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A geometric approach to evaluation-transversality techniques in generic bifurcation theory Aalto, Søren Karl 1987

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A Geometric Approach to Evaluation-Transversality Techniques in Generic Bifurcation Theory By  S0ren K a r l Aalto B.Sc,  A  St. Francis X a v i e r University,  Thesis Submitted The  1982  in P a r t i a l Fulfilment  Requirements  for the Degree of  M a s t e r of Science in The  Faculty of G r a d u a t e Studies Mathematics  We accept this thesis as conforming to the required standard  The  University of British C o l u m b i a September, 1987 © S 0 r e n Karl Aalto  of  In  presenting  degree  at  this  the  thesis  in  University of  partial  fulfilment  of  of  this  department publication  or of  thesis for by  his  or  her  representatives.  M rXT h r € M rVT C £ 1  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  DE-6(3/81)  D  <  ^ -  It  *S  ,  m~r  is  for  an advanced  Library shall make  it  that permission for extensive  scholarly purposes may be granted  permission.  Date  that the  agree  this thesis for financial gain shall not  Department of  requirements  British Columbia, I agree  freely available for reference and study. I further copying  the  by the  understood  that  head of copying  my pr  be allowed without my written  Abstract T h e study of bifurcations of vectorfields is concerned with changes in qualitative behaviour that can occur when a non-stucturally stable vectorfield is perturbed. In a sense, this is the study of how such a vectorfield fails to be structurally stable. F i n d i n g a systematic approach to the study of such questions is a difficult problem. O n e approach to bifurcations of vectorfields, known as "generic bifurcation theory," is the subject of much of the work of Sotomayor (Sotomayor [1973b],Sotomayor  [1973a],Sotomayor  [1974]). T h i s approach attempts to construct generic families of  k-parameter vectorfields (usually for k = l ) , for which all the bifurcations can be described. In Sotomayor [1973a]it is stated that the vectorfields associated with the "generic" bifurcations of individual critical elements for k-parameter vectorfields form submanifolds of codimension < k of the B a n a c h space X ( M ) of vectorfields r  on a compact manifold M. T h e bifurcations associated with one of these submanifolds of codimension-k are called  generic codimension-k bifurcations.  In Sotomayor  [1974]the construction of these submanifolds and the description of the associated bifurcations of codimension-1 for vectorfields on two dimensional manifolds is presented in detail. T h e bifurcations that occur are due to the parameterised vectorfield crossing one of these manifolds transversely as the parameter changes. A b r a h a m and R o b b i n used transversality results for evaluation maps to prove the K u p k a - S m a l e theorem in A b r a h a m and R o b b i n [1967]. In this thesis, we shall show how an extension of these evaluation transversality techniques will allow us to construct the submanifolds of X (M) r  associated with one type of generic bifurca-  tion of critical elements, and we shall consider how this approach might be extended to include the other well known generic bifurcations. For saddle-node type bifurcations of critical points, we will show that the changes in qualitative behaviour are related to geometric properties of the submanifold E is the pull-back of the set of zero vectors-or  0  of  X [M) r  zero section-by  x M , where E n  the evaluation map for  vectorfields. We will look at the relationship between the Taylor series of a vectorfield X (X,p)  at a critical point p and the geometry of £ of  a 0  t the corresponding point  X ( M ) x M . T h i s will motivate the non-degeneracy conditions for the r  saddle-node bifurcations, and will provide a more general geometric picture of this approach to studying bifurcations of critical points. Finally, we shall consider how this approach might be generalised to include other bifurcations of critical elements.  (ii)  Table of Contents  Chapter 1  Introduction  1 . 1  P a r t s  1 . 2  A n  Chapter 2  o f  E x a m p l e  S y s t e m  1  o f a B i f u r c a t i o n  T a n g e n t  2 . 2  J e t  B u n d l e ,  V e c t o r f i e l d s  B u n d l e s  T h e  3 . 2  D i f f e r e n t i a b i l i t y o f E v a l u a t i o n  4 . 1  Chapter 5  1  2  2  2  9  3  5  Evaluation Maps  3 . 1  Chapter 4  1  Global Formalism  2 . 1  Chapter 3  a D y n a m i c a l  B a n a c h  S p a c e  X  r  ( M ) M a p s  3  9  Maps and Transversality I m p l i c i t F u n c t i o n  T h e o r e m ,  T r a n s v e r s a l i t y  4  5  5  2  5  8  6  6  Bifurcations of Critical Points  5 . 1  C r t i c a l P o i n t s o f t h e  5 . 2  A n  5 . 3  C o n c l u d i n g  E x a m p l e - T h e  E v a l u a t i o n  M a p  S a d d l e - N o d e A g a i n  R e m a r k s  73  Bibliography  (iii)  1.1  1.1  Some Parts of a Dynamical System  The flow of a dynamical system is usually composed of several flow-invariant sets, including critical points, periodic orbits, and the stable and unstable manifolds of the critical points and periodic orbits, as well as other more complicated types of recurrent sets. By isolating each of these elements, we obtain a qualitative picture of the behaviour of a flow. When we study bifurcations, the easiest qualitative changes to consider are associated with changes in these elements because of the description of the dynamics in terms of critical elements is relatively complete and well-understood.  Definition of a D y n a m i c a l System  In the most general terms, a dynamical system is the deterministic evolution in time of the states of some state space. We shall restrict ourselves to dynamical systems defined by smooth vectorfields on spaces such as R  n  and other finite-  dimensional smooth manifolds. For example, suppose X is a smooth vectorfield defined on an open subset U of R  n  which vanishes off a compact subset of U.  Then X is a smooth map X : U — • R  n  which defines a differential equation  f = *MThe solution to this differential equation is a map  $ : £7 x  R — > t/,  -1-  where $(x, •) is the unique solution to the initial value problem -x(t)  The map  = X(x(t)),  z(0) = x.  $ is called the flow of the vectorfield X. Several properties of the flow  follow from elementary existence and uniqueness theory for ordinary differential equations—A solution to the above initial value problem exists for suitably small t, and any such solution can be extended to be defined for all real t. Furthermore, this solution is guaranteed to be unique and to depend smoothly (with the same smoothness as X) on initial conditions and time. Thus, it is is easily shown that the flow is a 1-parameter group of diffeomorphisms of U under composition in the following way; *(;S  + t) =  *(., )o<i>(;t). S  The flow of a dynamical system gives the time evolution of states/points/initial conditions in this way. Also, given a flow $ on a subset U of R", we may obtain the associated vectorfield X by differentiating with respect to time, i.e., X(s) = !*(0,x).  The derivative of the flow in the variable x is given by the variational  A full accounting of these results can be found in Abraham et. al.  J1983J,  equation  sect.  4.1.  In subsequent chapters of this thesis, we will be exclusively concerned with dynamical systems defined on compact smooth manifolds. This restriction is necessitated by the requirements of the evaluation transversality lemmas used in chapters 5 and 6. While there is no difficulty in defining a flow on a compact manifold, it is not immediately obvious what is meant by a vectorfield on a compact manifold, and so not obvious what the result of differentiating aflowwith respect to time would be.  1.1 We will define vectorfields on manifolds and derivatives of maps between manifolds in sect. 2.1. For the present we will assume that we are working with dynamical systems defined on R".  