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Melnikov's method with applications Chow, Yan Kin 2001

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M E L N I K O V ' S M E T H O D W I T H A P P L I C A T I O N S by CHOW Y A N KIN B.Sc. University of British Columbia, 1998 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 2001 © Chow Yan Kin, 2001 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer-ence and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Abstract This thesis gives a detailed discussion of Melnikov's method, which is an analytical tool to study global bifurcations that occur in homoclinic or heteroclinic loops, or in one-parameter families of periodic orbits of a perturbed system. Basic results of the Melnikov theory relat-ing the number, positions and multiplicities of the limit cycles by the number, positions and multiplicities of the zeros of the Melnikov function are proved. We then give several examples to illustrate the theory. In particular, we use the Melnikov theory to study the exact number of limit cycles in the Bogdanov-Takens system with reflection symmetry. We then extend the first-order Melnikov theory to higher-order and establish some results relating the number, po-sitions and multiplicities of the limit cycles by the number, positions and multiplicities of the zeros of the first non-vanishing Melnikov function. Next, we derive a formula for the second-order Melnikov function for certain perturbed Hamiltonian systems using Franchise's recursive algorithm. Finally, this formula is applied to an example. ii Table of Contents Abstract - ii Table of Contents iii List of Figures iv Introduction 1 Chapter 1. First-order Melnikov theory 5 1.1. Introduct ion 5 1.2. Der iva t ion of the first-order Meln ikov function 5 1.3. First-order Meln ikov theory 9 Chapter 2. Applications of first-order Melnikov theory 16 2.1. Per turbed linear centre 16 2.2. A p p l i c a t i o n to the Bogdanov-Takens bifurcation w i t h reflection symmetry 20 Chapter 3. Higher-order Melnikov theory 45 3.1. Int roduct ion 45 3.2. Higher-order theory 46 3.3. Der iva t ion of the second-order Meln ikov function 49 Chapter 4. An application of higher-order Melnikov theory 62 4.1. Per turbed linear centre 62 Chapter 5. Conclusions and future work 67 Bibliography 69 Appendix A. Derivation of (2.9) by the method of normal forms 71 i i i List of Figures 1.1. Illustration of the Poincare map 7 1.2. The displacement function d(e,fi) defined for e near 0, near a homoclinic orbit 15 2.1. Two hyperbolic limit cycles for system (2.6) with e = 0.3 and /ii = 0.28 35 2.2. A unique non-hyperbolic limit cycle for system (2.6) with e = 0.3 and n\ = 0.3. . . . 36 2.3. No limit cycles for system (2.6) with e = 0.3 and /zi = 0.32 37 2.4. Four limit cycles for system (2.7) with e = 0.00005. , 38 2.5. The partial bifurcation set and the corresponding phase portraits for the system (2.11) with the minus sign 39 2.6. Phase portrait of the system (2.13) with e = 0. 39 2.7. Phase portrait of the system (2.23) : 40 2.8. Trajectory when h>0. 40 2.9. Detailed phase portrait for system (2.23) 41 2.10. Bifurcation set and the corresponding phase portraits for the system (2.11) with the minus sign 42 2.11. The partial bifurcation set and the corresponding phase portraits for the system (2.11) with the plus sign 43 2.12. Bifurcation set and the corresponding phase portraits for the system (2.11) with the plus sign. 44 3.1. Illustration of the Poincare map ; 56 iv Introduction Bifurcation analysis is the study of the changes that occur in the structure of solutions of differential equations as parameters are varied. For example, consider parameterized families of ordinary differential equations i = /(x,/i), x £ 3 ? N , ( 1 ) depending on a parameter fj, 6 3? (or on several parameters fi E SR"), where / depends smoothly on x and /i. Let us denote the solution to the initial value problem consisting of ( 1 ) and the initial condition x(0) = xo by x(t) = <p(t, XQ, U). A S we vary //, phase portraits (collections of solution curves x(t) in W1) of ( 1 ) 'look pretty much the same' except perhaps at certain values of /i where there is a qualitative change in the phase portrait. These values are called bifurcation values. There are mainly two types of bifurcation: local bifurcation and global bifurcation. Local bifurcation is the study of qualitative changes in phase portraits that take place in a neighbourhood of a point. For example, saddle-node, transcritical, pitchfork and Hopf bifurcations are local bifurcations. Bifurcations that are not local are called global. Examples of global bifurcations are heteroclinic (or homoclinic) bifurcations and saddle-node bifurcations of cycles. Global bifurcations can be more complicated than local bifurcations, and some are not completely understood at present (e.g. global bifurcations in the Lorenz system). Before going further, we introduce some important types of orbits that can occur in systems ( 1 ) : equilibrium points, periodic orbits, homoclinic orbits and heteroclinic orbits. A point XQ is an equilibrium point of ( 1 ) if (j)(t,xo,fi) = XQ for all t. In other words, XQ corresponds to the intersections of ( 1 ) with x — 0. A periodic orbit of ( 1 ) is any closed solution curve which is not an equilibrium point. Closed solution curves correspond to periodic solutions, since 4>(-,XQ,II) defines a closed solution curve if and only if for all i 6 fi, <fi(t + T,xo,n) = <f>(t,xo,n) for some T > 0 . The smallest T for which the above equality holds is called the period of the periodic orbit 4>(-,XQ, fj,). 1 We call an equilibrium point or a periodic orbit F stable if for each e > 0 there is a neighbourhood U of T such that for all XQ G U and t > 0, we have dist((f)(t, xo,n),T) < e, where x(t) = <f>(t, XQ,fi) is the solution of (1) satisfying the initial condition x(0) = XQ. A periodic orbit V is called unstable if it is not stable. We call T asymptotically stable if it is stable, and for all points XQ in some neighbourhood V of F we have lim dist((j)(t, XQ, / i ) , T ) = 0 . Before talking about homoclinic and heteroclinic orbits, we define the stable and unstable manifolds of an equilibrium point. The stable manifold of an equilibrium point po is the set Ws{p0) = {x0\ <f>(t, x0,n) -> po as i - » o o } , and the unstable manifold of po is the set W"(Po) = {zo| <p(t,x0,iJ,) -+po ast^-oo}. A point q is a homoclinic point for (1) if there is an equilibrium point po / q such that q G VFs(po) n ^"(po), where Ws(po) and ^"(po) are the stable and unstable manifolds of Po- The orbit of a homoclinic point is called a homoclinic orbit. Thus, a solution curve T is a homoclinic orbit if and only if T C Ws(pa) fl Wu(po). In other words, the solution curve T approaches po both as t —> oo and as i —)• —oo. A point r is a heteroclinic point for (1) if there are two distinct equilibrium points po 7^  r , go 7^  r such that r £ Ws{p0) n Wu(<?o) (or VTs(qo) H W(p0)). The orbit of a heteroclinic point is called a heteroclinic orbit. A solution curve T is a heteroclinic orbit if T C Ws(po) n Wu(qo), i.e., the solution curve V approaches po as t - » 00 and approaches go as t —> —00. Periodic, homoclinic and heteroclinic orbits play an important role in the thesis. We study perturbed Hamiltonian systems in the plane, of the form 2 where H(x, y) is the Hamiltonian function, f(x, y, e, /i) and g(x, y, e, u) are functions depending smoothly on x,y,e,fi, /x = (/if,... , / i n ) 6 5?" and e is a small perturbation parameter. We make two assumptions. The first assumption is that for e = 0, the system (2) has a continuous one-parameter family of periodic orbits. Th of period Th with parameter h belonging to an interval / C fi equal to the total energy along the orbit. The second assumption is that for e = 0, the system (2) has a homoclinic orbit To to a hyperbolic saddle point XQ. If we add some small perturbations (i.e. make e ^  0 but small), what is the behaviour? For example, will the homoclinic orbit persist under perturbation and if so, for which parameter values? Will the homoclinic orbit break as we pass through some parameter values? How many of the limit cycles from the continuous family of periodic orbits are still preserved under perturbation? To answer these questions, we employ Melnikov's method, a global perturbation method due to Melnikov [15] and others, which gives us an excellent tool to study the global bifurcations that occur at homoclinic (or heteroclinic) loops or in a one-parameter family of periodic orbits of perturbed system. To apply Melnikov's method, we need to compute two quantities: 1. The Melnikov function along a periodic orbit, which is used to prove the existence of periodic orbits in system (2) for small e. 2. The Melnikov function along a homoclinic (or heteroclinic) orbit, which is used to prove the existence of homoclinic (or heteroclinic) orbits in system (2) for small e. The Melnikov function along a periodic orbit is very similar to the Melnikov function along a homoclinic orbit. You may ask: Is it always possible to compute the Melnikov function (along the periodic orbit) for system (2)? The answer is no but in many useful cases (as in our examples), the Melnikov function can be either computed explicitly or expressed as a linear combination of complete elliptic integrals. Although the computation of the Melnikov function can be somewhat technical, the benefits are great: we can determine the number, positions and multiplicities of limit cycles of (2) by the number, positions and multiplicities of the zeros of Melnikov function. If the Melnikov function of (2) is identically equal to zero across the continuous band of periodic orbits, a higher-order analysis is necessary. 3 The Melnikov type functions first appear in the 1890 paper [16] by H. Poincare, in the 1963 paper by V. Melnikov [15], in the 1964 paper by V.I. Arnold [2], in the book by A.A. Androdov et al. [1]. It also is given in textbooks, such as the one by Guckenheimer & Holmes [11]. It is therefore difficult to pin down the origins of Melnikov's method. It suffices to say that the idea of computing the displacement function, as well as its partial derivatives with respect to pa-rameters in terms of certain functions along periodic orbits, was used by many mathematicians working on the theory of dynamical systems at various times during the past 100 years. The thesis is organized as follows: In Chapter 1, we derive the (first-order) Melnikov function along a periodic orbit. We then prove several theorems concerning the exact number, positions and multiplicities of limit cycles of (2). We then derive the Melnikov function along a homoclinic orbit and give a theorem which guarantees that a unique homoclinic orbit for (2) exists for some parameters /z. In Chapter 2, we give several examples to illustrate the versatility and power of Melnikov's method. The examples are worked in the context of normal forms [11, p. 365-376] [8, p. 54-83]. In some of the examples, the Melnikov function can be computed explicitly, while in other examples the Melnikov functions are expressed as linear combinations of complete elliptic integrals. We analyze the two-parameter system x = y y = (j,ix + n2y ±x3 - x2y, where (x,y) G fi2, (jui,/i2) G fi2, studied by Takens [23] in his well known 1974 paper. In Chapter 3, we extend the first-order theory to higher-order. In particular, we use Frangoise's recursive algorithm [10] to give us a formula for the second-order Melnikov function for certain perturbed Hamiltonian systems in fi2. Finally, in Chapter 4, I give an example in which the first-order Melnikov function is identically equal to zero and so a second-order analysis is required to determine the number and positions of limit cycles. 