Parameters as in Figure 6.2. The circular and square shaded regions showthe parameter values that correspond to our bifurcation analysis.Chapter 6. Secondary Bogdanov-Takens bifurcations^ 129wheregi;^ — Cu] ±[Hart — Cud} •^(6.64).96j^[RIR + CG!?^(1/\2)[B11^C6I]^(e2j/eii)ai.^(6.65)From (6.60) we observe that the sign of i3j determines the orientation of t, and henceaffects stability types. If 3j > 0, there will be time reversal and the sign of 72 will bechanged, however if ,,3j < 0, then system (6.59) and (6.61) have the same dynamics.Also, from (6.62) and (6.63) it is clear that ai and /3.; depend only on el.; and ezi, soan explicit calculation of c3i is not necessary. The calculation of ezi is necessary forthe nondegeneracy condition and stability type of the solutions, but the calculation ofel.; is crucial to determine the dynamics. Because of the complicated form of ,3i wecould not find simple expression for the nondegeneracy condition in terms of the originalmagnetoconvection problem. However, for given set of magnetoconvection parameter itis very simple to check the condition numerically (see Table 6.2).To calculate e1 we note that when Tro,, = Deto,, = 0 we have= —11700611,^ (6.66)A2j = Wo[±1c6I—C6,].^ (6.67)Now letLA3j =^hat +^Wo[B11 — C611 •^ ( 6.68)After simplication at (it, v) = tt v;), we haveA3i = — 1170[±106I CU].^ (6.69)At this parameter value At =^therefore after simplification= 2e Wo[HC611[±1C61 — (B11)]^(6.70)Chapter 6. Secondary Bogdanov-Takens bifurcations^ 130Table 6.2: Values of aj , fij for a = 1,^.1, in= 107, and different values of 0. (Sincefor Q = 1007r2 there is only one bifurcation point for v > 0, the values for a2 and 132 areleft blank.)-61 al a2 in -2 13 1 171-2021007r2 —5820 — 51604 —30072 —7496 14.89 105551 —16.4550071-2 —8233 46.73 129794 —139.8where(C 7:3C 6) + GIL I 3 R — B211]2 A[+IC 61 — C61]For in sufficiently large, Bin — B911 = 0(e) and therefore= (6.71)sgn(ai) = scp.0(C2)(10 — B1)),^ (6.72)sgn(ct2) = —syliMC2)(1C1+ B1)).For both j = 1 and j = 2, under the non-degeneracy condition /3i 0„ we will havetwo different cases, according to the sign of cb , which have been discussed in [16, §7.3].Now we give a summary of their results and its implication for our problem. Along a1-parameter path transversal to the line = 0 there will be a subcritical (if ai > 0 )or supercritical (if aj < 0) pitchfork bifurcation as two other fixed points will bifurcatefrom the trivial solution. Also, by the Bendixson criterion there is no periodic orbit when-Y2 < 0. Along a 1-parameter path transversal to 1'4 = 0, > 0) there is a Hopfbifurcation of periodic orbits from the trivial solution.Case a (a; > 0): The periodic orbit created by the Hopf bifurcarion is destroyed in aheteroclinic (saddle connection) bifurcation along any 1-parameter path transversal toF5 = {(71,y2)^= —71/5 + 0(q), yi < 01.^(6.73)Chapter 6. Secondary Bogdan011-Takens bifiircations^ 131Case b (ai < 0): In addition to the periodic orbit created by Hopf bifurcation alongF4, two other periodic orbits bifurcate from nontrivial fixed points (+177T, 0) in Hopfbifurcations along the curveF6 = (71, 72) 'T1 = 721 71 > 01. (6.74)These two periodic orbits are destroyed while another periodic orbit will be created in aglobal homoclinic bifurcation, along any path transversal to: (71,72) :72 = (4/5)71 + 0(7;), 71 > 01.^(6.75)The two hyperbolic periodic orbits coalesce into a non-hyperbolic periodic orbit anddisappear in a saddle-node bifurcation of periodic orbits, along paths transversal toF8 = 1(71,72) = C71 + (q), C-Z,' 0.752, 71 > 01. (6.76)To apply the above analysis to our problem we observe that fixed points in (6.61)correspond to periodic orbits in (6.1). In particular the origin in (6.61) corresponds toSW0 periodic orbits, and non-trivial fixed points (±V--F., 0) correspond to two furtherperiodic orbits, which we denote by C2i, i = 1,2. Periodic orbits in (6.61) correspondto invariant tori in (6.11). In fact, the periodic orbits that bifurcate from F4 correspondto the invariant tori for which we have established existence in Chapter 5. In our case,since both aj and /3.; could be either positive and negative, several possibilities exist. InFigures 6.5-6.10 we have considered two of these cases. The other cases can be analyzedin the obvious way.In parameter region / in Figure 6.7, the only periodic solution in a. neighborhood ofthe SW0 solution is SW0 itself (the 51,17, solution exists, but it is not close), and thereare no invariant tori. Along the curve F3 there will be a. subcritical pitchfork bifurcationof periodic orbits = 1,2 from the SW0 solution and they exist in regions //, ///Chapter 6. Seconclaty Bogdanov-Takens bifurcations^ 132and /V. Along the curve r4 invariant tori bifurcate from the SIV0 solutions. Theseare unstable (of saddle type) and exist in region ///. (These are the same tori whoseexistence was proved in Chapter 5). Along F5 there is a heteroclinic manifold betweenthe periodic orbits Qi and as we cross r5 the invariant tori are destroyed in a globalheteroclinic bifurcation. See Figure 6.5, 6.6 and 6.7.In Figure 6.8 the only periodic orbits near the ST17 solutions are SW0 solutionsthemselves, for parameters in region I. Invariant tori denoted by T1 are created alongthe curve r, in Hopf bifurcations. These invariant tori correspond to those for whichwe established existence in Chapter 3, and they exist in regions //, ///, IV and V.Periodic orbits c bifurcate from SI470 solutions in pitchfork bifurcations, for parameterson r3. These periodic orbits exist in every parameter region except regions / and //.Two invariant tori denoted by T9 and T3 bifurcate from C2i, i = 1, 2 respectively, in Hopfbifurcations for parameters on F6. These invariant tori exist for parameters in region IV,and are destroyed, while another family of invariant tori, denoted by T4 are created in aglobal bifurcation along the curve r7. Hyperbolic invariant tori T1 and T4 persist until forparameters in region V they coalesce into non-hyperbolic invariant tori for parametersalong r8 and then disappear in a saddle-node bifurcation of tori. See Figures 6.9-6.13.We expect that if we restore the higher order terms to our truncated equations (6.12)most of the above behaviour persists. Since