Most of the definitions in this section are topological in  nature, so that corresponding generalizations to the case of dynamical systems on smooth manifolds is immediate. O r b i t s , Trajectories  7 (i) of a point x £ R  The trajectory  for a vectorfield X : R  n  2  n  —* R  n  is the  solution to the initial value problem  j lx{t) or in terms of the flow $x  7x(0) = x,  =X{ {t)),  t  lx  of X;  lx  {t)  = *x(x,t).  The or6i£ of the point x is the set of points in the (range of the) trajectory of x;  0{x)  =  { {t)\teR} lx  = The trajectories of X orbit of X.  partition R  n  e R}. into orbits—each point of R  is in exactly one  n  Also, the orbits are trivially the smallest sets invariant under the flow.  Critical Elements  Certain kinds of orbits are of particular interest. A critical  point  consisting of a single point. The trajectory of a critical point x € R constant solution  = x.  l {t) x  A periodic  point x whose trajectory ^j (t) x  T G R  such that ~i { ) T  x  —  orbit is the orbit of a periodic  is a periodic  trajectory-  (t  The smallest positive r such that I {T) X  + )= T  + T) =  *{x,t = *{x,t)  -3-  *{lz{T),t)  = >y {t), x  is just the point; i.e. a  which means there exists  the period of the periodic orbit/point/trajectory. Note that lx  n  is an orbit  — x is called  1.1 so we see that all points of the periodic orbit are periodic with the same period. The critical points and periodic orbits of a vectorfield X are collectively referred to as the critical recurrent  elements of X and denoted by  sets of the vectorfield  These are the most basic  X.  L i m i t Sets A trajectory *) of A' that remains bounded will have an orbit with compact x  closure. In the case of a dynamical system on a compact set or manifold, all orbits will have compact closure. For such orbits we may define the a-  and u— limit sets  of the trajectory ~i by x  {x)  U  = fl  { (t)\t>T}, lx  T>0  and a(x)  = P|  { (t)\t<T}. lx  T<0  The set a(x), (resp.u(x))  are where the trajectory through x ends up when / —>  oo (resp.t —» — oo). Intuitively, the orbit through the point x is "born" in a(z) and "dies" in w(x). Orbit  Structure  Critical elements are their own a - and LO -limit sets, and so exhibit a very strong form of recurrence. The a- and w-limit sets of any point are invariant under the flow. The orbit through any point x "joins" its w— limit set to its a— limit set in the sense that the trajectory 7 CJ(X)  Z  through x as the trajectory tends to the invariant set  as t -> —co and tends to a(x) as t —* +00. Of course, these sets  CJ(X),Q(X)  may be equal, such as in the case of critical elements. We may think of the orbit structure of the dynamical system, in qualitative terms, as consisting of recurrent sets that are joined by other "connecting" orbits. If S is a compact flow-invariant set for the flow $, then we may define the inset -A-  —  1.1  (resp. outset) of S is the set of all points x such that UJ(X) C S (resp. a(x) C S). We can obtain a great deal of information about the orbit structure by describing the recurrent sets of the flow and indicating which of these recurrent sets are joined by "connecting" orbits, i.e., orbits that are in the inset of one recurrent set and in the outset of another. Of course, the structure of the recurrent sets and the various connections can be very complicated. In the simplest case, the only recurrent sets would be the critical elements of the dynamical system, and the orbit structure of the system would consist of critical points, periodic orbits, and orbits connecting various critical elements. More general cases involve more complicated recurrent sets associated with less strong notions of recurrence. One important type of recurrent point is a non-wandering point. A point x is non-wandering if for any neighborhood U  x  of x and time T > 0, there exists a t > T such that some of the orbits starting  in U have come back to U ; i.e., x  x  The set of non-wandering points of a vectorfield X is denoted by fl{X). This is a comparatively general notion of recurrence, and difficult to understand well. Below we shall consider a family of dynamical systems where all of the recurrence is in the critical elements. Stable Manifolds of Critical Points  Since the trajectory of a critical point is constant, it is easy to compute the derivative of the flow with respect to initial conditions at a critical point. We recall that  if i is a critical point. Thus, j^$(x,t) the matrix j^X(x)  — exp(i • -^X(x)).  