4 Chapter 1 First-order Melnikov theory 1.1. In t roduct ion We introduce Melnikov theory in this chapter. With this theory we are able to determine global bifurcations that occur in perturbations of one-parameter families of periodic orbits, or of homoclinic or heteroclinic orbits. Assuming that when e = 0, the system (2) has a one-parameter family of periodic orbits, can we determine which periodic orbits are still preserved under perturbation? Finding the zeros of the Melnikov function, which is defined in Section 2, can allow us to do that. Therefore, a difficult global bifurcation problem is reduced to a reasonably straightforward computational problem. In Section 2, we derive the Melnikov function for (2) and show that in simple cases the periodic orbits correspond to zeros of the Melnikov function. This is a standard result. See, for example, Guckenheimer & Holmes [11] or Andronov et al. [1]. We then present a more general theory relating the multiplicities of periodic orbits to the multiplicities of the zeros of the Melnikov function, using the Implicit Function Theorem [19] or the Weierstrass Preparation Theorem [1, p. 388]. 1.2. Der i vat ion of the first-order Me ln i kov funct ion In this section, we derive a formula for the first-order Melnikov function for system (2). We assume that the functions f(x,y,e,fi) and g(x,y,e,/j,) in (2) depend smoothly on x,y,e,fi. Following Guckenheimer and Holmes [11, p. 184-188], we make the assumptions: Assumption 1.1 For e = 0,Jthe system (2) has a homoclinic orbit VQ : x — 70(i), —00 < t < 5 oo at a hyperbolic saddle point XQ. Assumption 1.2 For e — 0, the system (2) has a continuous one-parameter family of periodic orbits Th : (x,y) = (xh(t),yh(t)) = lh(t), 0 < t < Th of period Th with parameter h e / C fi equal to the total energy along the orbit. We are interested in knowing which periodic solutions give rise to periodic solutions after perturbations. For that purpose we consider a curve E normal to the family Th, parameterized by the value of h for the level curve of the Hamiltonian function H(x, y) = h, and we calculate the Poincare return map1 P(h, e, u) for initial values h in an open set of S. We assume that the return map P(h,e,u) is well-defined on an open set U containing / x {0} x fi™. Then the displacement function is also well-defined on U, and zeros of the displacement function d(h, e, /z) correspond to periodic solutions (see Figure 1.1). Let us call ThtCifi : (^^(i),2/^^(4)) = 7/i,e,/x(*) the trajectory of the system (2) starting at h on E. Then 1 The Poincare return map P(h,e,fi) is a mapping from S into itself. More explicitly, P(h,e,fi) is the value of the Hamiltonian function at the point where the trajectory of (2) first returns to S after starting from the point on S where the value of the Hamiltonian function is h. A periodic orbit corresponds to a value of h such that P(h, e, n) — h. d(h, e, fi) = P(h, e, /i) — h (1.1) + 9(^,£,M(i)'^,liW,0,/i)^-(x/l!£)M(t),yfei£)M(t)) dt + 0{e2), (1.2) 6 £=0 Figure 1.1: Illustration of the Poincare map. where we have expanded / and g in their Taylor series in e, and T(h, e, /*) is time it takes the trajectory I\)(£>AX to first return to. S. Formula (1.2) is exact but involves the orbit I \ £ J J of the perturbed Hamiltonian system (2), which we do not know explicitly. However, for small e > 0, the orbit of (2) differs only slightly from the closed orbit I\ (for (2) when e = 0). Therefore, we approximate by the closed orbit J?/, in order to find an approximation to formula (1.2). Using the smoothness of r/,)£)il with respect to the parameter e, we have T(h,e,fi) = Th+0(e), where T n is the period of the unperturbed closed orbit I\, and xh,tJt) = xh(t) + 0(e), yh,^{t) = yh(t) + 0(e), where (Z/I(*),2/A(£)) are the coordinates of IV Thus we see that Th,e,n lies in an 0(e) neigh-bourhood of 17/, and therefore tends to the closed orbit IV as e 0. This justifies the following 7 first-order approximation to formula (1.2): rT, f±h [ dH d(h,e,n) = e f(xh(t),yh(t),0,n) — (xh(t),yh dH + g(xh (t),yh(t),0,fi) — (xh (t), yh (t)) = e f h[-f(xh(t),yh(t),0,n)yh(t) Jo + g(xh(t),yh(t),0,n) x\(t)\ dt + O(e2), (*)) dt + 0{e2) since xh(t) = dH/dy(xh(t),yh(t)) and yh(t) = -dH/dx(xh(t),yh(t)). Now,.making change of variables dxh(t) = x'h{t) dt, dyh(t) = y'h(t) dt, and using our earlier notation Th : (x,y) = (xh(t),yh(t)) = jh(t), 0<t<Th for the closed orbit of the unperturbed Hamiltonian system (2), we have d(h, e,fi) = e f (g{x, y, 0, n) dx - f(x, y, 0, u) dy) + 0(e2). This leads to the following definition of the Melnikov function M(h,fi), which determines the displacement function d(h, e, u) to 0(e). Definition 1.1 Assume that Assumption 1.2 holds for all h £ I. Then d(h, e, n) = e M(h, u) + 0(e2) as e —> 0, where M(h,n) is the (first-order) Melnikov function for (2) given by M(h, n)= <b (g(x, y, 0, n) dx - f(x, y, 0, /*) dy). (1.3) Remark 1.1 For computational purposes, it is useful to write the Melnikov function M(h,/j,) as rTh r ftw f)H 1 dt. M(h,v.)= / Jo f (xh (t) ,yh(t),0,ii) ~Q^(xh (*), Vh (*)) + g(xh (t), yh (t), 0, a) — (xh (t), yh (t)) (1.4) 1.3. First-order Melnikov theory In Section 1, we know that the limit cycles correspond to the zeros of the displacement func-tion, and that the first-order contribution (in e) to the displacement function is essentially the Melnikov function. In order to show that limit cycles correspond to the zeros of the Melnikov function, and to know more 'information' about the limit cycles (i.e. uniqueness, etc.), we need to assume some non-degeneracy condition. For example, the simplest is ^(ho,/j,o) 7^  0, in which case we can apply the implicit function theorem to obtain a unique, hyperbolic limit cycle for small e. This result will be proved in this section. But before doing that, we need to define hyperbolicity and multiplicity of limit cycles. A limit cycle Th is hyperbolic if dP ^ ( W ) ^ i , where P(h, e,p) is the return map in Section 1.2, and h is the point on S where Th intersects. Hyperbolicity is important since it tells us about the stability of the limit cycles. For example, if 0 < tj£(h,e,(i) < 1, then the limit cycle is stable. If §£(/i, e,/i) > 1, then the limit cycle is unstable. We call a limit cycle non-hyperbolic if it is not hyperbolic. A non-hyperbolic limit cycle tells us nothing about the stability of the limit cycle (since linearization is not sufficient to determine stability). A limit cycle Fh is called a limit cycle of multiplicity k if dd d(k~^d d(h,e,fj.) = —(h,e,fi) = ••• = ^ - 1 ) CM,l*) = °> and If k = 1, then is called a simple limit cycle. We now prove a Theorem [1] giving the simplest conditions under which the perturbed Hamiltonian system (2) has a unique, hyperbolic limit cycle. 9 Theorem 1.1 Assume that Assumption 1.2 holds for all h G I. If there exists a ho E I and a fj-o G 3?n such that M(ho,no) = 0 and ^(/i 0,/io) / 0, then for all sufficiently small e ^  0, the system (2) has a unique, hyperbolic limit cycle T e which tends to the periodic orbit when e —» 0. Proof: Under Assumption 1.2, d(h,0,fi) = 0 for all h G I and fj. G 3?n. Define the function d(h,e,iM)) if e ^ 0 [ f (M,Mo) ife = 0 so that d(/i,e,/i0) = eF(M). (1-5) By Definition 1.1, we have F(h,e) = M(h,iA0) + O(e). Thus, F(h0,0) =M(/»o,/i 0) = 0, and - ( V O ) ^ — ( ^ , . 0 ) ^ 0 . Then, by the implicit function theorem [19], there exists a 5 > 0 and a unique function h = h(e), defined for |e| < 5, such that h(0) = ho and F(h(e),e) = 0 for all |e| < 5. It follows from the above definition of F(h,e) that for sufficiently small e, d(h(e), e, zi0) = 0 and for sufficiently small e ^ 0, dd/dh(h(e),e,iJ,o) / 0. Therefore, for sufficiently small e / 0, there is a unique isolated limit cycle F£ of (2) cutting the section S at the point /i = h(e). Using (1.1), and the fact that dd/dh(h(e),e,/j,o) i1 0, it follows that dP/dh(h(e), e, /in) 7^  1 and so the limit cycle Te is hyperbolic. Since h(e) = ho + 0(e), this limit cycle tends to the cycle Th0 as e —>• 0. • 10 In a more degenerate situation, for example, if M(/in,/io) = dM/dh(ho,Ho) = 0, then assuming d2M/dh2(ho,/j,o) ^ 0 and dMjd\i\(ho, /in) ^ 0, we can use the Weierstrass preparation theorem to obtain a unique non-hyperbolic limit cycle of multiplicity two for e ^  0. This result will be proved next. Theorem 1.2 Assume that Assumption 1.2 holds for all h G I. Then if there exists a ho £ I and a / J O € S" such that M(h0,no) = ^-(ho,f-o) = 0, d2M dM -^2-(/i0,Mo) ^ 0, — (h0,fJ.o) ^ 0 for some j = 1,... ,n, it follows that for all sufficiently small e, there are functions h(e) = ho + O(e), n(e) = /xrj + O(e) such that for sufficiently small e ^  0, the system (2) has a unique non-hyperbolic limit cycle of multiplicity two which tends to the periodic orbit T/i 0 as e —> 0. Proof: First we let no = (/i?,/i2? • • • ,/^n)- Without loss of generality, we assume that j = 1 (i.e. dM/dpi(ho,no) / 0). Under Assumption 1.2, d(h,0,n) = 0 for all h G I and /x £ W1. Define the function f ife^O { fe(h,0,fi) ife = 0 so that d{h,e,n) = eF(h,e,ii). (1.6) Also, F(h0,0,no) = M(ho,Ho) = 0, dF 8M — (ho,0,no) = -Q^(ho,Ho) = 0, d2F d2M -^2-(/jo,0,/io) = -^"(^0 , / io ) ^ 0, 11 dm dm Therefore, by the Weierstrass preparation theorem [1, p. 388, Theorem 69], there exists a S > 0 such that F(h,e,n) = [{h-ho)2 + A1(e,n){h-h0) + A2(€,n)]^(h,e,fi), (1.7) where Ai(e,fi), A2(e,/i), and $(/i,e,/i) are defined for |e| < S, \h — ho\ < 6, |/i — /in| < S; Ai(0,/i0) = A2(0,/i0-) = 0, $(/i0,0,/i0) 7^  0, and f^f(0,/i0) ± 0 since ^(/i0,0,Mo) # 0. It follows from (1.7) that dF — (h,e,ii) = [2(h-ho) + Ai(e,n)]$(h,e,n) <9<I> + [{h-h0)2 + A1{e,n)(h-h0) + A2(e,n)] — (h,e,fi), (1.8) and <92F 9$ -^(h,e,n) = 2^{h,e,ti) + 2[2(h-h0)+A1(e^)] — (h,e,li) + [(/» - /i0) + Ai(e, /*)(/i - /*,) + A2(e, »)}-Qtf(h, ^ A*)- (1-9) If 2(/i-/&0) + Ai(e,/z) = 0 and (/i - h0)2 + ^i(e, n)(h - h0) + A2(e, /i) = 0, it follows from (1.7), (1.8) and (1.9) that (2) has a multiplicity two limit cycle. Therefore, we set h = ho — Ai(e,/i)/2 and find from (1.7) that F(ho — A\(e, fi)/2, e, /x) = 0 if and only if the function B(e,fj,)^-i^(€,/i)+A2(e,/i) = 0 (since, by continuity, e,/i) / 0 for small |e|, |/i — /io|, |/J —• /^ o|)- Now, 5(0,/i0) = -|A?