Hopf and pitchfork bifurcations of periodicorbits are structurally stable, using similar methods as we did in Chapter 5, we couldprove that for a curve fl3 close to r:3 (see [16, theorem 4.3.1]) periodic orbits Ij , i = 1,2for (6.11) bifurcate from SITT0 solutions in a. pitchfork bifurcations of periodic orbits, sincethe higher order terms respect the required symmetry. Also, there are Hopf bifurcationsof periodic orbits about curves fi and f6 close to r4 and r6, and invariant hyperbolictori will bifurcate from SW, These invariant tori will have the same stabilitytypes as their corresponding periodic orbits of the truncated system (6.12).Chapter 6. Secondary Bogdanov-Takens bifurcations^ 133Figure 6.5: Magnification of the square region in Figure 6.4 with a2 > 0 , /32 < 0 (Casea).Chapter 6. Secondary Bogdanov-Takens bifurcations^ 134At (ii,, v) = (ii;, v) IY—/-II Along r,IWIII IVFigure 6.6: Phase portraits for regions I-TV and along r5 of Figure 6.5 and at(,u, v) =^vi*) of equation (6.60) with a2 > 0 and 2 < 0.Chapter 6. Secondary Bogdanov-Takens bifurcations^ 135C19S(a) v <c22SW0(b) v > v.;Figure 6.7: Bifurcation diagrams for Figure 6.5, corresponding to one-parameter pathsobtained by increasing it, for fixed v. Dots represent local bifurcations and the rectanglerepresents a global (heteroclinic) bifurcation. Solid lines represent stable solutions anddashed lines represent unstable solutions.Chapter 6. Secondary Bogdanov-Takens bifurcations^ 136Figure 6.8: Magnification of the circular regions in Figures 6.3 and 6.4 with al < 0, ,31 > 0(Case b).Chapter 6. Secondary Bogdanov-Takens bifurcations^ 137At (it, v) = (tit vi")C---1 H---Ø___III 11/ _.,-------*Iongr7NM/11$1° (01.V7cAlong r8 IIMmih611‘. dAITIVilil 114Figure 6.9: Phase portraits for regions 1-VI and along r7, r8 of Figure 6.8, and at01,0 =^iii*) of equation (6.60) with a < 0 and „(31 > 0.C-2111-22-Chapter 6. Secondary Bogdanov-Takens bifurcations^ 138(a) 1/ < Vi*ST470Figure 6.10: Bifurcation diagrams for Figure 6.8 corresponding to one-dimensional pathsobtained by increasing p, for fixed v. Dots represent local bifurcations, the rectanglerepresents a global homoclinic bifurcation and the cross represents the saddle-node bifur-cation of periodic orbits. Solid lines represent stable solutions and dashed lines representunstable solutions.Chapter 6. Secondary Bogdanov-Takens bifurcations^ 139We now indicate some differences that would be expected due to the '4'-dependenceof higher-order terms, which were neglected in the truncated normal form. When thedependence is restored, the phase portraits of Figures 6.6 and 6.9, represent approximatePoincare maps for (6.11), restricted to two-dimensional, invariant manifolds. The curvesF5 in Figure 6.6 and r7 in Figure 6.9 correspond to heteroclinic and homoclinic manifolds,but for maps such behavior is nongeneric. It is known that transverse heteroclinic andhomoclinic points (and consequently chaos) will exist generically in exponentially thinwedges in the (,a, v) parameter plane, near the curves F5 and r7, i.e., for generic higher-order terms these two curves will be replaced by exponentially thin parameter regions f5and 1;7 corresponding to the existence of transverse heteroclinic and homoclinic orbits,and away from these regions no such orbits exist [21]. Another situation where thedependence of higher-order terms would be expected to affect dynamics is for parametersnear the curve r8, which correspond to saddle-node bifurcation of tori. For generichigher order terms depending on kIf , the curve r8 will be replaced by a Cantor set 1;8that corresponds to quasi-periodic saddle-node bifurcations of invariant tori [31, Theorem1.1]. Moreover, it can be expected that near 18 there exist open sets of parameter values(called "bubbles" in [4]) that correspond to resonant and chaotic behavior.6.4 Approximate D4 symmetryUsing the asymptotic results of Chapter 4 on the normal form coefficients as in -4 ooin Case II, we observe that (6.3) approaches a small perturbation of a system with D4symmetry,0/2 = U[aRp + r(2.411 + Bn+ C11)12]-1:117(BI CI)/2± 0(r3 ± 11111'2 + kir 11^(6.77)1:12 = 1/[aRp + r(24 + B11 — C11)/2] + U147(BI — C1)/2Chapter 6. Secondary Bogdanov-Takens biThrcations^ 140+ 0(7'3 + Iftir2 + ifti2r)147/2 = W[aRit + r(AR + 131)] + UTTC1 + 0(7'3 + Ip17-2 + lp,12r)= Pw +Ignoring higher-order terms, system (6.77) has an additional symmetry generated by((I, T7,^)^(U,^),which corresponds to21, Z9, Z9) 24 (Z9, 29,^21)in (6.1). Using (3.45) this corresponds to the symmetry(4) , (1)1, (I)2, (1)2) —24 ((I)2, 4'9, (I) (T)i(6.78)(6.79)(6.80)in the original magnetoconvection equation. Using (2.21) and (2.22), this in turn corre-sponds to a. fixed translational symmetry in spatial variable74)(x, y, t) = (I)(x — )/2, y, t),^ (6.81)where A = lim,1 (2Ln1/m) satisfies (2.27). Therefore for large in in Case II we canconsider our magnetoconvection problem with sidewalls as corresponding to a small per-turbation of magnetoconvection in an infinite layer with the D4 symmetry generated bythe actions of -y,,3 and J, if we identify (I)(x,y,t) with (D(x + 2A, y, t).System (6.77) has been studied by Swift [41]. In system (6.77), there are at least threeinvariant subspaces V = TV = 0, U = TV = 0 and U = V = 0. In each of these subspacesthere are two pairs of periodic orbits that are denoted by "U", "V" and "TV" solutionsrespectively. The periodic solutions I-170(0, sit, v) and It (0, ii,. v) of our perturbed systemcorrespond to the pair of "IV" solutions of the unperturbed D4-symmetric system, andagree with them at the lowest order in E. Using the implicit function theorem, we canChapter 6. Secondary Bogdanov-Takens bifurcations^ 141prove the existence of periodic solutions corresponding to "U" and "V" solutions forsufficiently small c under the following nondegeneracy conditions. The perturbed "U"solutions exist ifICl2 — n(BC)^0, (6.82)and they have the form(C/i(7,b, it, v, E),^(0,^v,^(0,^v, E)) = (±U0 + 0(E), 0(E), 0(E)), i = 1,2where—2allp1U01= ^ , syn(p) = —syn(2A11 + Bft + CR).2AR + BR + CRSimilarly, the perturbed "V" solutions exist if1C12 +R(BC)^0, (6.83)and they have the form(NO, v, €), (V), v, E),^(0, v, E)) = (OW, ±Vo + 0(E), 0(E)), i = 1,2,where^—2anit^' sY71(11) =vi(2A11 + BR — CR).117°1 = 2A R + BR—CR These solutions have the same stability type as their corresponding unperturbed solu-tions. If the nondegeneracy conditions (6.82) and (6.83) are satisfied, no nonsymmetricsolutions bifurcate from "U" and "V" solutions (6.77). In the unperturbed case twonon-symmetric solutions bifurcate from the "W" solutions in a pitchfork bifurcation aswe crossed the parametric surface 1B12 = 0.Chapter 7ConclusionIn this chapter we summarize the results of Chapters 5 and 6, and also make someremarks on our magnetoconvection problem. For fixed magnetoconvection parametersa,( and Q, we have proved that the motionless conduction state loses its stability as weincrease the Rayleigh number I?. For L close to one of the 17„ in = 1, 2, , two standingwave solutions, which we denote by SW0 and SW, solutions, bifurcate in primary Hopfbifurcations of periodic orbits, and the SW solutions that bifurcate at the lower value ofR are asymptotically stable, and the other SW solutions are unstable. As we increaseR further there occurs a secondary Hopf bifurcation of invariant tori (e.g. unstablequasiperiodic solutions), which we denote by T1 solutions, from the branch of unstableSW solutions. After the T1 solutions bifurcate, both SW solutions are stable. We havealso proved that the tori T1 persist in wedges in the parametric plane.By considering a decreasing sequence in ( and an increasing sequence in Q, for fixeda, we were able to extend the regions of validity of our bifurcation analysis by consideringan alternate normal form. We showed the existence of Bogdanov-Takens singularities,and these codimension two singularities lead to more complicated dynamics such asfurther secondary and tertiary bifurcations of invariant tori and generically some openparameter regions corresponding to chaos. Figure 7.1 shows schematically the regionsin the parametric plane for which our bifurcation analysis of Case II is valid. The curvesr8 are the same as Chapter 6, and by reflection symmetry there are curves r,corresponding to bifurcations about the SW, solutions. Note that the curves r4 and r/4142Chapter 7. Conclusion^ 143(a)Figure 7.1: Schematic bifurcation sets for magnetoconvection equations in Case II, for Lnear L„, and R near R,„: (a) when r3 and F4 Only intersect once (ai < > 0); (b)when 1'3 and r4 intersect twice (cti < 0031 > 0 and a9 > 0 and /39 < 0). The shadedregions show parameter values for which the invariant tori T1 occur.Chapter 7. Conclusion144(b)Figure 7.1 (continued).Chapter 7. Conclusion^ 145near the origin correspond to the curves A1 and A2 in Chapter 5.In Figure 7.2 we give bifurcation diagrams for fixed v = L — L„, > 0, as we increasep = R — Rm. The bifurcation diagrams for v < 0 are the same as those for v > 0 if weinterchange the roles of SW0 and SW, solutions. If we compare the bifurcation diagramsin Figure 7.2(a, b) with Figure 5.6(b, c), we observe that they are the same, since for Vsufficiently small the analysis of Chapter 5 is valid. However, for v near vi" we have morebifurcations. Also, we observe that the bifurcation diagrams in Figure 7.2(b) and (c), andalso Figure 7.2(d) and (e), are rather different. This implies the existence of more globalor local bifurcations for v between 0 and vl* and between vi* and /4. It seems very hardto locate these bifurcations analytically, but it should be possible to get some resultsusing numerical methods and computer software packages like AUTO [11] to locate someof these bifurcations. AUTO is able to follow branches of periodic solutions and analyzechanges in their Floquet multipliers, as we change parameters. Also, we note that aslong as the point (p, v) in parameter plane lies below the curve of 1'4, for v > 0, the SW0solution is unstable, which is in agreement with our primary bifurcation results. For largem, the slopes of the curves r1, r8 are 0(m2). Therefore all of these curves lie veryclose to the line p = 0, and regions where our bifurcation analysis applies become smallas in oo.We have also shown that in the Case II limit as in oo our system becomes a smallperturbation of a system with D4 symmetry. Using this fact we found other periodicsolutions.This project by no means complete and it can be continued in number of ways. First,the results of Chapter 6 can be continued by proving that the results we have persist afterrestoring the higher-order terms (and therefore 'h-dependence) in the equations. Thereare some technical difficulties in achieving this, but in principle it could be done. Also,one could attempt to complete the bifurcation sets in Figure 7.1 as we mentioned above.• local bifurcationChapter 7. Conclusion^ 146(a) v = 0(b) v > 0,v near 0(c) 0 < v < , v near vi*Figure 7.2: Bifurcation diagrams for magnetoconvection equations in Case II.Chapter 7. Conclusion^ 147(d) v > ç, v near vr(e) vl* < v < v9* , v near v!2'SW,SW0(f) v > 74, v near v;Figure 7.2 (continued).Chapter 7. Conclusion^ 148One possibility is to try to exploit further the small perturbations from D4 symmetry.In our bifurcation analysis, we restricted to those magnetoconvection parameter valuesfor which the Hopf bifurcation was preferred. However, as it was mentioned in Chapter 2,there are parameter values for which steady state bifurcation is preferred, and also criticalparameter values for which the linearization of the convection problem has double zeroeigenvalues. The bifurcation analysis at these higher codimension singularities would bevery interesting but would need another thesis for a complete analysis.We simplified our original magnetoconvection problem in number of ways. First werestricted ourselves to two-dimensional flows by assuming the velocity, temperature andmagnetic field remain constant in third z direction. In three dimensions, depending on theshape of the container, there could be more than two different spatial Hopf modes for someparameter values. This would increase the dimension of reduced ordinary differentialequation and therefore a more complicated analysis would be needed. The calculation ofcenter manifold coefficients would also be much longer.We adopted boundary conditions that made our eigenfunction calculations possibleby hand. If we adopted different boundary conditions, the computation of the eigenfunc-tions for the linearized magnetoconvection equations would take long hours of computerprogramming and numerical calculations. The center manifold coefficients would alsoneed to be computed numerically. Also, we chose our boundary conditions so that thesystem has Z2 ED Z2 symmetry. This symmetry can be easily broken by perturbations thatdoes not respect the symmetry and might correspond to more physically more realisticsituations. For example, it can be arranged that (2.5) satisfy boundary conditions of theform= T0[1 OW] on 0,= T1 on y =Chapter 7. Conclusion^ 149where 0 is not an even function of x, so that there is no longer a reflection symmetryunder X -÷ -X.Bibliography[1] Arnold, V.I. [1983]. Geometrical Methods in the Theory of Differential Equations.Springer-Verlag, New York.