The eigenvalues of  are called the characteristic exponents of the critical point x. -5-  1.1  The characteristic exponents of a critical point are the various exponential rates of growth and decay for the linearization of the flow at the critical point. One might (correctly) expect that if none of these rates were zero, that the linear behaviour would be dominant near the critical point. For example, if all of the characteristic exponents of the critical point x have negative real part, then the linearization of the flow contracts all perturbations exponentialy and we can show that x is an asymptotically  stable or attracting critical point. This means that there is a neigh-  borhood V of x such that for any e -neighborhood U of x there is a T > 0 such x  that $(V ,t) x  C U for all t > T. In particular this means that x is the u>-limit of  all points in V , and that V is in the inset of x. x  x  If no eigenvalues of the critical point x have zero real part, then x is a hyperbolic critical point. In this case the linearization of the flow contracts perturbations in the subspace E  s  corresponding to eigenvalues with negative real parts and expands  perturbations in the subspace E  u  corresponding to eigenvalues with positive real  parts at various exponential rates. The stable manifold theorem tells us about the structure of the inset and outset of x\ Theorem (Local Stable Manifold Theorem). of the smooth vectorfield  X,  the subsets W*(x),W*(x)  are submanifolds  of U , t  point x. Furthermore,  If x is a hyperbolic  then there is an t -neighborhood  of U  £  which are characterized  U  €  critical point  of x such that  by  called the local stable and unstable manifolds of the critical the tangent spaces T W*(x),T W*(x) x  x  manifolds at the critical point are the subspaces E ,E S  U  of the local stable  mentioned  above.  A statement of this result with references is given in Guckenheimer and Holmes [1983], p. 13, Theorem 1.3.2, and is also given with proof as Theorem 27.1 of Abraham and Robbin [1967]. -6-  Points in W*{x)  are attracted to x along W*{x) at exponential rates. This  manifold is the set of points near x that are attracted to x without first wandering away. Points that are not in W/(x) may be attracted to x eventually, but will first have to leave the neighborhood U . t  Corresponding statements hold for W"(x).  Clearly, the local stable manifold is invariant under the flow in positive time. If we take the union  w*(x) = | J *(WY(X),O t£R  then W (x) is an injectively immersed flow-invariant submanifold, as it is an ex5  panding union of embedded submanifolds since the local stable manifold is invariant under the flow in positive time. W (x) s  is called the stable manifold of x, and is  in fact the inset of x as the orbit of any point in W  s  must eventually end up in  the local stable manifold, and so be in the inset of x, whereas points not in the stable manifold will never end up in W £ (x), and so must leave the neighborhood U  t  of x for arbitrarily large time. This characterizes the set of points that tend  asymptotically to the critical point x under the flow. Similarly we may define the unstable manifold of x. Thus, the inset and outset of a hyperbolic critical point have the structure of immersed submanifolds.  Periodic Orbits, Characteristic Multipliers  We would like to extend the above results to periodic orbits. Specifically, we want to know which orbits will be attracted to a periodic orbit i(t).  If r is the  period of -7, then and x is a point in the orbit of 7, then <J>(x,r) = x. We would expect that orbits near 7 would approach 7 along directions transverse to the orbit of 7 which are contracted by j^^i', )7  We need to consider the  asymptotic behaviour of perturbations transverse to the orbit of 7. Let S be an 11- 1 dimensional subspace (or submanifold) that intersects the orbit of 7 transversely at x ( X(x) does not lie in S ). Then we may define a diffeomorphism  1.1  -  of a neighborhood of x in S, called the  Poincare  map  of the periodic orbit 7.  