(0,/x0) + A2(0,/i0) = 0 (since Ai(0,/io) = A2(0,/xo) = 0) and f^ -(0,/io) = f^(0,/i0) 7^ 0 (9/ii 9/ii (since 8F/diJ.i(ho,0,/io) 7^ 0). By the implicit function theorem, there exists a (5 > 0 and a unique function /ii = /ii(e,/i2,... ,/in), defined for |e| < S, | / i 2 — / i 2 | < 5, . . . , | / i n —Mnl < >^ s u c n 12 that/x? = m(0,(4,... ,/z°) and5(e,/ii(e,/i2,... , Mn), M2, • • • ,/in) = 0 for |e| < 6, \/J,2~l4\ < ^ • • • , | / i n - A*nl < 5 - F o r l e l < w e d e f m e A * ( € ) = (A*l_(e, A*°> • • • . A *n ) . A * 2 . " - i A * n ) - T h e n > ii(e) = /io + 0(e) and (2) has a unique multiplicity two limit cycle T£ through the point h(e) = h0-A1(e,fi(e))/2 (1.10) on S. Using (1.6) and the fact that dF/dh(h(e),e, fi(e)) — 0 with h(e), fi(e) defined above, we have dd/dh(h(e),e,[i(e)) = 0. Now, using (1.1), we immediately get dP/dh(h(e),e,/i(e)) = 1 and so Te is non-hyperbolic. Finally, by continuity with respect to initial conditions and parameters, it follows that Te tends to the cycle Th0 as e -> 0 since Ai(0,/in) =0. • Remarks 1.1 1. The proof of Theorem 1.2 in fact establishes that there is an n-dimensional surface /ii = /ii(e, 112, • • • ,lin) through the point (0,/in) € $ln+1 on which (2) has a non-hyperbolic multiplicity-two limit cycle for sufficiently small e ^ 0. On one side of the surface where B(e,p.) < 0, the equation F(h(e),e,u) = 0 with h(e) given by (1.10) has two real solutions given by and on the other side where B(e,fi) > 0, F(h(e),e,/i) has no real solution; i.e., system (2) has two limit cycles if B(e,[i) < 0 and no limit cycle if B(e,fi) > 0. It follows from (1.8) that if h±(e,fi) are given by (1.11), then dF/dh(h±(e, fi),e, fi) ^ 0 respectively. Using (1.6), we have dd/dh(h±(e,fi),e,n) ^ 0. Now, using (1.1), we immediately get dP/dh(a±(e,/z),e,/i) ^ 1; i.e. system (2) has two hyperbolic (unstable or stable) limit cycles if B(e,fi) < 0. Therefore, the system (2) experiences a saddle-node bifurcation of periodic orbits as we cross this surface (see p. 22 for a description of saddle-node bifurcation). 2. IfM(h0,ii0) = dM/dh(ho,ii0) = ••• = d^M/dh^^tio) = 0, d^M/dh^(h0,p0) ^ 0, dM/d/ij(ho,fio) ^ 0 for some j — 1,... ,n, then it can be shown that for sufficiently small e ^  0, the system (2) has a unique non-hyperbolic limit cycle of multiplicity k which tends to the periodic orbit Fh0 as e —> 0. 13 3. If d^M/dhW(h0,no) = 0 for all k — 0,1,2,..., then dd/de(h,0, fi0) = 0 for all h e I and a higher-order analysis in e is necessary in order to determine the number, positions, and multiplicities of the limit cycles for small e ^  0. This type of higher-order analysis is discussed in Chapter 3. Besides the global bifurcation of periodic orbits from a continuous band of cycles, there is another type of global bifurcation that occurs in systems in fi2, namely, the homoclinic loop (or heteroclinic loop) bifurcation [11, 24]. The Melnikov theory for (2) also gives us explicit information on this bifurcation. Similar to what was done before, it can be shown (see Wiggins [24] for more details) that the distance d(e,fi) between the saddle separatrices Tse^ and r £ / i of (2) along a section £ to the homoclinic orbit To (in Assumption 1.1 above) at the point 70 (0) = an satisfies (see Figure 1.2) d(e,fi) = eM(/z) + 0(e 2), where M(/x) = ^^( 7 o(*),0, / i)^(7o(*)) + /(7oW,0,A*)^(7o(*))) dt (1.12) is the Melnikov function for (2) along the homoclinic orbit To: (xo(t),yo(t)) — jo(t), —00 < t < 00. From (1.4) and (1.12), we see that the Melnikov function along the periodic orbit is very similar to the Melnikov function along the homoclinic orbit To- The following theorem [24] gives us conditions under which the system (2) has a unique homoclinic orbit. Theorem 1.3 Under Assumption 1.1, if there exists a fiQ G fi™ such that M(/J,Q) = 0 and !^-(/io) 7^  0, then for sufficiently small e^O, there is a function fi(e) — //n + 0(e) such that the system (2) has a unique homoclinic orbit Fe which tends to the homoclinic orbit TQ as e —> 0. Proof: Under Assumption 1.1, d(0,n) F{e,u) = 0 for all fx € fi™. Define the function ' 4Ml i f e ^ o k f(0,/i) if 6 = 0. 14 £ = 0 Figure 1.2: The displacement function d(e,/i) defined for e near 0, near a homoclinic orbit. Also, F(0,/i 0) = M(/io) =0, 0M —-(0,/xo) = ^—(Mo) 7^  0. By the implicit function theorem, there exists a 8 > 0 and a unique function m = A*i (e, H2, • • • , (J>n) such that /i? = //!(0,/i2, • • • , M n ) a n ( i F { e ^ i ( e ^ 2 , • • • ,Vn),V2> • • • , M n ) = 0 for all |e| < 5, \H2~ l4\ <S, . . . ,\Hn - M n l < F o r l e l < <*> w e d e f i n e Me) = (A*l(e»A*2>-- - , M n ) » ^ 2 ' - - - i/O-Then, /j(e) = /in + 0(e) and (2) has a unique, homoclinic orbit F£. It then follows from the uniqueness of solutions and from the continuity of solutions with respect to initial conditions that the homoclinic orbit Te tends to To as e -> 0. • 15 Chapter 2 Applications of first-order Melnikov theory In this chapter, we give two examples to illustrate the usefulness of the Melnikov theory de-veloped in Chapter 1. The first example is the (2n + l)-th degree perturbed non-hyperbolic linear centre. The problem is determine the maximum number of limit cycles. To study this analytically, we compute the Melnikov function. The number, positions and multiplicities of the zeros of the Melnikov function is related to the number, positions and multiplicities of the limit cycles using Theorem 1.1 or 1.2. This example generalizes Theorem 76 in [1, p. 414]. The second example is the Bogdanov-Takens bifurcation with reflection symmetry. This occurs in the family of vector fields equivariant with respect to rotation by 7r in the plane which have an equilibrium with a double zero eigenvalue. The normal form of this bifurcation is x = y y = n\x + + ®x3 + bx2y, (o^O, b^O). This was first proved independently by Bogdanov [6] and Takens [23]. We study the system in a neighbourhood of parameter space near (/zi,/Z2) = (0,0), with emphasis on proving the existence of periodic orbits, homoclinic or heteroclinic orbits corresponding to different values of the parameters. 2.1. Perturbed linear centre In this section, we use the Melnikov theory developed in Chapter 1 to study the number, positions and multiplicities of the limit cycles that occur in a (2n + l)-th degree perturbed non-hyperbolic linear centre of the form (2.2). One reason this example is of particular interest 16 is that it is useful in the study of Hilbert's 16th problem, which asks for an upper bound on the number of limit cycles for nth degree polynomial systems in terms of the degree n [12]. For n = 1, the maximum number of limit cycles is 0 since linear systems do not have any limit cycles. However, even for the simplest class of nonlinear systems (i.e. n = 2), the maximum number of limit cycles has not yet been determined [4, p. 283-284]. There are quadratic systems with as many as four limit cycles [20]. However, for a perturbed non-hyperbolic linear centre of the form x = y + ef(x, y, e) y V 1 (2.1) V = .-x + egix^^), where / and g are polynomials of x, y with coefficients depending smoothly on small e, some useful results concerning the maximum number of limit cycles have been obtained [5, 21, 26]. Bautin [5] proved that the maximum number of limit cycles in a quadratically perturbed non-hyperbolic linear centre is three. He also proved that this maximum number can only be attained in a sixth- or higher-order analysis in e. That is, the &th-order Melnikov function, where k > 6, has at most three zeros. However, when n = 3 (i.e. a cubically perturbed linear centre), we still don't know the maximum number of limit cycles. We just know that this number is at least 11 [26]. One reason that the determination of the maximum number of limit cycles in system (2.1) is difficult is to find at which order k the maximum number of limit cycles will 'stabilize'. Therefore, one way to make this problem easier to study is fix the order k. Since we know the Melnikov function when k — 1, we can determine the maximum number of limit cycles from a first-order analysis of an n-th degree perturbed non-hyperbolic linear centre. This motivates us to consider the perturbed system x - y + e(n±x + n2x2 + h Li2n+ix2n+l) (2.2) y = -x, where \i — (/xi, H2, • • • , M2n+i) S 3? 2 n + 1. For e = 0, (2.2) is a Hamiltonian system with H(x, y) = (x2 + y2)/2; it has a one-parameter family of periodic orbits Xh(t) — V2hcost, yh{t) — —V2hsmt, 17 with the parameter h G (0, oo) being the energy level along the periodic orbit. The Melnikov function is given by r2ir M(h,n) = Jo dH dH f(xh(t),yh{t),0, ii) — (xh(t), yh{t)) + g{xh(t),yh(t),0, p) — (xh(t),yh(t)) dt = / xh(t){mxh(t)+fi2x2h(t) + --- + fi2n+lx2hn+l{t))dt. Jo Using the fact that f2w xkh{t) dt = [2h)kl2 f2w cosk tdt = 0 for k is odd, we get r2n M(h,fi) = 2h (iiicos2i + 2//3/icos4i + ••• + fj,2n+1(2h)n cos2n+21) dt = ^+l^---+"M2:ti)i2hr)' (2-3) where (£) is the binomial coefficient n \ n ! ^kj k\(n-k)V Since h = a2/2, where a G (0, oo) is the positive rc-axis intercept of the unperturbed periodic orbit, (2.3) becomes ,2 / Ml , 3 . . . „ . 2 , , M2n+i (2n + 2\^2n M(a, M ) = 2™° + + - + I ) a * ) . (2.4) Using Theorems 1.1 and 1.2 we obtain the following result: Result 2.1 For sufficiently small e / 0, the above system has at most n limit cycles. Fur-thermore, for e 0, it has exactly n hyperbolic limit cycles asymptotic to circles of radii TJ, j = 1,... ,n as e —>• 0 if and only if the nth degree equation in a2 Mi 3. 2 / i 2 n + i (2n + 2\ 2 n T+gMaa +--- + ^ ^ n + 1J« =0 (2-5) has n positive roots a2 = r j , j = 1,... , n . Now, we illustrate with a concrete example with n = 2. Consider the system £ = y + e(nix - 2x3 + 3x5) (2.6) y = -x. 18 Using (2.4) with M i = M i : M3 = -2, fi5 — 3 and all other m = 0, we get M ( o ^ ) = 2 ™ 2 ( f - | « 2 + ^ 4 ) -We need to solve 15 Solving, we have or So we have the following results: Result 2.2 1. 7/0 < /ii < 0.3 and e ^ 0 is sufficiently small, (2.6) has exactly two hyper-bolic limit cycles asymptotic to circles of radii as e —> 0. 2. For all sufficiently small e, there is a function n\(e) = 0.3 + O(e) such that (2.6) has a unique limit cycle of multiplicity two, asymptotic to the circle of radius r = v2/5 as 3. If M i > 0.3 and e ^ 0 is sufficiently small, (2.6) has no limit cycles. The results above are illustrated by the numerical computations shown in Figures 2.1, 2.2, 2.3. The result obtained above (c.f. Result 2.1) allows us to construct systems with as many limit cycles as we like. For example, suppose that we wish to find a system of the form (2.2) with exactly four limit cycles asymptotic to circles of radius r = l , r = 2, r = 3 and r = 4. To do this, we simply set the polynomial (a2 — l)(a2 — 4)(or2 — 9)(a2 — 16) equal to the polynomial in (2.5) with n = 4. So we have e->0. (a-,2 l)(a2 -4) (a2 -9) (or ,2 1.6) 19 or 3 5 35 63 a 8 - 30a6 + 273a4 - 820a2 + 576 = ^ + -/i3a2 + j ^ a 6 + Equating coefficients, we have (ig = 256/63, (17 — —768/7, /J5 = 4368/5, (i% = —6560/3 and (i\ = 1152. For e^O sufficiently small, Result 2.1 implies that the system x = y + e (1152a; - ^™x3 + ^ x 5 -7-fx7 + ?§x9) y = -x has exactly four limit cycles asymptotic to r=l, 2, 3, 4 as e —>• 0. The four limit cycles for this system with e = 0.00005 are shown in Figure 2.4. 2.2. Application to the Bogdanov-Takens bifurcation with re-flection symmetry We investigate the parameterized family of vector fields x = y (2.8) y = mx + Li2y ± xz - x2y, where / J =.