[2] Bedford, T., Swift, J., eds. [1988]. New Directions in Dynamical Systems. CambridgeUniversity Press.[3] Chandrasekhar, S. [1961]. Hydrodynamic and Hydromagnetic Stability. Oxford Uni-versity Press, Oxford.[4] Chenciner, A. [1988]. Bifurcation de points fixes elliptiques, III. Orbites periodiquesde `petites' periodes et elimination resonnante des couples de courbes invariantes.Publ. Math. IHES, 66, 5-91.[5] Chow, S.N., Hale, J.K. [1982]. Methods of Bifurcation Theory. Springer-Verlag, NewYork.[6] Crawford, J.D., Knobloch,E. [1988]. Degenerate Hopf bifurcation with broken 0(2)symmetry. 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To find explicit formulae we use (4.33)-(4.38); firstlet us give M(., (V) = M(4), (V, L„,) in vector form explicitly from (3.44), as( o-(Qtly [Ob.,+ L"_,aby + u au' Olt' _2 9X \Oy I Ox Ox + v ay + L '" ax[—Ob., + L,_2_2.011+[u_av' + v_al+-Oy '" Ox Ox ay ay11 10(1),V) =L , x(A.1)where x is chosen so that the divergence of the first two components vanishes. Using(A.1) and (3.39)-(3.40) we observe that P2110(4) • , (I)k) = 0 for j,k = 1, 2, due to elementarytrigonometric identities, and we calculate0-(I - 13)1_110(4)1,(1)1) =0sin(27y) 2P1(1 +^)071-2 cos(m7r.x)\ Pi(( + iwi) /(A.2)153Appendix A. Normal form coefficients 154—(I — P )Alo ( 4)2, 4)2) =Sinceith(4)j, (DJ) = 1110(4)1,•and007r sin(27y) 2P2(1 + iw2)0cos((in + 1)7x) P2(C iW2)= 1, 2,(A.3)1110(4) j, (Dj) = Alo(4),, j), j = 1, 2,we have— (I — P)04-0(4)1,4)1) + M0(4)1,(1)1)) =— P)P/102,(1)2) +^2,,D2)) =007r sin(27y)P1(1 + 4)0272( cos(in7x)Pi ((2 + 4) /0071 sin ( 27y) P9(1 + 4)0272( cos( (in + 1)7s)P2(C2 + (4)(A.4).^(A.5)Appendix A. Normal form coefficientsWe also have—(I — P)^) + M0(4)2, (1)1)) =7r2Bi4B0B2 Sill( (2M,^1)71-42) sin(27ry)B3 cos( (2m, + 1)742) sin(271-y)B3 (27n + 1) .B4 Sill( (2771 + 1)7142) ^ sin( (2m + 1) x/2) cos(2y)4Bo(2m, + 1)7r2B1 sin((2in + 1)71-42) sin(2y)cos((2in + 1)742) cos(27ry)7r2A1^  cos(71-x/2) cos(27ry)Ao71-2A1 .^ sin(7rx/2) sin(27ry)4A0A2 sin(7x12) Sin(27Y)A3 cos(7rx/2) sin(27ry)A3.A4 sin(7rx/2) —^sin(7rx/2) cos(271-y),^(A.6)155whereAo = 7r2(1/4L2„, + 4),Bo = ((2M /2Lin)2 + 4),(A.7)(A.8)^aCQ7(P2 — P1) ^(2m + 1)7r(A.9)111(177, + 1)/31 P2((^iw2)^4arn(rn^1)'= (2m, + 1)/31, (A.10)7r(2m, + 1) [inP2(1 + iw2) + (in + 1)P1(1 + iwi)) A2 =-^ (A.11)47n(171,^1)P1P2(1^iW1)(1^i(-02)[—mP2(1 + 1(.02) + (in + 1)P1(1 + iwi )} B2^4in(in + 1)P1 P.2(1 +^)(1 + iw2)(A.12)72 [Pi +^) — P2(+^)1 B3171(771^1)P1P2K^i(4,1)((^iL4)2)'^(A.13).43 = (2m, + (A.14)72 [P1 (ç+^) + P2( + iw2)] A4 =^ (A.15)4in(m, + 1)P1 P(( +^)(( +B4 = (27/1. + 1)2-44.^ (A.16)Appendix A. Normal form coefficients^ 156We now solve (4.23), using (A.2):(1(0^2iP1C-01) W2000 =00it sin(27y)2P1(1 +072 cos(m7rx)\^(( +(A.17)i.e.,a [(Al — 2iP1w1 /a)u — VI + R,„.0ey + (Q(VI x b) x(Al — 2iPIQI)0 u • ey((Al —^C21 )b + V x (u x ey)where II/2000 = (11, 0, b)T has the general form( 0 )0it sin(27y)2Pi (1 -I-072 cos(rn7x)P1(( + ic4;1)(A.18)el sin(m7x) cos(27y)—(inci /2) cos(rnmr) sin(27y)c9 cos(m7s) sin(27y) + c3 sin(27y)^ (A.19)c4 sin(m7x) sin (27y)mc4 /2 cos(m7x) cos(27y) + c5cos(mirx) jand x = c6 cos(m7x)+c7 cos(27y)+c8 cos(in7rx) cos(27y) is chosen so that the divergenceof the first two components of T2000 become zero. Substituting (A.19) into (A.17), weAppendix A. Normal form coefficients^ 157solve for the coefficients^c5 in (A.19) to get 0071 sin(27ry)"2000 = (A.20)2/1(1 + iwi)(zuri + 2iwi)047r2L72„ cos(m7rx)pr((+iw,)07,2.1(+8iw1L,)where WI 47r2/ Pi. Similarly, we solve7r2P2(1 +00.^ sin(271y)zw9)0(K0 — 2iP2w2) T0020 = (A.21)971- cos((m + 1)7rx)P2(( + iw2)to obtainW0020 =00^sin(27ry)2/31(1 + iw2)(m2 + 27102)0^ cos((m + 1)7rx)\^+ iw2) ((m + 1)2w2( + 8iw2gi)71472L2771(A.22)Appendix A. Normal form coefficients^ 158where W2 = 472/P). We also solveKODuo° =to obtain0sin (27y)Pi(1 + con0272(9^ cos(m7rs)((2 + wr)00071(A.23)1„^ sin(27ry)47.Pi (1 + col-)02L2171(A.24)W 1100 =COS (777,7137)Topi ((2 w?)■and similarly,KoT oolihas solutionT ow ' =0071^ sin(27y)P2(1 -H-0.4)0272(P9((2 + b);. ) cos((iii + 1)7x)00^ sin(27y)47rP2(1 + w:=3)1(A.25)(A.26)2L.„^ cos((m + 1)715)\ (in + 112 P2K2 c4)Appendix A. Normal form coefficients^ 159By (4.29) we have(K0 — iP1w1 — iP2w2)^=^-^pm(D1,4,2)-FA10(2,4)in, (A.27)where — 15) (M0(4'i (p2) +^(c12, (1)1))is given by (A.6), therefore 'I'010 has the formcos(7x/2) cos(27y)(c1 /4) sin(n-x/2) sin(27y)=^c9 sin(742) sin(27y)C3 cos(7x12) sin(27y)—(c3/4) sin(7x/2) cos(27y) + e4 sin(742)cos((2m + 1)7x/2) cos(27y)(2m, + 1)c/1 sin((2m + 1)742) sin(27Y)4sin((2m + 1)7x/2) sin(27y)c'3 cos((2m + 1)71-s/2) sin(27y)(A.28)(2m + 1)c'3 sin((2m + 1)742) cos(27y) + c'4 sin((2m + 1)7x/2)4andx = [c5 sin(7x/2) + c15 sin((2m + 1)742)] cos(27ry) + C6 sin(7142) + c16 sin((2m + 1)7x/2).To simplify our notation, let1/1 = A0 + iP1w1 + iAw2,^T = B0 + iP1w1 + iP9w2,^(A.29)719 = o-A0 + iP1w1 + iP2w2 ,^= o-B0 + iP1w1 + iP9co9,^(A.30)113 = CA0 + iP1w1 + iP9w9,^413 = (Bo + iP1w1 + iP9w9,^(A.31)2T^9 (2111 + 1 )7 9114^C(^ )- +^+ iP2w2, tP4 = (( ^iP2(.02. (A.32)2L„,Appendix A. Normal form coefficients^ 160Now by using (A.6), (A.27)——719cland (A.28) we get(Qc4 + e6aCQA0e3^a7c5+=--==0,72,41(A.33)(A.34)(A.35)ihei) +27r^2LFaRmc2 + 27ro-c5Ao72A14^,4A0—111e2 + e1/4 = A2, (A.36)—7/3c3 — 27c4 = A3, (A.37)-1/4C4 = A4, (A.38)(Qc14 + c'6 = 0, (A.39)—k1/9c, + ci(QBoe/31 (2m, + 1)o-7rc'5 = 72BI/B0, (A.40)27r 2E72n(2771 + 1)T2cC + (27n + 1)72B4a R.,,e9 + 27 o- c; = (A.41)4 4B0— tP 1 02 + eC (2in + 1)/4 = B2, (A.42)-T36 - 27c,C — B3, (A.43)—kli4c4 = B4. (A.44)Explicit calculation of pressure term is not neccessary, and we solve the system (A.33)—(A.44) of algebraic equationsClto find c1,...,e4,^cc, ..., c'4, to getu(QA0A3^aRrn A2Aii4 +(A.45)+277)3^4/4ni11172A0 + a(QAo^aR„i472^1)3ci — 4A2164711C2 = (A.46)41/127c1 + A3C3 = (A.47)113C4 = (A.48)Appendix A. Normal form coefficients^ 161cl(QB0B3 (27n + 1)o-R1B2Bi /4 +^27T3^44k1J1 T2B0 cr(QB0 (27n + 1)2o-R1cr‘472^‘1,3^164 Ti(27n + 1)cii — 4B22 71-cii + B3T3--__ 4, -Ci4 =^13 AP 47C2C3(A.49)(A.50)(A.51)(A.52)We observe that um can be obtained from Ti010 by changing c...)9^--co2 and thereforeT1001 =di cos(7x/ 2) cos(27y)(di /4) sin(742) sin(27y)d2 sin(742) sin(27y)^ (A.53)d3 cos(7x/2) sin(27y)—(d3/4) sin(7x/ 2) cos(27-y) + sin(7x/2)cos( (27n + 1)742) cos(27y)(27n + 1)c/1^ sin((277-7 + 1)7x/2) sin(27y)4d'9 sin((2m + 1)7x/2) sin(27y)d'3 cos((2m + 1)7x/2) sin(27y)(27n + 1)4sin( (2m + 1)742) cos(27y) + c'4 sin( (2m + 1)742)andx = [d5 sin(7x/2) + 4 sin( (2m + 1)7112)] cos(27y)+ d6 sin(7x/2) +d sin((2m +1)7x/2).where di, c1 , i = 1, ..., 4 are^i = 1, ..., 4 under the change co9^—w9. Also, by (4.30)and (4.31) we have^= 100)We now compute the terms .110(^40, etc. that appear in (4.17)—(4.22). Tosimplify our notation, from now on we give only /C10(., •), which is the same as A/0(., •)471-2.7(Q (2L, /m)3 cos(m7rx/2) cos(y)4L,1P?((2 + 4)(( +7r2o-(Q(2L,„ fm)24P?((-2 + )(( +sin(mirx/2)sin(71-y)w? 1 sin(m7rx/ 2) sin(7ry)4P1(1 + w?)7(2L1/m)3 cos(m42) sin(y)4L771P1 ((2 + 4)71-(2L„iim,)34P1((2 + f)sin(m7rx/2) cos(7ry)wIfio(4)1,^wo) = (A.56)Appendix A. Normal form coefficients^ 162except for terms that are eliminated in the inner products with €1);!, j = 1,2, i.e.,(*ro(., .), (D;) = ( icloc.,.),(1);) . (A.54)We first calculate/^o-(Q(2L71,/(771, +1))3 sin((in + 1)7rx/2) cos(y) L;i2,9x/8xax/aY41,2(11+ co3) cos((m + 1)7rx/2) sin(7ry)71-(2L71,/(m + 1))3 .sm((m, + 1)7rx/2) sin(7ry)4L711P2((2 + 4)7r(2L,,./(in + 1))2 cos((m, + 1)7rx/2) cos(71-y)4P2((2 + 4)(A.55)where x = ccos((in + 1)7rx/2)cos(7r1) is chosen so that the divergence of the first twocomponents of (A.55) is zero. If we denote the first two components of /1-10(cD2,M9, then we should have1))27rdiv(114i, M2) =^w?)((iwo^Plc) cos((m, + 1)x/2) cos(y) = 0,a(Q(2L„,/ therefore71-2U(Q(2Lm./(771^1))3^ sin((m, + 1)71-x/2) cos(y),41,P?((-2 w?)(( + icol)7r2o-(Q(2L4(in + 1))2Al2 = cos( (m, + 1)7r.x/ 2) sin(7ry).4Pl2((2 + wf)(C +Similarly,A( 4)21111 0011) =4Lr1P2((2 + w3)(( + iw2)Toon) byAppendix A. Normal form coefficients^ 163We also have11-40 (1111001 W1 ) = 1C10(W 00111 W2 ) (A.57)\ 0 /011-10 W 2000 (I)1^= 21-4-o ( T00201 W2) = 00-ICI- 0(4)2/ W2000) —/^71-4cr(Q(2L711/(in +1))3 sin((in + 1)7x/2) cos(y)2PA7,((2 +^(w2( + 2L)2(2L1/(77/ + 1))2)74cr(Q(2L111/(in + 1))2cos((m + 1)7x/2) sin(7y)2PIL,1(( + iw2)(zz`2( + 2ico2(2L7„^+ 1))2)71-3(2L„,./ (in + 1))22P23((2 + con (7-v2( + 2iw2(2L7„ / (in + 1 ) )2 )7r2cos((rn, + 1)7x/2) sin(71y)2PI(1 + W2 )(W7 -I- 21:c02)71-3(2L71/(7n + 1))3sin( (Tn + 1)7x/2) sin(7y)(A.58)2/1(( + ib.)2)(732( + 2icv2(2L71/(rn, + 1))2) cos((in + 1)7x/2) cos(7y)(A.59)and740-(Q(722L,„/m)3 cos(rturx/2) cos(7y),^2P?(1 +^+^) sin(m7rx/2 sin(7y)^3f2^/77l)sin(m7rx/2) cos(7y)2P?(( +^)(w1( +^(2/,m/m,)2)(A.60)+ w?)(coi( +^(2L,-,1/m)2)740-(Q(2Lm/m)273(2L/m)sin (m7rx/2) sin(71-y)4L711Pi3(C2 w3)(w2( 2ic.o9(2L,/m)2)71- 2cos(m/Tx/2) sin(w-y)2ML„,(( + ic.oi)(711( + 2ico1(2Lni/771)2)2Appendix A. Normal form coefficients^ 1641C/10(4)1 1 2000)o-Rn, [^1 971'-Cm/2{P1(1 ± iL01) I_^4P1(1 + W?) 2A2(1 +1101)( + 2iwi)]Now using (4.33) and (4.37) we have=ia(Q(2.L.,„1/in)472w2P?(( +L-11)((2 + Wi2)(W1( + 2i(.4.4 (2Lmirit)2) } 'C5 ---7-- C„14.1 /2 { P2 o-R,„(1 + i [ 1^ 2ci.)2)^4P2(1+ w)^211(1+ ic.02)(7a2 + 2iw2)]7iC/(Q(2L1/(M, + 1))472W2 Pl(( + iw2)((2 + w3)(a72( +2ico2(21,/(m + 1))2) } •(A.62)where Cm, Cm,“ are given by (2.55).To compute C7, C3 C4 and C6 we also need0(A .61)0 (A.63)01(lo(Tocul, (Di) =^c1)2) =Appendix A. Normal form coefficients^ 165■00and1A-/-10(4)2, T1100) = 4L„,131(1 + co?) cos((in + 1)7rx/2) sin(7ry) (A.64)0In the following equations, ci, di, c, d, i = 1, ..., 4 are as in equations (A.45)-(A.52). Wehave:sin((in + 1)742) cos(7ry)(in + 1)D1^ cos((in + 1)7rx/2) sin(7ry)271 {(2771^1)C2^C12]COS((711^1)7rx/2) sin(y)4msin((in + 1)77x/2) sin(7ry)(in + 1)Ei2 cos((in + 1)7rx/2) cos(7ry){ 0- C(2 ((277? + 1)c3 + 6 + 8(c4 c14))P2^8m — )sin((in + 1)7142) cos(7ry)(in + 1)D1cos((in + 1)742) sin(iry)2it ((2m + 1)d2 +-9) cos((m + 1)737/2) sin(7y)4:71.sin( (in. + 1)7142) sin(7ry)(7n +1)Ei2 cos( (in + 1)742) cos(7ry)1Clo ( (pi^low =It-40(43i, io io) =whereit ((2m + 1)ci +^7r(in + 1)(2m + (ci 4m 32771471 ((2in + 1)c3 + 6) — 87r(c4^c).8777,(A.65)(A.66)(A.67)(A.68)Appendix A. Normal form coefficientswhere166.6 7r2 f^aCQ [(2in + 1)d3 + d'3 + 8(d4 + d'4)] (A.69)8in (( +P27r ((2m^1)di +^7r(m ± 1)(2in + 1)(d14m 321711,„7r R2in^1)d3^d'3 — 8(d4 + di)]. (A.70)8772D9 sin((in^1)7x/2) cos(7y)Iflo(T (1)].) =(in ± 1)/391)7rx/2)cos((in +^sin(7y)(A.71)iv [(2in + ?)ci — cid 1)7rx/2)cos((iii,^sin(7y)16/31(1 —E2 Sill( (771 + 1)7x/2) sin(7y)(rn^1)E9 1)7x/2)cos((rn +^cos(7y)2whereD2 =.72r--9I 720-(Q [(2m + 1)c3 + cid^72o-CQ(27n^1)c3 (A.72)2711Pi (( — ic.01^327711.2mPi (( —71-20-(Q [(2in + 1)3c'3 + 8 (c4 + (2m + 1)2c)]3 2771L2m PI (( — 1:W1 )iv ((2m + 1)ci^c'1)^7(772 + 1) [(2772^1)Ci — Cid8m 3247r2[(2m +1)ci^c'dE2 (A.73)81711i(( D2 sin((in^1)7rx/2) cos(71-y)(in + 1)n2^ cos((in + 1)7x/2) sin(7y)27^+ 1)di — di) cos((in ± 1)7x/2) sin(7y)16P9(1^)E9 sin( (m. + 1)7.42) sin(7ry)(in ± 1)É9cos((m^1)71-x/2) cos(7y)2A-4(T low, (1)1) = (A.74)whereD3712 { ^Pm 1)C3 C-13 — 8(C4 Ci4)} 8(7T/^1)((^iW2)71 [(2"7 + 1)ci + cid^7r77/(2m + 1)(c7 — 4(n7 + 1)^32g,(777 + 1)7 [(277t + 1)c3 + c1 + 8(c.1 + ci4 )18(77t + 1)E3(A.78)(A.79)Appendix A. Normal form coefficients^ 167whereD2^72 f 72a(Q[(2m, + 1)d3 + d]^72a(Q(2771 + 1)d3 P2^277/P1K + 7:W1)^3217/LPIK71-2(r(Q [(2m + 1)34 + 8 (d4 + (27n + 1)2(4)] 327771F1P7 (( + iW1)71 “2"/ 1)d1^7r(Tn + 1) R2171 ± 1)d1 — d'd}8m 32L.?„,712 [(2m + 1)d1 +877/Pi (( + iwi )(A.75)(A.76)D3 cos('7r212) cos(7ry)772D3 .^ sin(777,7rx/2) sin(7y)2((2m + 1)c2 + c)sin(777,742) sin(7y)4(777, + 1)E3 cos(7777112) sin(7y)77-7E3 .