Points y of S are denned by n • (y — x) = 0, where n is the normal vector of the subspace S. Then n-($(x,T)-X)  — 0 and since ^ ( n - ( $ ( i , r ) - = n-J^$(x,r) =  n • A'(x) 7^ 0, then by the implicit function theorem, we have that there is a map r(y) in a neighborhood U of x such that n • (4>(y,7"(t/)) - x = 0. This means that $(y,T(y)) 6 5 . If we restrict y to S, then we end up with a map from a neighborhood of x in S to a neighborhood of x in S. This is the Poincare map of the periodic orbit 7 which we shall denote by 0. We have for y in S near x  e(y) = *(y,'(y)), 0 : S|(Snt/) -> 5. The Poincare map is smooth (as smooth as the vectorfield X) as it is the composition of smooth maps. Furthermore, 0 is a local diffeomorphism by the inverse function theorem. This follows as we may compute d d - 0 ( x ) • v = -*(*, =  for t» in S. Since ^ $ ( x , r )  r).v  —<S>{X,T)-V  +  +  d d -*(x,r) • - r ( x ) • v ( —  T ( X ) - V ) X ( X )  is onto R , then the kernel of J^0(x) must consist n  of vectors that are mapped along the direction of X(x) by invariant under ^ $ ( x , r ) by the following; *(x,r) = * ( * ( i , 0 ) , r ) , so that differentiating with respect to time gives ^ * ( x , r ) = X(*(x,r)) = X(x)  = JU(*(x,0),r) = £ # ( . , r ) . £•(,,(>) = £d>(x,r).X(x). -8-  J ^ $ { X , T )  .  But X(x) is  1.1  Thus, the kernel of J^0(x) is along the direction of X(x), which is not in 5. The Poincare map  of a periodic orbit describes the behaviour of nearby orbits in  directions transverse to the orbit. The image of a point y in S under 0 is the point of first intersection (in positive time) of the orbit through y with S. The asymptotic behaviour of points under 0 indicates the asymptotic behaviour of the associated orbits. If the successive images of point approach x, then the orbit through this point is asymptotic to the periodic orbit 7. the Poincare map  But the main reason for introducing  is that there is a version of the stable manifold theorem for fixed  points of diffeomorphisms. We  expect that nearby orbits will be attracted along  directions that are expanded by the derivative ^ 0 ( x )  of the Poincare map,  and  repelled along directions that are contracted by ^ 0 ( x ) , by analogy with the stable manifold theorem for critical points. We Theorem  (Stable M a n i f o l d  diffeomorphism eigenvalues  T h e o r e m for  with fixed point  of unit modulus),  state the theorem  x. If x  Diffeomorphisms).  is a hyperbolic  (Df  fixed point  then there is a neighborhood  U  Let  f  be a has  of x such that  no the  sets = {v  L( )  W  X  ioc( )  w  x  e  = {y^  U\f (y) n  U\f (y) n  - » i o « n - 4 + o o and - 0 0 and  -> x as n  f {x) n  f (x) n  are submanifo/ds in U, and the tangent spaces  T Wf (x)(resp.  manifolds  to eigenvalues  greater  at x are the subspaces than  corresponding  x  oc  € U, n >  o}  € U, n<  o}  T W^ [x)) x  of Df  oc  (x) of  to these modulus  1 (resp. less than ]).  All of the observations we made for the local stable and unstable manifolds of a critical point hold for W^W"  .  The eigenvalues of the Poincare map or Floquet  multipliers  are called the characteristic  multipliers  of the periodic orbit. A periodic orbit is hyperbolic if the  corresponding fixed point of its Poincare map -9-  is a hyperbolic fixed point. We define  1.1 the stable and unstable manifolds of a hyperbolic periodic orbit 7 as the unions of all orbits passing through the local stable and unstable manifolds of the Poincare map of the 7. These are injectively immersed submanifolds which are the inset and outset of the periodic orbit.  Morse-Smale Systems Now we define a family of vectorfields for which we have a fairly description of the orbit structure. A vectorfield is a Morse-Smale  complete  vectorfield if it  satisfies: (1) There are a finite number of critical elements, and each is hyperbolic. (2) A l l stable and unstable manifolds of critical elements must intersect transversely, and (3) T h e non-wandering set Q consists only of critical elements. T h i s defines a family of vectorfields whose orbit structures are relatively simple. Given a Morse-Smale vectorfield X, Definition.  