((11,112) a r e parameters. This system possesses a reflection symmetry under (x, y) i->-(—a;, —y). In other words, the vector field is equivariant with respect to a rotation in the plane by 7r. This bifurcation with reflection symmetry occurs frequently in applications. In general, the symmetry occurs due to the geometry of the system or to assumptions made for the model. When such a system undergoes a bifurcation, it is a general principle that the presence of symmetry in the system modifies the generic behaviour that would be expected in a system with no symmetry present (Arnold [3]). Moreover, people have studied the effect of small deviations from symmetry in the system (2.8), which appear in some models of chemical reactors [7, 17]. This is necessary in order to give a theoretical interpretation of experimental data like those of [17]- . We will focus on the most difficult parts of the analysis of the system: 1. The determination of the exact number and positions of the limit cycles in the system. r 20 2. The determination of the parameter values for which a homoclinic (or heteroclinic) orbit exists. However, in order to apply the Melnikov theory developed in Chapter 1, we need to reduce the system (2.8) to a perturbed system in the form of (2) by rescaling the variables and parameters. This will be done in detail in this section. The Bogdanov-Takens bifurcation with reflection symmetry results from unfoldings1 of the normal form x = y (2-9) y = ax3 + bx2y. In the Appendix, we see that this normal form is obtained from a generic system with reflection symmetry at an equilibrium point with a double zero eigenvalue. Without loss of generality, we may assume the equilibrium point is the origin, and the vector field is of the form , 0 1 \ x= ( x + F3(x) + 0(\x\5), 0 0 / where ^(x) is a homogeneous polynomial of degree (exactly) 3. In [23], Takens proves that all possible types of dynamical behaviour that can occur in C°° perturbations of the system (2.9) with reflection symmetry are determined by the unfolding of (2.9) given by x — y (2.10) y = n\x + M22/ + ax3 + bx2y. By rescaling the variables (i.e. x -» (—\a\^/b)x, y -» (\a\%/b2)y, fii —> (\a\2/b2)/j,i, \i2 -> (—|a|/6)/i2, t —> (—6/lal)*)> the system becomes x = y (2.H) y = \ixx + fi2y ± x3 - x2y. 1system (1) is caled an unfolding 'of the vector field fo(x) if it is embedded in a parameterized family of vector fields (1) with f(x,n0) = /oOc)-21 Note that the case with the plus sign is only possible if a > 0, and the case with the minus sign is only possible if a < 0. Also, note that time reversal occurs if b > 0 (that is why we do not need to consider the system (2.10) when b > 0). Before going on any further, we need a dictionary that establishes 'names' for the local bifur-cations which occur in the Bogdanov-Takens system. The local bifurcations are well known and discussed more fully in [11, 24]. In the following, we give the list of these (with a brief description of each) which occur in the system (2.11). 1. Saddle-Node Bifurcation: The normal form for a saddle-node bifurcation in 5R2 is x — a — x2 y = ay, (a ^ 0). This bifurcation describes the simultaneous creation or annihilation of a pair of equilib-rium points as the parameter a varies. One equilibrium is a saddle and the other is a stable or unstable node. The saddle-node bifurcation is sometimes called the fold bifurca-tion. Although this bifurcation does not occur in the system (2.11), we need this notion in order to understand the saddle-node bifurcation of periodic orbits, which does occur in system (2.11). 2. Pitchfork Bifurcation: The normal form for a pitchfork bifurcation in SR2 is x = ax ± x3 V = ay, (a ^  0). Note that the above system possess a reflection symmetry under (x,y) H-> (—x, —y) (thus the origin must always be an equilibrium point). This bifurcation describes the creation or annihilation of a pair of new equilibria (for the parameter a on one side of 0) from an existing one (i.e. the origin). By symmetry, each equilibrium of the pair of new equilibria must be of the same type. 3. Hopf Bifurcation: The normal form for Hopf bifurcation in fi2 is x = — y + x(a ± (x2 + y2)) y =. x + y(a±(x2 + y2)), 22 or in polar coordinates r = ar ± r 3 , (r > 0) 9 = 1. We see that a family of periodic orbits emerges from an equilibrium (i.e. the origin) as the eigenvalues of the equilibrium cross the imaginary axis at a = 0. We are now in good shape to study system (2.11) in detail. Since this thesis focuses on global bifurcations and Melnikov theory, we just state (without proofs) the local bifurcations which occur in system (2.11). Those who are interested in the details of the local bifurcations (e.g. how to prove the existence of Hopf bifurcation) are encouraged to look at [11, 24]. First, consider the case with the minus sign in system (2.11). For Hi < 0 there is only one equilibrium point at the origin, and it is a hyperbolic sink for H2 < 0, a hyperbolic source for H2 > 0 and a non-hyperbolic linear centre for H2 = 0. For HI > 0 there are three equilibrium points, a hyperbolic saddle at the origin, and hyperbolic sources at (i^/Tii,0) for H2 > / Z I , hyperbolic sinks at (iyZ/JT,0) for /j,2 < HI, and non-hyperbolic linear centres at (±^/H~I,0) for /j,2 = Hi-There is a pitchfork bifurcation at points on the /i2-axis with H2 / 0. For /ii < 0 there is a supercritical Hopf bifurcation at points on the pi-axis, and for HI > 0 there is a subcritical Hopf bifurcation at points on H2 = Hi- The partial bifurcation set and phase portraits is shown in Figure 2.5. We know that for all n\ > 0, the unstable limit cycles are generated (via Hopf bifurcation) at fi2 < fx\ (fi2 close to ni). What happens to the unstable limit cycles as /J,2 continually decreases from Hi (HI > 0 fixed)? Do the limit cycles exist forever? Later in this chapter, we prove that this is actually not the case. In fact, the unstable limit cycles expand monotonically as H2 decreases from Hi until they intersect the saddle point at the origin and form a 'double' homoclinic orbit at some parameter value HI — MMI)I i-e-> there exists a homoclinic loop bifurcation curve in the /izi/i2-plane given by Hi = MMI) f° r Mi > 0- To prove the existence of the homoclinic-loop bifurcation curve requires the use of the Melnikov theory developed in Chapter 1. However, since the system (2.11) is not of the form (2), we use the rescaling of 23 variables and parameters x = eu, y = e2v, ' n\ = e 2 7, /i2 = e 2 A, t—¥t/e, (2.12) given by Takens in [23], in order to reduce the system (2.11) to a perturbed system (i.e. (2)) to which the Melnikov theory developed in Chapter 1 applies. Substituting the rescaling trans-formation (2.12) into the system (2.11), we get where we have set 7 = +1 (so that /ii > 0 as on p. 373 of [11]). The system has three equilibria: (0,0), (1,0) and (—1,0). Linearization shows that (0,0) is a hyperbolic saddle point, and that (±1,0) are hyperbolic sinks if A < 1, hyperbolic sources if A > 1 and non-hyperbolic linear centres if A = 1. For e > 0, there is a subcritical Hopf bifurcation at these points on A = 1. For e = 0, (2.13) is a Hamiltonian system with Hamiltonian H(u,v) = v2/2 — u2/2 + u4/4. The phase portrait with e = 0 is given in Figure 2.6. Note that the 'double' homoclinic orbits TQ correspond to the level set H(u, v) = 0. Due to the reflection symmetry, we only need to compute the Melnikov function M ( A ) along TQ". Before doing so, we need to find an expression for the homoclinic solution 7Q"(*) (i.e. TQ : (uQ(t),vo{t)) = 7Q~(*))- Since corresponds to the level set H(u,v) = 0, this is represented by u v (2.13) v = u — u3 + e(Xv — u2v), V = ±u\ 1 —— so Solving the above differential equation, we get V2 cosh(t)' Also. V = U = VQ 24 therefore + n\ - ( ^ - /o t anh(*A 7 0 [ t ) ~ [coSh(tY V cosh(t) j' We now use (1.12) to compute the Melnikov function. /OO / flTJ QTJ \ ^(7otW,0,A)^(7o+(t)) +/(7o+W,0,A) —(7o+(*))J /oo (\v20(t)-u20(t)v2(t))dt -oo y_oo V cosh2(t) cosh4(t)/ 4 16 _ 3 A ~15' where the integral above can be evaluated using an integral table or a symbolic computation program such as Maple. We see that M(A) = 0 if and only if A = | , (2.14) therefore, according to Theorem 1.3, for all sufficiently small e ^  0, there is a function A(e) = 4/5 + 0(e) such that the system (2.13) with A = A(e) has two homoclinic orbits Tf which tend to TQ as e -> 0. We next consider the difficult question of the exact number of periodic orbits that persist for sufficiently small e ^  0. The proofs for these are essentially due to Carr [8] and Cushman and Sanders [9]. The main ideas are to compute the Melnikov function, which is expressed in terms of elliptic integrals, and then use Picard-Fuchs analysis to determine the number of zeros of the Melnikov function (e.g. the number of limit cycles) corresponding to different values of the parameters. Now, using (1.3), the Melnikov function along the periodic orbit 7/1 (t) for system (2.13) is given 25 by M(h,X) = f (g(u,v,0, X)du - f(u, v,0, \)dv) JTh = ® (Xv — u2v)du = A ® vdu— 0 u2v du / r h ./rh = Xh(h) - I2(h), where h(h) = (p vdu, 12(h) = f u2v du. • h Let R(h) = I2(h)/Ii(h), then M(h,X) - 0 if and only if the function M(h, A) = A - R(h) (2.15) vanishes (since by Green's Theorem, h(h) = (f v du = ff dudv > 0, h> —\, Jrh JJ Q(h) 4 where Q(h) is the region surrounded by r^). In what follows, we determine the number of zeros of the function M(h,X) for h G (—1/4,0) and h G (0,00) respectively. In order to do so, we derive and analyze the Picard-Puchs system (the system of differential equations that h(h) and I2(h) satisfies) and the Riccati equation (the differential equation that R(h) satisfies). The analysis is similar to the ones given in Carr [8] and Cushman and Sanders [9], but is given in more detail. Note that the interior periodic orbits correspond to the level sets of the Hamiltonian H(u,v) = h for h G (—1/4,0) and the exterior periodic orbits correspond to H(u,v) = h for h G (0,00). The "double" homoclinic loops correspond to h = 0. Considering v as a function of u and h, and differentiating H(u, v) = h with respect to h, one obtains I =l- <"«> oh v 26 Also, differentiating H(u, v) = h with respect to u, one obtains v ^ - u + u3 = 0. (2.17) au Multiplying (2.17) by uk/v and integrating by parts along T^, we get the following identity: - <t> du+ <p du = k<p uk~lvdu. (2.18) Jrh v JTh v JTh Now, using (2.16) and the identity (2.18) with A; — 1, we get dh h'dh h i ^ If , 1 / t i 2 , 1 / K 4 - (b vdu <p —du + — <b —au 2 JVh 2 j T h v 4 JTh v - I — 0) vdu+ <f> —du  Jr. * Jrh v •* JTh \ h - \ l h 1 + ltrh-dU 1 _ Idh l / r r ",2 2 1 ~ 2 dh + 4 3A _ Idh 4 Adh' (2.19) Similarly, using (2.16) and the identity (2.18) when k = 1 and 3, we get / i ^ 2 - = /i — du dh JTh v If 1 f u4 1 f u6 = 5/2-K/i+f)+K3/2+i'Miu+£v'i") Taking into account (2.19) and (2.20), we arrive at the following lemma: Lemma 2.1 (Picard-Fuchs system) The integrals h(h) and h(h) satisfy the following sys-tem of differential equations: h{h+\)I[ = {\h + \ ) h - ^ h ( 2 2 1 ) h{h + \)I'2 = -\hh + \hh. 27 Now we can easily derive the Riccati equation that R(h) satisfies. Using (2.21), •4h + 4 f t A ) - U , ' + 4 ) A + 16if + + 1 ) * + A * - ^ 2 + ( l A - i ) f l - i f t -Therefore, we have the following lemma: Lemma 2.2 (Riccati equation) The function R(h) satisfies the Riccati equation Ah(Ah + 1)R' = 5R2 + {8h - A)R - Ah. (2.22) We can also write (2.22) as a system with respect to (h,R): h = Ah(Ah + l) V ; (2.23) R = hR? + {8h-A)R-Ah. The vector field is sketched in Figure 2.7. The system (2.23) has 4 equilibrium points: So = (0,0), N0 = (-1/4,1/5), Si = (-1/4,1), Ni = (0,4/5). Linearization at the equilibrium points shows that SQ and Si are hyperbolic saddle points, while No and Ni are hyperbolic (degenerate) nodes, respectively stable and unstable. Note that the sets {h = —1/4} and {h = 0} are invariant sets. Now, we need to determine which 'points' R(h) actually passes through. This is done in the following lemma. Lemma 2.3 (a) lim f t_ >_i R(h) = 1. (b) R(0) = 4 5 ' Proof: (a) Recall that R{h) = I2(h)/Ii{h), where Ii(h) = (p vdu, I2(h) = <p u2vdu. 28 It is easy to see that lim h(h) = lim I2(h) = 0. h^-\ h-+-\ Now, using the Green's Theorem, we have lim R{h) = lim hr+-\ h->-±h{h) = lim ^ — h ^ - \ f r h v d u = hm ry ' —. (2.24) J J Q ( h ) d u d v By applying the Mean Value Theorem for double integrals to (2.24), we have lim R(h) = lim u^(h) 4 4 for some points (uo(h),vo(h)) in Q(h). As h ^ —1/4, Q{h) will 'shrink' to the points (±1,0) and so lim R(h) = 1. (b) Recall that H = 0 corresponds to "double" homoclinic orbits for system (2.13) when e = 0. Now, using (2.15) with h = 0, we have M(0,A) = A-i?