^ sin(77-77:712) cos(7y)21'-4-0(21 T1010) = (A.77) ../53 cos(77/7x/2) cos(7y)7T/D32  7_777(7777u/2) sin(7y)—71 ((2777+ 1)712 + (/'2)sin(m742) sin(7y)4(m + 1)E3 COS(177,7/112) sin(7ry)777E3 .2^ sin(m7rx/ 2) cos(7y)11-10(4)21 T011 0 ) = (A.80)71.2 [ 0-(Q [(2m + 1 )(73 + (113 — 8(c14 + (114)]D38( 77i + 1 )( C + 7102 )(A.81)E472 [(2in + 1) + c']8P2(in + 1) (( — iw2)(A.85)andAppendix A. Normal form coefficients^ 168where7r [(2777 + 1)di +^772(2771^1)7(di — di)]4(m + 1) 32/4,2(77/ + 1)7r {(2772 + 1)d3 + (7'3 + 8(d4 +=8(771 + 1)(A.82)whereD4 cos(777,7x/2) cos(7y)InD4^sin(m7x/2) sin(7y)2((2777, + 1)ci — c/i) . si1(777,7x/2) sin(7y)16P9(1 — iw9)E4 COS(71/71X/2) sin(7y)rnE4 .^ sin(777,7x/2) cos(7y)2ICI0(1110101 (1)2) (A.83)(A.84)D4 {R-20_,,,(2m.+1),3+,,,,_^72a(Q(27/7 + 1)c3Pi^2(777 + 1)P2(( — iw2)^32(7n + 1)4P2K — icv2)72(7“:2 [(27n + 1)3c/3 — 8 (c4 + (2m + 1)2G14)] 32(777, + 1)g,.P2(( — 7102)71[(2711,^1)Ci + Cid^ni7 [(27n + 1)ci — c'd8(m + 1) 324 A cos(m7x/2) cos(7y)inA^ sm(m7x/2) sin(7y)27 ( 27n + 1 )(71^dl'1)sm(m7x/2) sin(7y)16P9(1 + ico9)cos(7777312) sin(7y)m E.-74 .^ sm(777,742) cos(7y)2II-10 (41 01 10 1 (1)2 ) (A.86)Appendix A. Normal form coefficients^ 169where=Pi^2(m + 1)P2(( + ico2)^32(m ±1)4P2K + ic<>2)72 raw^+1),3 7r2o-(Q(2m, + 1)d3(A.87)7r2o-(Q [(2n1 1)34 — 8 (d4 + (2m + 1)2d14)]32(m + 1)4P2(( + iw2)7r [(2m 1)11 +^m7r [(2m, + 1)di —8(m + 1) 3247r2 [(2m^1)d1^dC]8P2(ln 1)(( iW2)We observe that the inner products(A4(4)2, 'Plow), 4)1) and (1lo(T1oo1,(1)2),(1)*1)are obtained fromand^(/0(4)2, 'Flom), (1)i),010(4)2, Tim), (Vi)respectively, by changing w9^—w9. Similarly,(Mo((1)1, Tom), (1^and) (Iti0(T0110,(1)1),(1)are obtained fromand (/110(Tioio,(1)1),(b;)(Alo(4)1,^(I);)by changing col^—col. Therefore we also have(Mo(T2000, (1)2), (1)^(Mg Woo2o, (Di), (1)i)000\ 01(A.89)(A.88)Appendix A. Normal form coefficients^ 170Too9o) =00^ s n(7ry)2PA1 + iw2)(w2 + 22ch,2)00alsoand(A.90)0Iffo(4)2, T2000) =7T 22P?(1 + iwi)(rai + 2k-11.)sin (7y) (A.91)Finally, we observe that(A10(T0110, (DA (DI) and (1\10(4)2, Tom), VI)are obtained from(Mo(Tioio, 4)2), 4)7) and (A/0(4)2, Tim), (1)*1),respectively by changing w2^--w2 and^i = 1, ..., 4. Therefore by (4.34)-(4.36) and (4.38) we can now computec2= C„/2^o-R„,^  [7r [Pm + 1)(c2 + (12) + + d'2] ,P1(1 +i1)^4(m + 1)((2m + 1)ci +^( 2777^1)di +^) + ^1 ^116P2(1 - 7102)^16P2(1 iW2)^4P2(1 W3)J+ 2L2m Pi [D3 + D4 + n3 ± n4 +  a(C271-  tE3 E4 E3 Ed71' 2 711^ K + iw1 )P1(A.92)Appendix A. Normal form coefficients^ 171o-Rm^((2m 1)(c.2 + (12) + c/2 + d'2)C4 = Cni+112 {P2(1 + iw2) 4m7r ((27-n + 1)ei + cC.)^7T ((2772 ± 1)(11 +^1 16P1(1 —^16/31(1 + 4P1(1 + 4)1(A.93)2.14P2p2(Crc(±(27riw2) (Ei E2^+^/^ [Di + D2 + D1 ± -152 +(77/ + 1)72C3^Cni/2^2^D3 + +2aPi r^cjC(27  (E3 E4)]{'71 771 ((^iW1)P17r ((2m + 1)71 —o-Rm^[^/T2(A.94)P1(1 + /IVO 2P22.(1 iW2) (W2 + 2114)2) + 16P2(1 +iw2)7T ((27n + 1)d2 + d'2) }4(m + 1)^= C /2 f ^2I;P2^a(Q7r^+ )]C6^_m+i, _ t (in + 1)72 [ 1^2^P2 (( iW2) 2^o-Rm 7r ((2in + 1)di —P2(1 + tiCA.12) 12/J1(1^i:W1) (W1 + 2iW1)^16P1(1 +71- ((21T/^1)d2^(112)14m(A.95)where Dj E , i =1,...,4 are Di, E , i = 1, ..., 4 under the change c,../2^—w2, and overbardenotes the complex conjugation.A.2 Limiting values of normal form coefficientsIn this section we calculate limiting values the normal form coeficients Ci, ..., C6 as moo, for both Cases I and II described in §2.5.Appendix A. Normal form coefficients^ 172A.2.1 Case IWe first need certain auxilliary limits. Using the equations (4.61)-(4.68) in Chapter 4we find that as in^oo in Case I we havec11 = lirn Lnici = 0,^ (A.96)772 -4 0071ulna C2 = ^C21 =^P2(1 + iw)(r.-o +2i) where zu = 47r2/P,^(A.97)cvC31 = lirn L711c3 = 0, (A.98)771-1.00C41 = liM C4 = 0,^ (A.99)771-400(A.100)d11 = lim 1,,,di = 0,7/1—>oo(A.101)c'21 = urnc'2 = 0,in—>,oc)c 1 = liM L171C/3 = 0,^ (A.102)771-400272A2c'41 = urn c'4 = rn-+oo^P2(( ± iW) (L7'( ± 2iWA2) '^ (A.103)Aic.471-dii = lim Ltali =  (A.104)171-)00^p((-2+ w2)(472 4_ Q),1d21 . lim d2 =  ^ (A.105)771-›,00^471-P(1 + w2) '2Aiunr2(131 = lim Ln,(13 =  ^ (A.106)/71.-> 00^(P(472+ Q)((2 + w2)'d4i =--- lim d4 = 0,^ (A.107)77/ -3.00(1111 = 1 i M L711 d' = 0, (A.108)7-n-00U,21—^liin (1'2 = 0,^ (A.109)m -4004 1 = lim L771(1/3 = 0, (A.110)ni--400A2d1^lirn (1'4 =  ^ (A.111)711-4 00^2P((2 +co2)'where P,o) and A are given by (2.59), (2.60) and (2.27). Then (A.96)-(A.110) imply that720.(QA6441lim L77 1 D1 = — lim L771D3 = ^712 -4 00 111-'00^ 2P(<"(A.112)lim Li„Eini-*00Jim Lm.D277/ -+ 00liM LmE277/ -*00— lina L111E3 = —71-A41/2,UI 00RAUCQAC/41 — liM L1D4 =in -+ colim L,1.E4 = 0,in -+ oo(A.113)(A.114)(A.115)2P2(C — iw)Jim L7n^=/it 00lirn LD2 =nz -400— lim Ln1E3711--). 00— Jim712-4 00urn Lm Di =711-4 00lim Lr,,E1 =M-4 00liM m n 2 =nz -*coJim^ 72(111Ln1k2 = — Jim L7 E4 =772 -)00 777-> 00 (A.123)8(( + iw)'72(1 + iW)(( iW) Jim Om = 11111 C11 =771 -400^in -*00 2A2Pw(w — i(5 ) (A.124)Appendix A. Normal form coefficients^ 173andlirnDi = —nz-+ ooli-n L„,n3M. -*00(72/p)^a(Q(—d31 + 2)d41) 4(( iw)7rdir 2 } (A.116)—7rAcr41/2 — nd31/4,(72/P) ro.,Q(_d3i± (2/A)cf41)P((±iiw)7rdi (A2 — 1) 14A2^f2g all4P(C + ic.0)li-n L7E2 = — lirn L 7 E4 =M-4 00^ in--4,00We also have(A.117), (A.118)(A.119)= —Win { Cr(C2(d31 + 2M,141) d7rii/2} 0.120)4(( + iw)= —71-1\d1/2 + 7d31/4,^(A.121){7r2o-(Q(d3 + (2/A)(141) = (72/p)P(C + iw)irdii (A2 — 1) f1(A.122)+ (422 From (2.49) we getwhere 6 = 1 + a +^Using (4.39)-(4.44), (A.96)-(A.110) and (A.112)-(A.123) we getlirn^= 0,?It -4 00e9 I^liM= -^771-->00^P2(1 +^2iw)7r(A.132)where^= 47r2/P,^(A.133)Appendix A. Normal form coefficients^ 174lirn C1 -, urn C5 = A + B,^(A.125)^L„,--00^L„,—>colirn C2 = urn C4 = A, (A.126)L„,-400^L„,—+.lirn C3 = lirn C6 = C,^ (A.127)^L—*oo^L.,--+oowhere712o-R0(( + iw)  1^1^R.24A2P2w(w — iS) [ 4P(1+ w2) 2P2(1 + iw)(w + 2i7rwldiiia(QA271-4(1 + iw) + 4P3((2 + w2)(w( + 2iwA2)(w — i6.)'^ (A.128)—7r2(( + iw)(1 + iw) f^o-Ro ^[7(021 + d21) +^4A2Pw(w — iS)^1 P(1+ iw)^2^4AP(1 + iw)1^1± o-Q(2A2c141^271-2.9-(Qc1 . a(Q(/1A2+ 4P(1 + w2) i^((-2 + w2)^P(( — iw) 1- ( + iw^7rd11(3A2 — 1)^7r2o-(Q(Ad31 — 2d41)^71-or(QC  1+^ + (A.129)4A^P(( + iw)^4P(( + iw)2 f 'A + BA=C= —72(( iw)(1 + iw)^aR04A2Pw(w — i6)^P(1+^ iw) [2P2(1 + iw)(w + 2iw)^7rd114AP(1 + iw) + 7d21/2]^7r2o-(Q)(c131 + 2d41) P(( + iw)++o-(Qd1A2^aCQA71-c/ii^7rdi 1 (3A2 — 1) }( + iw^4P(( + iw)2^4A^.