(0). We see that M(0, A) vanishes if and only if A = R(0). Using the fact that A = 4/5 (i.e. (2.14)), the result follows. • Using the above lemma, it follows that the graph of R(h) is the stable manifold of S\. It joins Si to the node Ni (see Figure 2.7). We need to find the properties that R(h) has. This is summarized in the following lemma. L e m m a 2.4 The function R(h) has the following properties: (a) R'(h) -> -1/2 as h -> -1/4, and R'(h) -» -oo as h -4 0. 29 (b) R(h) - » +00 as h ->• +00. (c) R(h) > 1/2 forh>0. (d) R'(h) has a unique zero for h = h* > 0, is negative for —\<h<h* and positive for h> h*. (e) R"(h*) > 0. That is, R(h) attains the minimum value at h = h*. P r o o f : Some of the proofs given below are similar to the ones given in Carr [8], but with more explanations and clarity. (a) Consider just the interval h G [—1/4,0]. Since system (2.23) is symmetric with respect to the point (—1/8,1/2), it is sufficient to show that R'(h) —> —1/2 as h -> 0 with R(0) = 0. Let R(h) = Ch + 0(h2). Substituting R{h) and its derivative into (2.22), we easily get C = —1/2 and the result follows. The second part of proof is straightforward. (b) Without loss of generality, we assume h > 0. Recall that ,4 du (2.25) (by symmetry), where £ = £ ( / i ) is the unique positive root of (2.26) (see Figure 2.8). Let u = £z in (2.25), we get h Similarly, 30 Now, let g(z) = y / | ( l - z 4 ) + ( z 2 _ 1 ) ) for 0 < z < 1. The function g(z) defined, above has two critical points z = 0 and z = l/£. It is easy to see that z = 0 is the (local) minimum and z = l /£ is the global maximum. Therefore, p(.z) < g(l/() for 0 < z < 1. Since h —> oo if and only if £ - » oo (see (2.26)), we have ii < ki£3 for some positive constant k\. It is easy to obtain that I2 > k2£b for some positive constant k2. So we have lim R(h) = lim = +°°-(c) Recall that system (2.23) has four nullclines: h = 0, /i = —1/4 and the two branches of the hyperbola 5i?2 + (8/1 — 4)i? — 4h = 0. Let us denote these two branches by R(h) and i?(/i) (see Figure 2.9). Note that the phase plane is divided into nine different regions by the invariant sets {h = 0}, [h = —1/4} and the two branches R = R(h), R = R(h). The vector field is vertical on the two invariant sets and the vector field is horizontal on the two branches. Using the fact that R(—1/4) = 1 and differentiating the equation 5.R2 + (8h — 4).R — Ah = 0 with respect to h and set h = —1/4, we easily find that '^(—1/4) = —1. Similarly, using the fact that j^ (0) = 4/5 and differentiating the equation bR2 + (8h — 4)R — Ah = 0 with respect to h and set h = 0, we find that .R'(O) = —3/5. Using these two results together with part (a), we see that the graph of R(h) must stay in Region lfor— l/A < h < 0 and must go to Region 2 for 0 < h <S 1. In Regions 1 and 2, dR/dh < 0. But R(h) ->• + 0 0 as h + 0 0 (part (b)) and R(h) —>• 1/2 as h -> + 0 0 . Therefore, there exists h* > 0 such that R(h*) = R(h*), that is, R'{h*) = 0 and R'(h) > 0 for h > h*. Also, since R'{h) < 0 and R(h) -> 1/2 as h -)• +cc, we have R(h*) = R{h*) > 1/2. Hence R(h) > 1/2 for h > 0. Finally, note that i2(/i*) (= Rmin) is determined numerically to be 0.752 (d) Proved in part (c). 31 (e) Using Lemma 2.2 to compute the second derivative of R(h), we have 47i(4/i + l)R" = R'(10R - 24h - 8) + (SR - 4). Substitute Ti = h* into the above equation and using part (c) and (d), we immedi-ately get R"(h*) > 0. • Using the above lemma, we see that R(h) has the form shown in Figure 2.9. Recalling that M(h, A) vanishes when A = R(h), we have the following results: Result 2.3 1. If A > 1, then A - R(h) > 0 V7i E (-1/4,0). This implies d(h, e, A) > 0 and all orbits spiral outward. 2. If A < 4/5, then A - R(h) < 0 VTi E (-1/4,0): 77m s'mpZies d(h,e,\) < 0 and a// or&ifc spiral inward. 3. 7/4/5 < An < 1, i/ien An — R(h) = 0 for some unique ho E (—1/4,0). This implies d(h,e,X) = 0 for unique A(e) = An + 0(e) and h(e) = /in + O(e). [/sing Theorem 1.1, the system (2.13) has a unique, hyperbolic periodic orbitTe which tends to as e —>• 0. ./Vow/, if h > h(e), then d(h,e,X) > 0. T/m£ means orbits spiral outward. If h < h(e), then d(h, e, A) < 0. That means orbits spiral inward. Therefore, the periodic orbit is unstable. 4- If XQ > 4/5, then XQ — R(h) = 0 for some unique ho S (0, oo). This implies d(h, e, A) = 0 for unique A(e) = An + O(e) and h(e) = ho + 0(e). Using Theorem 1.1, the system (2.13) has a unique, hyperbolic periodic orbit Y€ which tends to T/j0 as e —> 0. Now, if h > h(e), then d(h,e,X)< 0. That means orbits spiral inward. If h < h(e), then d(h,e,X) > 0. That means orbits spiral outward. Therefore, the periodic orbit is stable. 5. If Rmin < An < 4/5, then there are 2 values of h, h\o and h2o (0 < h\o < h2o) such that An — R(h) = 0. This implies there are 2 periodic orbits for hi(e) — h±o + 0(e), h2(e) = h2o + 0(e), respectively, and A(e) = An + 0(e). Now, if h < hi(e), then d(h,e,X) < 0. That means orbits spiral inward. If h\(e) < h < h2(e), then d(h,e,X) > 0. That means 32 orbits spiral outward. If h > /12(e), then d(/i, e, A) < 0. That means orbits spiral inward. Therefore, the periodic orbit is unstable at h = h\(e) and the periodic orbit is stable at h = /12(e). 6. If Ao = Rmin ~ 0.752, then An — R{h) = 0 for unique h = h* > 0. This implies d(h, e, A) = 0 for unique A(e) = Ao + 0(e) and /i(e) = /i* + 0(e). Using Theorem 1.2, the system (2.13) has a unique, non-hyperbolic periodic orbit of multiplicity two which tends to Th* as e —>• 0. Now, if h > h(e), then d(h,e,X) < 0. That means orbits spiral inward. If h < h(e), then d(h,e,X) < 0. That means orbits spiral inward. Therefore, the periodic orbit is "semi-stable" (i.e. saddle-node bifurcation of periodic orbits). 7. If X < Rmin, then X — R(h) < 0 V7i £ (0, co). This implies d(h,e,X) < 0 and all orbits spiral inward. We see that, even though the computation of the Melnikov function is somewhat technical, the benefits are great: we determine the exact number, positions, and multiplicities of the limit cycles from the zeros of the Melnikov function. It is time now for us to return to system (2.11) with the minus sign. The homoclinic loop bifurcation, which occurs at A = 4/5 + O(e) for (2.13), corresponds to M2 = ^ + oo4) (Mi>0) for (2.11) with the minus sign. Also, the saddle-node bifurcation of limit cycles, which occurs at A si 0.752 + O(e) for (2.13), corresponds to M2 « 0.752/ii + 0(4) (m > 0) for (2.11) with the minus sign. Using Result 2.3, we can determine the exact number of periodic orbits, corresponding to different values of the parameters in system (2.11) with the minus sign. For /J,2 > Mi) there is a unique (stable) limit cycle around the three equilibria. For 4/JI/5 + 3 0(/ii) < M2 < Mi) there are two (unstable) limit cycles around the two equilibria (±y/jli,0) respectively, while there is a unique (stable) limit cycle surrounding the two limit cycles and 33 3 the three equilibria. For /Z2 = 4/^ i/5 + O(nl), there is a pair of symmetric homoclinic orbits, while there is a unique (stable) limit cycle surrounding the homoclinic orbits and the three 3 3 equilibria. For RminHi + 0 ( M i ) < M2 < 4//i/5 + 0 (Mi )> there are two limit cycles (smaller one 3 unstable, larger one stable) surrounding the three equilibria. For p2 — RminUi + 0(n{), there is 3 a unique multiplicity two (semi-stable) limit cycle. For fi2 < RminHi + 0 ( M i )> there is no limit 3 cycle. As fi2 increases past RminHi + O(nl), the system (2.11) with the minus sign undergoes a saddle-node bifurcation of periodic orbits. The bifurcation set and the corresponding phase portraits for the system (2.11) with the minus sign is shown in Figure 2.10. Next, we consider the system (2.11) with the plus sign. This case is actually simpler. For / i i > 0 there is only one equilibrium point at the origin, and it is a hyperbolic saddle. For HI < 0 there are three equilibria, a hyperbolic source at the origin for \i2 > 0, a hyperbolic sink at the origin for [j,2 < 0, a non-hyperbolic linear centre at the origin for \X2 = 0, and two hyperbolic saddle points at (±y/—fj,i,0). There is a pitchfork bifurcation at points on the /i2-axis with \i2 ^ 0. There is a supercritical Hopf bifurcation at points on the pi-axis with \i\ < 0- The partial bifurcation set and phase portraits is shown in Figure 2.11. We know that for all HI < 0> a n asymptotically stable limit cycle is generated (via Hopf bifurcation) at \i2 > 0 (fi2 close to 0). What happens to the stable limit cycle as /j,2 increases from 0 ( / i i < 0 fixed)? Do the limit cycles exist forever? In fact, the asymptotically stable limit cycle expands monotonically as fi2 increases from 0 until it intersects the saddle points at (±\/—Mi>0) a n d forms a 'double' heteroclinic orbit at some parameter value [i2 = M M I ) = —/xi/5 + 0(|/^i12) (HI < 0), i.e., there exists a heteroclinic loop bifurcation curve in the Hi^-plane given by ^2 — ^ ( M I ) f° r M i < 0- The proof is similar to the case with the minus sign treated above, and is omitted. Similarly, we can prove that the asymptotically stable limit cycle is unique when pi < 0, 0 < ^2 < M M I ) - ^ n e bifurcation set and the corresponding phase portraits for the system (2.11) with the plus sign is shown in Figure 2.12. 34 oo o o d CN © © d d oo d Figure 2.1: Two hyperbolic limit cycles for system (2.6) with e = 0.3 and n\ 35 Figure 2.2: A unique non-hyperbolic limit cycle for system (2.6) with e = 0.3 and 36 Figure 2.3: No limit cycles for system (2.6) with e = 0.3 and = 0.32. 37 38 Figure 2.5: The partial bifurcation set and the corresponding phase portraits for the system (2.11) with the minus sign. Figure 2.6: Phase portrait of the system (2.13) with e = 0. 39 Figure 2.7: Phase portrait of the system (2.23). Figure 2.8: Trajectory when h > 0. 40 Figure 2.9: Detailed phase portrait for system (2.23). 41 My Figure 2.10: Bifurcation set and the corresponding phase portraits for the system (2.11) with the minus sign. 42 -M, Figure 2.11: The partial bifurcation set and the corresponding phase portraits for the system (2.11) with the plus sign. 43 44 1 Chapter 3 Higher-order Melnikov theory 3.1. Introduction In Chapter 1 we saw that the number, positions and multiplicities of the limit cycles of (2) are determined by the number, positions and multiplicities of the zeros of the Melnikov function M(h,/j,). But what if M(h,/j,o) = 0 for some po € 5Rn ? In this case a higher-order analysis in e is necessary. For instance, if the first-order Melnikov function that we derived in Chapter lis identically equal to 0, then we need to look at the second-order Melnikov function, which we hope is not identically equal to 0, and relate the number, positions and multiplicities of the zeros of it to the number, positions and multiplicities of the limit cycles. Similarly, if the first- and second-order Melnikov functions are both identically equal to 0, then we need to look at the third-order Melnikov function. So, in general, we suppose that if the first k — 1 Melnikov functions are all identically equal to zero, then we need to look at the fcth-order Melnikov function. You may ask: How to relate the number, positions and multiplicities of the zeros of the fcth-order Melnikov function to the number, positions and multiplicities of the limit cycles? The answer is that this relationship is the same as the relationship between the number, positions and multiplicities of the zeros of the first-order Melnikov function and the number, positions and multiplicities of the limit cycles. For example, if the fcth-order Melnikov function Mfc(/i, p) is equal to zero and satisfies some non-degeneracy condition (i.e. dMk/dh(h, p) ^ 0), we can apply the implicit function theorem to obtain a unique hyperbolic limit cycle for small e. This kind of result, which is an extension of the results obtained in Chapter 1, is presented in Section 2. In Section 3, we use a recursive algorithm of Frangoise [10] to derive a second-order 45 Melnikov function to which the theory developed in Section 2 applies. 3.2. Higher-order theory Although the theorems in Chapter 1 are sufficient and convenient for a first-order analysis, they do not apply to a higher-order analysis of (2). In this section, I present the general results concerning higher-order Melnikov theory, which extend Theorems 1.1, 1.2. The following theorem, which is an extension of Theorem 1.1, gives us conditions under which the system has a unique, hyperbolic limit cycle for small e ^  0 when a higher-order analysis is required. Note that Theorem 1.1 is equivalent to the following theorem with k = 1. Theorem 3.1 Assume that Assumption 1.2 holds for all h E I. If there exists a po E$ln such that dd d^k~^d d{h,0,L-o) = d^ (ft,OiA*o) = ••• = g (fc-i) (M,Mo) = 0 for all hE I and if there exists a ho E I such that d^d d^k+V)d ^-^y(/i0,0,/i0) = 0 and ^ ^ ^ ( ^ 0 , 0 , Mo) # 0, then for all sufficiently small e ^ 0, the system (2) has a unique, hyperbolic limit cycle Te which tends to the periodic orbit Th0 as e —> 0. Proof: Define the function F(h,e) = { { E S T ( M , M O ) if 6 = 0 so that d{h,e,'no) = ekF(h,e), where 1 d^d F(h,e) = -^M(hAno) + 0(e). 46 Also, 1 d^d OF 1 d(k+1U Then, by the implicit function theorem [19], there exists a 8 > 0 and a unique function h = h(e), defined for |e| < 5, such that h(0) = ho and F(h(e), e) = 0 for all |e| < 8. The rest of the proof is exactly the same as of Theorem 1.1. • The next theorem gives us an extension of Theorem 1.2 for determining a unique, non-hyperbolic limit cycle of multiplicity two from system (2) when a higher-order analysis is required. Theorem 1.2 is equivalent to the following theorem with k = 1. Theorem 3.2 Assume that Assumption 1.2 holds for all h E I. If there exists a po E 9fJn such that dd d^^d d(h,0,/i0) = — (h,0,no) = • • • = g ^ f c , ! ) (h,°>Mo) = 0 for all he I and if there exists a ho E I such that Q{k)d d^k+v>d — (h0,0,Ho)=0, ^ - ^ ( / i 0 , 0 , / i o ) = 0, g e ( f c ) g h 2(^o,0,Mo) #0) ge(fc)gM (fro,0,Mo) 7^  0 for some j = 1,... ,n, then for all sufficiently small e, there are functions h(e) = ho + O(e), p(e) = po + O(e) such that for sufficiently small e ^  0, the system (2) has a unique non-hyperbolic limit cycle of multiplicity two which tends to the periodic orbit Th0 as e —>• 0. Proof: First we let po = ( M ? > M 2 > • • • ,Mn)- Without loss of generality, we assume that j = 1 (i.e. ge((l,|g d t (fro>0)Mo) 7^  0). Define the function f ^ f i l ife.^0 F(h,e,p) = { { £g>f(M,/z) if e = 0 47 so that d(h,e,iJ,) = ekF(h,e,tJ,), where 1 d^d F(h,e,p) = --^(h,0,fi) + O(e). Also, 1 d^d OF 1 d( f c + 1 )d ^ ^ 0 ' W ) = ^ P ^ ( / 1 0 ' 0 ' W ) = 0 ' d2F 1 d^k+2U dF 1 d^k+l">d ^ ( ^ ' ° ^ o ) = ^ a e ( ^ { / l o ' ( ) ' / i o ) 7 ' a Therefore, by the Weierstrass preparation theorem [1, p. 388, Theorem 69], there exists a 6 > 0 such that F(h,e,'fj) .= [(h-h0)2 + Ai(e, li)(h — h0) + A 2(e, /i)]$(n,e, /i), where Ai(e, M ) , .4.2 (e,M), and e, M ) are defined for |e| < 6, \h — ho\ < 6, | M — Mol < <^  - A I ( 0 , M O ) = ^2(0,Mo) = 0, i s $(/i0,0,Mo) 7^ 0, and f^(0, Mo) 7^  0 since J -^(ho,0,Mo) 7^ 0. The remainder of the proof exactly the same as of Theorem 1.2. • Remark 3.1 7/d(M,Mo) = f (M,Mo) = • • • = gSf (M,Mo) = 0, g(k)d d(k+^d d(k+m~~^d Q(k+m)d d(k+Vd de(k)dh(m) (^o,0,Mo)^0, ge(fc)g #0 /or some; = 1,... ,n, rTien it can 6e shown that for sufficiently small e ^ 0, t/ie system (2) has a unique non-hyperbolic limit cycle of multiplicity m which tends to as e —> 0. 48 3.3. Derivation of the second-order Melnikov function In Section 2, we have seen that limit cycles correspond to the zeros of the displacement function. The A;th-order contribution (in e) to the displacement function is essentially the fcth-order Melnikov function (if the first k — 1 Melnikov functions are identically equal to zero). However, this kind of result can be useful in applications only if we know the formula for the second (or higher) order Melnikov function. In this section I will use Frangoise's recursive algorithm [10] to derive an explicit formula for the second-order Melnikov function M2(h,[i) in terms of certain integrals along the periodic orbits T^ in Assumption 1.2. This formula applies to perturbed Hamiltonian systems of the form where ^ = y, Qf^ = —U'(x), U'(x) is a polynomial of degree one or higher, / and g are functions depending smoothly on x,y,e,/j,, and e is a small perturbation parameter. Note that for e = 0, the system (3.1) is a Hamiltonian system with Assume that Assumption 1.2 holds for all h G I. The first-order Melnikov function M(h,iT), which we denote by Mi(h,fi) in this section, can be written as where Q(h) is the simply connected region surrounded by IV If Mi(h,[i) = 0 for all h G I and x = %• + ef{x,y;e,n), (3.1) H(x,y) = yj-U(x). (by Green's Theorem) LI G fin (this occurs, for example, when ^ + | | is an odd function with respect to y), we need to look at the next term M2(h, /z) in the expansion of the displacement function d(h,e, /J): d{h,e,fj.) = e + 0(e) = e2M2(/i,ii) + 0(e3). 49 In order to compute the second-order Melnikov function M2(h,p), provided that the first-order Melnikov function M\(h,n) = 0, we develop the idea of [10]. Given a perturbation (f,g)T in (3.1), we let u = g(x,y,e, p)dx — f(x,y,e,n)dy. Expand / and g in Taylor series in e and thus ( Og — (x,y,0,n)dx ^ f ^\ - -^(x,y,0,n)dyj +••• = u>0 + eu>i + • • • , and the system (3.1) can be rewritten as dH - e u = 0, since OH OH dH — etc = -rr-dx + -^—dy — e(gdx — f dy) ox ay [OH \ J (OH \ , = \-0x--t9)dX+{-0y- + e ! ) d V implies that = 0 (OH \ dy ( OH \ dx {-oy7 + ef)Tt = {-0x- + e9) TV and therefore dx dH i f dt ~ dy e J ' djl = -9K + e a dt dx ^ey-With this notation, the first-order Melnikov function M\(h,/z) can be written as Mi(h,n) = <£ w 0 . In the following, I will state and prove the theorem which gives us a formula for the second-order Melnikov function for (3.1). Theorem 3.3 Under Assumption 1.2, if M\(h,p) = 0 for all h G I and p £ W1, then the displacement function for the system (3.1) is d(h, e, H) = e2M2(h, fx) + 0(e3), (3.2) 50 where the second-order Melnikov function M2(h,jj) is given by M2[h,y) = & [Gih(x,y,/j,)P2(x,h,n)-Gi(x,y,n)P2h(x,h,n)]dx + F{x,y,n) ^(x,y,0,n) + ~{x,y,0,p) df, dx de(x,y,0,n)dx - ~(x,y,0,n)dy (3.3) where ry rx F(x,y,(j.) = / f(x,s,0,p)ds- / g(s, 0,0, n)ds, Jo Jo OF G(x, y, p) = g{x, y, 0, fi) + -Q^(X, y, p), (3.4) (3-5) Gi(x,y,fi) denotes the odd part of G(x,y,p) with respect toy, G2(x,y,p) denotes the even part ofG{x,y,n) with respect to y, G(x,y,fi) = Gi{x,y,n)-\-G2{x,y,p), Gi{x,y,n) = yGi{x,y2,n), G2(x,y,fi) = G2(x,y2,n), 1 dGx Gih(x,y,n) y dy (x,y,n), P2{x,h,n) = f G2{s,2h + 2U(s),n)ds, Jo and P2h(x,h, p) denotes the partial derivative of P2(x,h, p) with respect to h. Before proving the above theorem, let us introduce some of the main ideas from [10]. The following definition is taken, with some modifications, from [10]. Definition 3.1 We say that the Hamiltonian function H satisfies condition (*) if for all poly-nomial one-forms1 to such that w = 0, there exists a continuous function g and a locally Lipschitz continuous function R such that to = g dH + dR. (*) If TJ satisfies the condition (*), we can compute, using the algorithm in [10], the first derivative of the displacement function which is not identically 0. In [10], Frangoise showed that the 1UJ is a polynomial one-form in 3i 2 if u can be expressed as f dx + gdy, where / and g are polynomials in x, y. 51 Hamiltonian function H(x,y) = (x2 + y2)/2 satisfies the condition (*). We now want to extend this specific Hamiltonian to a larger class so that any Hamiltonian of the form H(x,y) = y2/2 — U(x) satisfies condition (*). This result is proved in the following lemma. Now, for convenience of notation, we write M2{h) = M2(h, LI), G\h{x,y) = Gih{x,y, p) and so on (since LI is regarded as a "constant" in the formula for the second-order Melnikov function). Lemma 3.1 Assume that Assumption 1.2 holds for all h G I and that M\(h) = 0. Then H satisfies the condition (*). That is, there exists a continuous function go{x,y) and a locally Lipschitz continuous function Ro(x,y) such that UQ can be expressed as where F, G are defined by (3.4) and (3.5) respectively. We need to show that Gdx can be expressed as go dH + dRo- Given a point (x, y), let TH be a closed smooth curve passing through (x, y). Suppose xmin and xmax are the minimal and the maximal values of x respectively on TH-Now we let xmin = a(H) and xmax = b(H), where a(H) and b(H) are the roots of H + U(x) = 0 (i.e. H(a{H),0) = H{b{H),0) = H). Also, let B(x,H) = \y\ = ^ 2H + 2U(x). Take a path LOQ = go dH + dRo-Proof: Recall that = g(x,y,0)dx - f(x,y,0)dy = Gdx- dF, F(x,y) C TH which begins at (a(H),0) and ends at (x,y). Also, T ( X ^ has the same orientation as TH- Let (3.6) First, assume that y > 0. Ro, which is defined in (3.6), can be written as (3.7) Therefore, RQ can be considered as a function of x and H. Now, dR0(x,H) = dRo (x, H) dx + dRo (x, H) dH, dx dH 52 where dR H) = G(x, B(x, H)) = G(x, y), (3.8) dx and (using the Chain Rule), d R o ( m n <m m tm m^><m+ T dG/dyfoBfoH)) -QJJ{X, H) = -G{a(H),B(a{H),H))a (H) + J B(fH) As B(a(H), H) = 0 and G{x, 0) = 0 (by definition) , (3.9) becomes O H ^ h ) = h{H)—mm—d(i dg/dy(t, B(£, H),0)+ df/dx(£, B((, £0,0) ^ o , „ _ fx dG/dy(t,B(Z,H)) Ja{ 7a( which can be written as a i i o f df/dxit, r>, 0) + dg/dy(Z, r,, 0) ^ d ef Similarly, for y < 0, Ro can be written as rb(H) i(H) Jb(H) which is a function of x and H. Now, MB) fx Ro= G(Z,B{Z,H))dt+ G(£,-B{£,H))d£, (3.11) ya ffdRo(x, H) = ^ ( x , #) + |^(x , i ? ) dH, where dRp dx and (x, H) = G(x, -B(x, H)) = G(x, y), (3.12) dH (x,H) = -G(a{H),B(a(H),H))a'(H) + G{b{H),B(b(H),H))b'{H) Ja(H) -O ( . £ , . « ) _ r dG/dy{S,-B{S,H))^ ( 3 1 3 ) 53 As B(a(H),H) = 0, B{b{H),H) = 0 and G(x,0) = 0, (3.13) becomes dRQ. [»W dGJdy{Z,B{Z,H))J& '[* dG/dy(Z,-B(j,H)) „ m ^ H ) = Ja{H) —mjf)—v-Jw—mw)—* * W dg/dy(£,B(S,H),0)+df/dx{t,B(£,H),0) pi I ntr T T \ "S> Jb(. = f dg/dy(£, -B&H), 0) + df/dx(S, -B(£, H),0) 10(H) B(£,H) which can be written as dR0 f df/dx(t,ri,OJ+dg/dy(Z,v,0),cdef = f 9f/dx^V,0)+d9/dy{^,0)d^_go{xy) ( 3 M ) Jrtr._„, V So in any case we have dR0 = G{x,y)dx - gQ(x,y)dH. Since M i (ft) = i w 0 •>rh — <p Gdx, the hypothesis Mi (ft) = M[(h) = 0 implies that lim Ro{a(h),y) = lim Ro(a(h),y) = 0, y—>o~ 2/->o+ lim g0(a(h),y) - lim g0(a(h),y) = 0. 