= 71 —k/2Q = 111 k/2Q,A.2.2 Case IIIn Case II, when97F-(A.130)(A.131)for fixed a, 61 and 0 < k < 2, the limits in equations (A.96)-(A.111) are replaced byAppendix A. Normal form coefficients=^IllE L7nC3 = 0,nz—>oo^n'175(A.134)641 = urn c4 = 0,771 -),00 (A.135)Cl 1 = lirn Ln2c14 = 0,rn -4 00^rn(A.136)C21 lim c/9 = 0,712-)00 (A.137)-•/C31 IiM LMICI3 = 0,IA -400^n(A.138)C41 lirn c= (A.139)-)00^6D2p2d11 lim 1,;,"„-tdi = —71-5t.i/PCD,1(A.140)d21 lirn d9= (A.141)-17/ -400 47P(1 +2)'d31 (A.142)rd3 =Ut^ 00 (Q15C0'd41 1 i m (14 = 0,—*00(A.143)alld21lirn Lmdi = 0,rn—>oolirn d^= o,772^00(A.144)(A.145)dim lim Ld'3 = 0,171-> 00(A.146)di41 lirn d'4 = (A.147)771->. 00 21)i,2'where P , ci) and 5 satisfy (2.72)-(2.74). If k = 2 in (A.131), all the limits given by(A.132)-(A.147) are the same, except for (i41. In this case (A.146) is replaced by(141 = lim (14 = ^T/1 -4 00^- .^ (A.148)8P(Co + 2i()If k> 2,and C6 become unbounded. Therefore we only consider the Case II with k < 2. Then(A.135)-(A.115) imply thatIhn^=771 -)00lirn L,n„lE1 =00 ".— urn LD3rn—>oo— urn 1E3TI1 >DO720-(06'41 2Pic:o—71-k44/2,(A.149)(A.150)d-41 becomes unbounded as in^oo, and this in turn will imply that C2 C4, C3andlirn LTD1 = — Jim/-/2 -400 -^ -+^-lim LrnE1 — urnm->oo^m-ioolim iD2 = - 11111rn oo 721^7/1 -)00 7rcii 1^— 1 )160■2(A.155)lim En1E7 2 = - Jim LE4 =rn -4 00 rn^971 -400 112^ 8icI, •However, for k = 2 we have(A.156)(72/P) {0-(Q(—(231 + 4A041 + (LH)) =^7rcinr } ,k /^ 2^(A.157)4cD. (7205) fa(Q(—d31 + 4A041 + d41)) 7rciiir}z^(A.158)46.5. —71-54(r4i + (441) — 71(431/4,^ (A.159). i ,1^j—7,441 + (.41) — 7(131/4, (A.160). /,72 ,i5)^7r2a(Q(—(131 + (2/)(J1 ±  d41)1:PcI)liiii L"1 .15 1In -3 oo rnLM b3712-500^772Jim LmEiIn -400 721Jim L„,"1Jim LinID2rn--+ooAppendix A. Normal form coefficients^ 176^lirn LrnD2 = — Jim^=M. -3. 00^71-Jim LZE2 =771 -) 00ntlaCQ5■6'41 2.152iColim LZE4 = 0,112 -*00(A.151)(A.152)- (2/P) {a(Q(—c1:31 + 25v-F41)71d-11/2} ,(A.153)▪ —7r5vi1/2 — 71(i3/4,^ (A.154)(712/P)^72C1(Q(-431^(2/A)(1141)ii5CD4)—urn Lmb4172 00^11271dii(A' — 1) 145■2• (712/J3)^720-(Q(—(1,31 + (2/A)^a41)7rai^ 2 — 1) }4A2(A.161)(A.162)and this implies that the limiting behaviours of the normal form coefficients are qualita-tively different from those in Case I, and in Case II with 0 < k < 2, since d.41 is not reallirn L151 =nz-+00liM 1/777:E1711--).00IiM L ^=711-),00- liM 1/7,11, n3711-3.0C- lim 1,E3772-400- lim L9„7,1 7;14711-100Appendix A. Normal form coefficients^ 177for k = 2. In fact it is easy to show that when k = 2 (A.157)-(A.162) imply thatliM C9^liM C4, and lim C3^lim C6.m-+co^771-9•00^ 771-}00^in-400 (A.163)unlike in Case I, and in Case II with 0 < k < 2. For this reason, from now on we restrictourselves in Case II to only 0 < k < 2.We also have:_-_-= 17(711i2,4114 5d)2{ :::::I/341,± 2 .(1‘41)^7rd-ii /2} ,((AA..116654))4ic;)= (72/p) racQ(,,.+(2/.41)+ Irciii (5■2 _ 1) 1fiPcD(A.166)4A2^,(A.167)From (2.55) we get72(111 liM LME2^11111 L 7 E =in -+00 711.-*00 "I^ 47r2(1^iCo)ilirn 0771 = urn O7i+1 =711-)00^711-Y00 (A.168)22P(c1) - i(o- + 1))Now using (A.132)-(A.147) and (A.149)-(A.150) we finally getlirn C1 =L',;',' -+ooliM C2L7,1-400 -lina C3 =Lg:-*oourn c5 = A +liiii C1 = A,/4,q-4coliM C6 = a,L cc(A.169)(A.170)(A.171)wherei7r2o-Po^14 .A2P2(cD - i(a +1)) 1.4P(1 +^▪ 2P2(1+^+ 2idd+ o-(Q74 (1 + icD)8cD3P3(d., - i(o- +1))' (A.172)Appendix A. Normal form coefficients^ 178A i7r2 CIP0^  71-(a21^(121)1^1^ + -C j^45■2P2(C.o — i(o- + 1)) [^2 4AP(1 + ic)^4P(1 + o2)71-2(1^17120-(Q(2-C/41^24, + AC131) 4_ 7raCQC111 4PX‘2((i) — i(0" + 1) [^ii5J) 4 PCD27rciii(35■2 — 1)^A2o-(Qd-'41(A.173).+^4A. zwa = i7r2c/Ro^7r-9^7r in ^7 d-2114A2P2[cZ, — i(o- + 1)] [2P2(1 + i 'W) ( it + 2iW)^4AP(1 + i +c.D)^2i7r2(1 + icD)^1 7r2o-(QA (ci3i + 2c1'41)^7rd1 1 (3 .A2 — 1)4/55■2(a) — i(o- + 1)) [^iPa.)^4;\0-(01'415■2^aCQA7rdi 1  1+^. (A.174)7.4.:,^i4Pc.D2^j .Appendix BNumerical values of normal form coefficientsIn this section we give more numerical results. In Tables B.1—B.6, numerical values inCase I of normal form coefficients for a = 1,( = .1 and for Q = 72, 10472 and 10672 andfor increasing values of 771, are given. As in Chapter 4, the symbols oo in the tables showthe corresponding values of coefficients as in cc. In our numerical calculations we havealso calculated the values of C1, C5 for a wider set of parameter values (i) a =1,( = .01,(ii) a = 1, = .5, (iii) a = 10, .01 and increasing values of Q and in. We have alsoconsidered, for fixed a decreasing sequence of values of a and an increasing sequence ofvalues of Q such that Pjcvi, j = 1, 2 remains fixed. Because of lack of space we give onlya sample of these numerical calculations. In all of these calculations we find that C1Rand C5R are negative. Because the numerical calculations of C2 and C4 were very timeconsuming, we have looked at the values of C1R C2R for large in for several differentset of values of a,( and Q. In all of these calculations we found that C2R < 0 andCiR - C2R > 0. These values are important in stability results of Chapter 5. We alsogive another sample of our numerical calculation of normal form coefficients in Case II.179Appendix B. Numerical values of normal form coefficients^ 180Table B.1: Normal form coefficients (Case I) for a =1,( = .1, Q 72.In 2/,,,/in Rm 100a1 100021 2.007 950.3 .1822 - .1517i .2871 - .8320i101 1.407 811.5 .2555- .2828i .5720- .2881i10001 1.400 811.5 .2563- .2854i .2563- .2854i100001 1.400 811.5 .2563- .2854i .2563- .2854ioo 1.400 811.5 .2563- .2854i .2563- .2854iTable B.2: Normal form coefficients (Case I) for a = 1,C = .1, Q = 72 (continued).771 Lb2R L Mb 1 I L 771b211 -2.882 5.294 3.942 -2.8277101 -3.893 3.927 3.379 3.33710001 -3.909 3.910 3.358 3.358100001 -3.910 3.910 3.358 3.358oo -3.910 3.910 3.358 3.358Table B.3: Normal form coefficients (Case I) for a = 1, = .1, Q = 72 (continued).in C1 C2 C4 C51 -.3061 + .7467i -1.868 - .4878i -1.714 - .3182i -.6399 + .3554i101 -.4912 + .6478i -1.673 - .3706i -1.665 - .3708i -.4973 + .6422i10001 -.4942 + .6451i -1.668 - .3706i -1.667 - .3706i -.4943 + .6450i100001 -.4943+ .6450i -1.668 - .3706i -1.668 - .3706i -.4943+ .6450ioo -.4943+ .6450i -1.668 - .3706i -1.668 - .3706i -.4943+ .6450iAppendix B. Numerical values of normal form coefficients^ 181Table B.4: Normal form coefficients (Case I) for a = 1,( = .1, Q 1047r2.771 2L,„ im, .1R1 1000a1 1000a21 .6145 7596 .2637- .7716i .5153- .5004i101 .4229 6919 .4417- .7250i .4460- .7203i10001 .4208 6919 .4438- .7227i .4439- .7226i100001 .4208 6919 .4438- .7227i .4439- .7227ioo .4208 6919 .4439- .7227i .4439- .7227iTable B.5: Normal form coefficients (Case I) for a = 1,C = .1, Q = 10472 (continued).m LbiR L2mb2R Lmbii Lmbn1 -5.734 15.91 95.51 204.7101 -9.839 9.989 124.6 126.410001 -9.913 9.915 125.5 125.5100001 -9.914 9.914 125.5 125.5oo -9.914 9.914 125.5 125.5Table B.6: Normal form coefficients (Case I) for a = 1,( = .1, Q = 10472 (continued).771 C1 C2 C4 C51 -.0045+ .0211i -.0132+ .0528i -.0209+ .0349i -.0213+ .0274i101 -.0115+ .0259i -.0260+ .0625i -.0222+ .0632i -.0117+ .026110001 -.0116+ .0260i -.0242+ .0632i -.0241 + .0632i -.0116+ .0260i100001 -.0116+ .0260i -.0242+ .0632i -.0242+ .0632i -.0116+ .0260ioo -.0116+ .0260i -.0242+ .0632i -.0242+ .0632i -.0116+ .0260iAppendix B. Numerical values of normal form coefficients^ 182Table B.7: Normal form coefficients (Case I) for a = 1,( = .1, Q = 10672.m 2Ln1/iii 10-7R, 104a1 104a21 .2778 .5795 .1430 - 1.0887i .4582 - .9322i101 .1904 .5651 .2873 - 1.071i .2922 - 1.068i10001 .1895 .5651 4.2897 - 1.070i .2898 - 1.070i100001 .1895 .5651 .2897 - 1.070i .2898 - 1.070ioo .1895 .5651 .2898 - 1.070i .2898 - 1.070iTable B.8: Normal form coefficients (Case I) for a = 1,( = .1, Q = 10672 (continued).in L1b1R L2mb2R Lnibll Lmb211 -2.938 13.943 245.8 825.4101 -6.163 6.308 351.3 359.310001 -6.234 6.236 355.2 355.3100001 -6.235 6.235 355.3 355.3oo -6.235 6.235 355.3 355.3Table B.9: Normal form coefficients (Case I) for a = 1,C = .1, Q = 10672 (continued).in 100C1 100C2 100C4 100051 -.06848 + .5881i -.2550 + 2.072i -.3500 + .8118i -.7566 + 1.603i101 -.2637+ 1.049i -.4901+ 1.936i -.4136+ 1.928i -.2730+ 1.064i10001 -.2683 + 1.056i -.4545 + 1.948i -.4536 + 1.948i -.2684 + 1.056i100001 -.2685+ 1.057i -.4540+ 1.948i -.4540+ 1.948i -.2685+ 1.057ico -.2685 + 1.057i -.4540 + 1.948i -.4540 + 1.948i -.2685 + 1.057iAppendix B. Numerical values of normal form coefficients^ 183Table B.10: Normal form coefficients (Case I) for a = 1,( = .01, Q = 72.Tit C1 C51 -.2247 + 1.814i -.5729 + 1.724i11 -.3288 + 1.801i -.3711 + 1.792i101 -.3468 + 1.797i -.3516 + 1.796i1001 -.3489 + 1.797i -.3494 + 1.797i10001 -.3491 + 1.797i -.3492 + 1.797ioo -.3491 + 1.797i -.3492 + 1.797iTable B.11: Normal form coefficients (Case I) for a = 1,<- = .01. Q = 1007r2.m C1 C51 -.0837+.2139i -.0755 + .1997i11 -.0802+ .2078i -.0791 + .2059i101 -.0798 + 2069i -.0796 + .2067i1001 -.0797+ .2068i -.0797 + .2068ico -.0797 + .2068i -.0797 + .2068iTable B.12: Normal form coefficients (Case I) for a = 1,( = .01, Q =--- 10472.in 10C1 10051 -.1006 + .4180i -.2597+ .3921i11 -.1705 + .4252i -.1926 + .4218i101 -.1803 + .4240i -.1828 + .4236i1001 -.1814 + .4238i -.1817+ .4238i10001 -.1816 + .4238i -.1816 + .4238ioo -.1816 + .4238i -.1816 + .4238iAppendix B. Numerical values of normal form coefficients^ 184Table B.13: Normal form coefficients (Case I) for a = 1,( = .5, Q = 100R-2.in C1 C51 -.06479 + .1288i -.08207 -I- .1111i11 -.07989 + .1154i -.08240 + .1119i101 -.0811 + .1138i -.0814 + .1134i1001 -.0812 + .1136i -.0812 + .1135i10001 -.0812 + .1136i -.0812 + .1136ioo -.0812 + .1136i -.0812 + .1136iTable B.14: Normal form coefficients (Case I) for a = 1,( = .5, Q = 1047r2.in 10C1 10051 -.0904 + .3310i -.4095 + .5270i11 -.2110 + 4417i -.2559 + .4681i101 -.2306 + .4540i -.2357 + .4570i1001 -.2329 + .4553i -.2334 + .4556i10001 -.2331 + .4555i -.2331 + .4555ico .2331 + .4555i -.2331 + .4555iTable B.15: Normal form coefficients (Case I) for a = 1, = .5, Q = 10672.in 10C1 10051 -.01411 + .1185i -.1538 + .3537i11 -.0476 + .2103i -.0476 + .2103i101 -.0546 + .2242i -.0565 + .2278i1001 -.0555 + .2258i -.0557 + .2262i10001 -.0557+ .2262i -.0557 + .2262ioo -.0556 + .2261i -.0556 + .2261iAppendix B. Numerical values of normal form coefficients^ 185Table B.16: Normal form coefficients (Case I) for a = 10-6, = .01, Q = 4000072.in CI C51 -2.298 + 7.327i -6.313 + .8211i11 -4.286 + 6.332i -4.938 + 5.676i101 -4.578 + 6.069i -4.652 + 5.994i1001 -4.611 + 6.036i -4.618 + 6.028i10001 -4.614 + 6.032i -4.615 + 6.031i100001 -4.615 + 6.032i -4.615 + 6.032ioo -4.615 + 6.032i -4.615 + 6.032iTable B.17: Normal form coefficients (Case I) for a = 10-6,C = .01, Q = 10672.in Ci C51 -.0644+ 1.263i -.2487+ 1.232i11 -.1289 + 1.258i -.1314 + 1.257i101 -.1300 + 1.258i -.1303 + 1.258i1001 -.1301 + 1.258i -.1301 + 1.258ioo -.1301 + 1.258i -.1301 + 1.258iTable B.18: Normal form coefficients (Case I) for a = 10-6, = .01, Q = 10872.m 100C1 100051 -.1788 + 12.66i -.4366 + 12.62i11 -.2577+12.65i -.2889 + 12.64i101 -.2710 + 12.65i -.2746 + 12.65i1001 -.2726 + 12.65i -.2730 + 12.65i10001 -.2728+12.65 -.2728 + 12.65ioo -.2728+ 12.65 -.2728 + 12.65iAppendix B. Numerical values of normal form coefficients^ 186Table B.19: Normal form coefficients (Case II) for a = 1 , = .01, .■Q = 71-2, k =1.in 2Lr1/M Rm 103a1 103a21 2.024 786.5 1.975.4116i 3.133 - .9750i101 1.419 660.3 2.802 - .1252i 2.821 - .1260i10001 1.412 659.1 2.814- .0722i 2.814- .0722i100001 1.412 659.0 2.814- .0681i 2.814- .0681i100000001 1.412 658.9 2.814- .0663i 2.814- .0663ioo 1.412 658.9 2.814 - .0663i 2.814 - .0663iTable B.20: Normal form coefficients (Case II) for a =^= .01, () = 7r2, k = 1(continued).772 L2mbiR ab2R ',mkt Lin1)211 -2.597 4.794 .8240 -.6923101 -3.484 3.515 .4655 .467310001 -3.496 3.497 .4657 .4657100001 -3.496 3.496 .4657 .4657100000001 -3.496 3.496 .4656 .4656oo -3.496 3.496 .4656 .4656Table B.21: Normal form coefficients (Case II) for a = 1 , = .01,e1 = 7r2, k = 1(continued).77/, Cl C2 C4 C51 -.2247+ 1.814i -6.969 - 2.731i -12.79 - 4.230i -.5729 + 1.724i101 -.1133+ 1.773i -24.93- 1.101i -23.02- 1.050i -.1135+ 1.773i10001 -.0909 + 1.771i -32.18- 1.290i -31.94 - 1.284i -.0908 + 1.771i100001 -.0886+ 1.771i -33.21- 1.352i -33.19- 1.351i -.0886+ 1.771i108 + 1 -.0883 + 1.771i -33.33- 1.359i -33.33- 1.359i -.0883 + 1.771ico -.0883 + 1.771i -33.33 - 1.359i -33.33 - 1.359i -.0883 + 1.771iAppendix B. Numerical values of normal form coefficients^ 187Table B.22: Normal form coefficients (Case II) for o- = 1, = .01,e? = 7r2, k = 1(continued).in C3R C31 C611 C611 9.188 —4.566 11.40 —10.87101 24.91 —.6175 22.91 —.664410001 32.11 .1286 31.86 .1230100001 33.14 .1968 33.12 .1963108 + 1 33.26 .2044 33.26 .2044oo 33.26 .2044 33.26 .2044

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