2/->0~ 2/->0+ Thus, i?o(^ , y) and o^(^ ) ?/) are single-valued functions. This ensures that they have the required continuity. The proof is now complete. • In Frangoise's paper [10], he derives an algorithm for computing the higher-order Melnikov function for the system x = y + ef(x.y) y M y ) (3.15) y = U'(x) + eg{x,y) 54 Note that in (3.15), / and g are independent of e. But in general, / and g may depend on e (i.e. system (3.1)). In the following, I will use [10] to derive an algorithm for computing the higher-order Melnikov function for system (3.1) (i.e. / = f(x,y,e) and g = g(x,y,e)). But before doing that, let us first derive a formula for the second-order Melnikov function. Using the above lemma, we are able to prove the following theorem which gives a formula for the second-order Melnikov function (c.f. [10]). Theorem 3.4 Assume that Assumption 1.2 holds for all h E I and that M\(h) = 0. Then the second-order Melnikov function is given by M2(h) = I (ff0w0 + wi). (3.16) Proof: We recall the construction from [10]. Fix h E I and denote by a the smallest solution of the equation H(a, 0) = h. Let Pn = (o^  0) and choose a line segment S containing Pn that is normal to the trajectory of (3.1) at Pn. For sufficiently small e ^ 0, let P i = (QI,0) £ S be the point of the Poincare first return map (see Figure 3.1). Let d(h, e) = H(ai, 0) — H(a, 0) be the displacement function. Now, using Lemma 3.1, UIQ = go dH + dRo yields (1 + ego)(dH - eu) = (1 + eg0)(dH - eu0 - eV + 0(e3)) = dH-e{uj0 ~g0dH) - e2(ffow0 + wi) + 0(e3) = dH -edRo-e2(gou)o + uJi) + 0(e3). (3.17) Integrating (3.17) along r e , where r e (= T(e, h)) is the trajectory of (3.1) connecting Pn and P i (see Figure 3.1), and recalling that dH — eu = 0 on the trajectory r e , we get 0 = j dH-e f dRo-e2 I (g0uo + wi) + 0(e3). (3.18) Now, since d{h,e) = H(Pi) - H(P0), (3.18) becomes d(h, e) = e [ dRo + e2 [ (5ow0 + wi) + 0(e3). (3.19) We want to find a bound for Jr dRo. For e sufficiently small and since Ro{x,y) is locally Lipschitz continuous, there exists a constant c > 0 such that / dRQ = \Ra(ai,0) -Ro(a,0)\ < c\ax - a\. (3.20) 55 Figure 3.1: Illustration of the Poincare map. Now, by the Mean Value Theorem, there exists a P* between Po and Pi such that \dH. \H(au0)-H(a,0)\ for e sufficiently small. Therefore, dx (P*) c*i -a\ (3.21) / dR0 = |e| / dRo < c\e\\ai-a\ (by (3.20)) c|e||d(M) = 0{e\ (by (3-21)) (3.22) because d(h, e) = 0(e2) by hypothesis. Finally, using the fact that Tt tends to the closed orbit Tk as e -» 0 and using (3.22), (3.19) becomes ,e) =e2 / (5owo + a;i) + 0(e3). • d(/i Next, we derive a more general version of Frangoise's recursive algorithm which applies for systems of the form (3.1). Note,that Theorem 3.4 is equivalent to the following theorem with k = 1. 56 Theorem 3.5 Assume that Assumption 1.2 holds for all h £ I and that M\(h) = Mk(h) = 0 for some k>l. Then Mk+i(h) = i fifc, Jrh where £lo = u>o, (3.23) O n = ujn + T^^  gi^Jj, 1 < n < k (3-24) i+j=n—1 and the functions gi, 0 < i < k — 1, are determined successively from the representations Oj = 9idH + dRi with gi, Ri as in Lemma 3.1. Proof: First, by integration of the equation dH - eu = dH - eu0 + 0{e2) . along r e and noting that dH — eu = 0 on r e, we immediately obtain f dH = e f uj0 + O(e2). Using the fact that Jr dH = d(h, e) and r e tends to the closed orbit as e —> 0, we have d{h,e) = ei co0 + O{e2), Jrh. and therefore Mi(h) — jTh OJQ (the usual first-order Melnikov function). We now make the following induction hypothesis (c.f. [10]): there exist continuous functions 90i9\i • • • 19k-2 such that for all n = 1,... , k, Mn(h) = <p O n _i = 0, where 57 and Q,n-i = w n _ i + ^ gtujj i-\-j=n—2 for n = 1,... , A;. Using this relation for n = k, and applying Lemma 3.1, there exist a continuous function gk_i and a locally Lipschitz continuous function Rk-i such that fifc-i = 9k-\dH + dRfc-i = c; f c_i + ^ ftOj. (3.25) This proves that the functions gk-i, Rk-i c a n be constructed by recurrence. Next, we multiply dH — eu by 1 + ego + e2g\ H h ekgk-i, which gives - e(u>o - ffodtf) - e 2(wi + g0uo - g\dH) - e f c(w f c-i + go^k-2 H + 9k-2^Q ~ 9k-\dH) - ek+1{uk + ffowfe_i + • • • + gk-iuo) + 0(ek+2), which can be written as (using (3.25)) dH - edRo - e2 dRi ek dRk-i - ek+1nk + 0(ek+2). Therefore, (1 + ego + • • • + ekgk_1)(dH - ew) = dH -edR0-e2dRx ekdRk_x - ek+1Qk + 0(ek+2). Integrating the above equation along T e, where Te is the trajectory of dH — eu = 0, and noting that f dH = d(h,e), [ (edR0 + e2dR1 + --- + ekdRk-i) = 0(ek+2) (the second estimate follows from the fact that d(h,e) = 0(e f c + 1)), we get d(h,e) = ek+1 [ nk + 0(ek+2). 58 Finally, Te coincides with the closed curve Th (up to 0(e)) and therefore we obtain d(h,e)=ek+1 I nk + 0(ek+2), which proves the theorem. • ' We derived a formula for the second-order Melnikov function (c.f. (3.16)). Is this formula useful in practice? The answer is no since go(x,y), which is given by (3.10), is an elliptic integral in general (and so the calculation of 50 is v e r y messy, which we do not want to go into it). However, if ^{x, y, 0) + fajix, y, 0) is an odd function of y, then go(x;y) is a polynomial, so the complexity is due to the even part of | | + Therefore, we need to express the second-order Melnikov function M2(h) in a more convenient form (i.e. Theorem 3.3) which will be useful in any case. In the following, I will derive Theorem 3.3 using (3.16). From Lemma 3.1, too can be written as UQ = godH + d(Ro — F). Substituting the above expression for UQ into (3.16) and using the fact that dH = 0 on Th, we get M2(h) = i [gQd{R0-F)+ul] = £ g0dR0 - I g0dF+ £> UJX. (3.26) Jrh Jrh Jrh Applying integration by parts to the second integral on the right-hand side of (3.26), we get M2{h)=<f godRo+f Fdg0 + f wi- (3.27) Jrh Jrh Jrh Using the definition of go, we see that on Th, = Jdf/dX(x,y,0) + dg/dy(x,y,0))dx y Also, we know that uo\ = | | (x, y, 0) dx — ^ (x, y, 0) dy. Therefore, it remains to find a convenient form for the first integral on the right-hand side of (3.27). In order to do so, we split the function G(x,y) into odd and even parts with respect to y: G(x,y) = G\(x,y)+G2(x,y) (i.e. G\{x,—y) = 59 -Gi(x,y), G2(x,-y) = G2(x,y)). We write Gi{x,y) = yGi(x,y2), G2(x,y) = G2(x,y2). Now, Ro = f G(£,r?K = I (Gi((,v) + G2(^V))d( = f vGi((,V2)d(+ f G2fori2)dt = f vGi(Z,r)2)dt+ fX G2(£,2h + 2U{Z))dt =' pi(x,y)+P2(x,h). (3.29) Also, def dRo go = dh = ~t{x^-^h)- < 3 ' 3 0 ) Note that pi(x, y) is an elliptic integral and P2(x, h) is a polynomial in x (and hence continuous with respect to x). Using the hypothesis that M\(h) = M[(h) = 0, we obtain lim pi(a(h),y) = lim pi(a(h),y) = 0 = pi(b(h),0) y^t0~ J/-+0+ and lim ^(a(h),y) = lim.^(a(h),y) = 0 = ^(b(h),0). y->o~ on 2/->o+ an an The above imply that jf d^L(x,y)d\p1(x,y)] = 0, (3.31) a n [ X ' V ) = 0. (3.32) jf ^(x,y)d[P2(x,h)]+P2(x,h)d Also, it is easy to see that jf d^-(x,h)d[P2(x,h)]=0. (3.33) 60 Using (3.29), (3.3.0), (3.31), (3.32) and (3.33), we can now find a convenient form for §rkg0dR0 easily. We have g0dR0 = - -^-(x,y) +-~(x,h) d[pi(x,y) + P2(x,h)] = -L ^(x, y) d[Pl (x, y)} + ^ {x, y) d[P2(x, h)} + ^ ( s , h) d[Pl(x,y)] + ^ (x, h) d[P2(x, h)} = / P2(x,h)d dp dh dP2 dh (x,h) d[pi(x,y)} Now, using the definition of pi{x, y), we see that on T ,^ (3.34) d[pi{x,y)] = Gi(x,y)dx, (3.35) d ^(x y) dh [ x , y ) dGl( ldG1( (x,y)dx = --^—{x,y)dx. dh^°'~ y dy Substituting (3.35) and (3.36) into (3.34), we immediately get (3.36) <j> g0dRo=<f> [Glh(x,y)P2(x,h) - Gi{x,y)P2h(x,h)]dx, so that Theorem 3.3 is proved. Finally, note that without loss of generality in (3.3), we can take a primitive with P2(0, h) = 0. 61 Chapter 4 An application of higher-order Melnikov theory In this chapter, we apply the theory described in the last chapter to an example. We consider the quadratically perturbed non-hyperbolic linear centre studied by Bautin [5]. As in the examples of Chapter 2, the problem is determine the maximum number of limit cycles. In order to do so, we compute the first-order Melnikov function. However, if the first-order Melnikov function is identically equal to zero, then we need to compute the second-order Melnikov function. The number, positions and multiplicities of the zeros of the second-order Melnikov function is related to the number, positions and multiplicities of the limit cycles using Theorem 3.1 or 3.2. 4.1. Perturbed linear centre In this section, we use the Melnikov theory developed in Chapters 1 and 3 to study the number and positions of the limit cycles that occur in a quadratically perturbed non-hyperbolic linear centre of the form (4.1). This example is of particular interest since it concerns Hilbert's 16th problem, which asks for a bound for the number of limit cycles of a polynomial system in terms of the degrees of the polynomials that define the system. This problem is not solved, even for quadratic systems [4]. Hilbert's 16th problem was studied intensively by many authors, see, for example, [5, 18, 25]. In [5, 18, 25], they study the determination of an exact upper bound of limit cycles under perturbations of certain specific polynomial systems. The strongest theoretical result obtained so far is the following deep theorem of Il'yashenko [13]. Theorem 4.1 (Il'yashenko) A polynomial system has at most a finite number of limit cycles. 62 The above discussion motivates us to study the 6-parameter family x = y + Xix - X3x2 - (2A2 + \b)xy + XGy2 y = -x + Xiy - X2x2 + (2A3 + XA)xy + X2y2, (4.1) where Aj(e) = J2<jLi^ije''-> * = >6 and e is a small perturbation parameter. This is the same system studied by Bautin [5], who proved that there are at most three limit cycles in (4.1) and that a sixth- or higher-order analysis is required to produce that number. The hardest part in his proof is to determine at which order k (the order of the Melnikov function) the maximum number of limit cycles will 'stabilize'. That is, to determine the order of the Melnikov function M/t for which the next Melnikov functions will have the same maximum number of zeros. Therefore, one way to make this problem easier to study is fix the order k. Since we know the Melnikov functions when k = 1 and 2, we can determine the maximum number of limit cycles obtained from a first- and a second-order analysis of a quadratically perturbed non-hyperbolic linear centre of the form (4.1). In order to do so, we compute the first- and second-order limit cycles from a first-order analysis and that there is at most one hyperbolic limit cycle from a second-order analysis. Our results are consistent with those obtained by Bautin [5]. Bautin, however, goes on to a sixth-order analysis. For e = 0, (4.1) is a Hamiltonian system with H(x, y) = (x2 + y2)/2, and it has a one-parameter family of periodic orbits with the parameter h G (0,00) being the total energy along the orbit. We compute the first-and second-order Melnikov functions of system (4.1). The first-order Melnikov function is given Melnikov functions using the formulas derived in Chapters 1 and 3. We show that there are no Xh{t) = V2hcost, yh{t) — —V2hsmt, (4.2) by (4.3) From (4.1), we find that f(x, y, 0, A) = Ana; - A3ia;2 - (2A2i + A5i)a;y + X6ly2, 63 g(x, y, 0, A) = Any - A 2ix 2 + (2A3i + A4i)xy + A 2iy 2. Substituting into (4.3), we have Mi(h,X) = <p [(Aiiy-A 2ix 2 + (2A3i + A4i)xy + A2iy2)dx - (Xnx - X31x2 - (2A2i + A5i)xy + A6iy2) dy] = ^11 f ydx — Xn f xdy (All other integrals vanish) p2n /»27r = X u / y2h(t)dt + Xu / x\(t)dt Jo Jo r2-K r2w nh I s'm2tdt + 2An/i / cos2tdt Jo Jo (4.6) = AnXuh. (AA) Since h = a2/2, where a G (0, oo) is the positive x-axis intercept of the unperturbed periodic orbit, (4.4) becomes Mi (a, A) = 27rAna2. (4.5) We see from (4.5) that the system (4.1) has no limit cycles from a first-order analysis. Note that Mi (a, A) = 0 for all a > 0 if and only if An = 0. Therefore, for a second-order analysis it suffices to consider the following system x = y + e[eXi2x - X3ix2 - (2A2i + X51)xy + A6iy2] y = -x + e[eXi2y - X2ix2 + (2X31 + Xn)xy + X2iy2]. From (4.6), we find that f(x, y, e, A) = eAi2rc - A 3ix 2 - (2A2i + A5i)rcy + A 6iy 2, g(x, y, e, A) = eAi2y - A 2ix 2 + (2A3i + A4i)o;y + A 2 i y 2 , F(x,y, X) = -X31x2y - ^(2X21 + A5i)xy2 + ^A 6iy 3 + ^A 2ix 3, G(x, y, X) = XA1xy - ^A 5 xy 2, 64 Gi{x, y, A) = \41xy, G2(x, y, A) = - - A 5 i y 2 , Gih{x, y, A) = X41X P2(x,h,X) = -X5i(hx - —), P2h(x,h,X) = -X51X. Now, using Theorem 3.3 to compute the second-order Melnikov function, we have M2(/i,A) = - A 4 1 A 5 1 X" (hx - ^-) - x2y y 6 dx f 1 1 1 - 4 (-A3ix2y - -(2A21 + A51)xy2 + -A 6 1 y 3 + -A2ia:3) dx f (-Asly + XiXx) — + A12 <t> {ydx-x dy) /2ir j {hx2h(t)--x{{t)-x2(t)y2h{t))dt /2-K 2 (X3lX51x2h(t)y2h(t) - -(2X21 + X5l)X41x2h(t)y2h(t) ^A5iA6iy^(t) + ^A2iA4i4W)^ r2ir A12 / (y2h(t) + x2(t)) dt Jo (All other integrals in (4.7) vanish). + Substituting Xh{t) = V2hcost, yn[t) = —V2hs'mt into (4.8), we get M2(h,X) = - A 4 1 A 5 1 f2n 2 / (2/i2 cos21 - -h2 cos41 - 4h2 cos21 sin21) dt Jo 3 r2ir - / (4A3iA5i/i2cos2isin2i - 2(2A2i + A 5 i)A4in 2 cos 2 isin 2 i Jo 10- 0A51A61/1 2 sin4t+ -A 2iA 4i/t 2 cos41) dt o o + X12 / (2/isin2t + 2/icos2t)dt Jo 4TTAI2/I - 7rA 5i(A 3i - A6i)n2. (4.7) (4.8) (4.9) Since h = a2/2, where a G (0,00) is the positive x-axis intercept of the unperturbed periodic orbit, (4.9) becomes M2(c*,A) = 27rAi2a2 - Y A 5 I ( A 3 I - A6i)o:4. (4.10) We see from (4.10) that the system (4.1) has at most one hyperbolic limit cycle from a second-order analysis. Finally, note that if the second-order Melnikov function is identically equal to 65 zero, then we need to consider the third-order Melnikov function. For details on the derivation of the third- or higher-order Melnikov function, see, for example, [5, 18, 25]. 66 Chapter 5 Conclusions and future work In the Introduction we asked the question: What is the exact number, positions and multiplic-ities of limit cycles in a perturbed planar system of the form (2), assuming that Assumption 1.2 is satisfied? We then showed that they can be determined by the number, positions and multiplicities of the zeros of the (first-order) Melnikov function, which was derived in Chap-ter 1. In Chapter 1, we established some non-degeneracy conditions which guarantee that a first-order Melnikov analysis is valid. Also, we established a more degenerate condition which guarantees that a unique non-hyperbolic limit cycle of multiplicity two exists in (2). Chapter 2 contains some examples which illustrate the first-order Melnikov theory. In particular, we analyzed the Bogdanov-Takens bifurcation with reflection symmetry, with the emphasis on the determination of the number of limit cycles corresponding to different values of the parameters. In Chapter 3, the first-order theory was extended to higher-order. In particular, we derived a formula for the second-order Melnikov function for certain perturbed Hamiltonian systems. This formula is useful if the first-order Melnikov function is identically equal to zero. This formula is then applied to a quadratically perturbed non-hyperbolic linear centre in Chapter 4, the results of which agree with those obtained previously by Bautin [5]. Although the Melnikov function can be computed for any system of the form (2), finding the zeros of the Melnikov function is sometimes formidable, especially when a higher-order analysis is required. For example, consider the system x = y (5-1) y — — x + x3 + \\y + \2X2y, 67 where A;(e) = Y^jLi , 2 = 1,2 and e is a small perturbation parameter. If An = A21 = 0, then the first-order Melnikov function is identically equal to zero and a higher-order analysis of (5.1) will be necessary to determine the number of limit cycles that are still preserved under perturbation. In fact, it is possible, using the second-order Melnikov function derived in Chapter 3, to determine the number of limit cycles from a second-order analysis of (5.1). Again, the non-trivial part is to determine the number of zeros of the second-order Melnikov function M2(h, A). This determination of the number of zeros is beyond the scope of this thesis. The other problem that may arise is that both the first- and second-order Melnikov functions are identically equal to zero. It is possible to use Frangoise's resursive algorithm (e.g. Theorem 3.5) to compute the higher-order Melnikov functions, but since Frangoise's method requires calculations of growing complexity at each successive step, in practice only the first few Melnikov functions can be derived by using Theorem 3.5. Other approaches to higher-order Melnikov functions share similar problems. Much work has been done on the determination of the number of limit cycles in a quadratically perturbed non-hyperbolic linear centre. In [5], Bautin shows that a quadratically perturbed non-hyperbolic linear centre has at most three limit cycles, and that a sixth- or higher-order analysis is required to produce that number. His proof is to derive a 'structural' result about the form of the displacement function d(a, e) for small a, e, and based on this structure, determine what order analysis in e (i.e. what order Melnikov function) is required to produce that number. On the other hand, much less is known about the number of limit cycles in a cubically perturbed non-hyperbolic linear centre. For example, what is the maximum number of limit cycles in a cubically perturbed non-hyperbolic linear centre, and what order Melnikov function is required to produce that number? These questions are still not completely answered. It is known that the maximum number of limit cycles is greater than or equal to 11 [26]. Li and others [14] in fact constructed an example of a cubic system with eleven limit cycles. Much mathematical research has been devoted to Hilbert's 16th problem in the past and will probably continue to be in the future. 68 Bibliography A.A. Andronov et al. 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Sur les equations de la dynamique et le probleme des trois corps. Acta Math., 13:1-270, 1890. 69 [17] I. Rehberg and G. Ahlers. Experimental observation of a codimension two bifurcation in a binary fluid mixture. Phys. Rev. Lett., 55:500-503, 1985. [18] R. Roussarie. Cyclicite finie des lacets et des points cuspidaux. Nonlinearity, 2:73-117, 1989. [19] W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, New York, 1964. [20] S. Shi. A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. Sinica, 23:153-158, 1980. [21] K.S. Sibriskii. On the number of limit cycles in the neighbourhood of a singular point. Differential Equations, 1:36-47, 1965. [22] S. Strogatz. Nonlinear Dynamics and Chaos. Addison-Wesley, 1994. [23] F. Takens. Forced oscillations and bifurcations, Applications of Global Analysis. Comm. Math. Inst. Rijksuniversiteit Utrecht, 3:1-59, 1974. [24] S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, 1990. [25] H. Zoladek. Quadratic systems with centre and their perturbations. J. Diff. Eq., 109:223-273, 1994. [26] H. Zoladek. Eleven small limit cycles in a cubic vector field. Nonlinearity, 8:843-860, 1995. 70 Appendix A Derivation of (2.9) by the method of normal forms In this Appendix, we use the method of Poincare normal forms to derive (2.9) (see Wiggins [24] for more details). Consider the system x = Ax + F3(x) + 0(\x\5), x = (x,y)e$l2, (A.l) where \ 0 0 and F3(x) is a vector field consisting entirely of third-degree terms. Note that in order for (A.l) to preserve reflection symmetry, the higher-order terms need to be odd (i.e. fifth-order terms). The idea of Poincare normal form is to introduce a near-identity coordinate change x = u + h3(u), u = (u,v) € fi2, (A.2) where h3(u) = 0(\u\3), to reduce the system (A.l) to its 'normal form'. Substituting the transformation (A.2) into (A.l), we get (I + Dh3{u))u = Au + Ah3(u) + F3(u) + 0(\u\5) Now, since (J + Dhsiu))-1 = I - Dh3(u) + 0(\u\% 71 it follows that ii = Au +Ah3(u) - Dh3(u)Au +F3(u)+0(\u\5) = Au + F3{u) + 0(\u\5), where F3(u) = Ah3(u) - Dh3(u)Au + F3(u). (A.3) The point is that we can choose h3(u) to make F3(u) as simple as possible. Ideally, this would mean choosing h3(u) such that F3(u) = 0; however, this is not always possible. In order to get a clear understanding of how ^ 3 (u) is chosen, let us view the function LA [h3 (tx)] = Ah3 (u) - Dh3 (u)Au (A.4) as a linear transformation on the space H3 of all third-degree (homogeneous) polynomials. Since x = (x,y) 6 we consider LA as a linear operator on the eight-dimensional vector space Span < x° 0 2 z y zy 0 y° 0 2 x y xy' Then, h3 6 H3 is given by h3(u) = hou3 + f2\U2V + fnuv2 + / 0 3 v 3 gsou3 + 92\u2v + g\2uv2 + go3vz We need to compute LA[h3{u)}. We get / 93ou3 + (921 - 3/ 3o)^ 2w + (312 - 2f2i)uv2 + (g03 - fn)v3 LA[h(u)} =\ \ -3g30uzv - 2g2iuvz - g l 2tr Therefore, LA{H3) = Span { 0 u° -3u2v .0 2 U V uv 0 VJ 0 uv* (A.5) 72 and we see that H3 = LA(H3) 0 C 3 , where C3 = Span \ This shows that any system of the form x = Ax + F3(x) + 0(\x\5), x = {x,y)e^2 with A = and F3 € -£13 can be reduced, by a nonlinear transformation of coordinates a; — u + h3(u) with h3 € H3 given by (A.5), to the normal form ii = v + 0(\u,v\5) (A.6) v = au3+ bu2v+ 0(\u,v\5). It can be shown that if a ^ 0 and b 7^  0, then the higher-order terms (i.e. fifth-order terms) do not affect the qualitative nature of the non-hyperbolic equilibrium point at the origin provided that they respect the reflection symmetry. Therefore, we can delete these terms in studying the bifurcations that take place in a neighbourhood of this non-hyperbolic equilibrium point. That is, we can just consider the truncated normal form ii = v (A.7) v = au3 + bu2v. 73 

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