The phase diagram  we may define its phase diagram  T of a Morse-Smale  vectorfield  X  T, is the set of  critical elements of X with the following partial order: If C\, 01 are critical elements of X,  then o\ < 01 if W (o}) n W (o )  an orbit joining  s  u  2  o\  7^ 0- In other words, o\ < 02 if there is  to 02 that is "born" in 02 and dies in  o\.  T h e phase diagram of a Morse-Smale vectorfield gives us a great deal of information about the flow of the vectorfield.  -10-  1.2  1.2  A n Example of a Bifurcation  In this section we will examine the bifurcations that arise from the failure of a critical point of a vectorfield to persist smoothly under perturbations of the vectorfield. The usual approach to bifurcation theory involves dynamical systems depending on parameters and the analysis of qualitative changes that occur in the dynamics as the dynamical system is perturbed by varying the parameters. As an example, let us consider a vectorfield X on a 1-dimensional manifold ( i ? , or its 1  one-point compactification S — i f we insist on compactness) depending on a scalar 1  parameter /z . Then X  is a function  X :R  x R  1  Assume that X is at least C  2  1  —• R  1  : (X,LI) .—> X ( x , / x ) .  .  Non-Degenerate Critical Points  Now, suppose that xo is a critical point of X(.,LIQ) , i.e., that X ( x n , M o ) = 0. We  may  examine what happens to the critical point xo as the parameter // is  varied near HQ by looking at the solutions of X(x,/z) = 0 that are near (xo,/xo) . For example, if J ^ X ( x n , M o ) ^ 0, then by the implicit function theorem of calculus, there is a smooth (at least C ) function x{yi), defined near fin 2  X(X{H),LI)  s  u  c  n  that  = 0. Furthermore, the implicit function theorem states that the curve  (x(/z),/Lt) is the unique solution of  X(x,n) = 0 in some neighborhood of (xn,Mo)-11-  1.2  Thus, the critical point x  0  varies smoothly as the parameter \i is varied near no  and no new critical points appear near XQ . This is shown in the graph of Fig. 1.  Critical points of this type are said to be non-degenerate.  Degenerate C r i t i c a l Points  However, if the critical point xo is degenerate in the sense that j^X(xo,  Ho) = 0  then it may fail to persist as the parameter n is varied. If ^ X ( x n , M o ) ^ 0 then the implicit function theorem may be applied as before to conclude there is a smooth function /x(x) such that X ( x , j i ( x ) ) = 0 near Xo. Furthermore, fi'(xo) = 0 as j^X(x,fj,(x))  + j^X[x,  n(x))fx'(x) = 0 by implicit differentiation. The graph of such  a curve (x,//(x)) is shown in Fig. 2.  We can obtain some qualitative information about the bifurcation occurring at (xo,/io) from the graph of Fig. 2. When // < no we see that the vectorfield X(-,/z) has two distinct critical points near xo. At the bifurcation value of the parameter, HQ,  there is only one critical point of  X(-,HO)  indicated by the graph,  XQ.  For  parameter values above fio there are no critical points of X(-,/z) near XQ. There is an obvious change in qualitative behaviour as the parameter /j, is varied through the bifurcation value no  3 5  the number of critical points of the vectorfield near xo  changes. This change in behaviour is caused by two distinct critical points of the vectorfield coalescing and annihilating each other.  T a y l o r Series Conditions  Of course, the above qualitative analysis depends on a qualitative property of the graph of Fig. 2—that it is concave at xo . A sufficient condition for this concavity is n"{ o) x  0. In terms of the Taylor series of the vectorfield X -12-  at (xo,Mo) the  "f Figure  1.2.2  -13-  A.2  sufficient condition above is equivalent to the following: Q X{x ,Li )=0, 0  — X [ X  0  ox  0  , L I  0  )  =  0,  ^  0,  Q  —  X(XO,LLO)  d -^ ( o^o) ox* 2  a  Indeed,  X(X,LI(X))  —  d  n  d  x  ± 0.  x  0, so that  d X{x,n{x)) + ^-X(x,n{x))n'[x)  OX  = 0,  OLL  and also d2  d  2  dx dxdfx — A'(x,//(x)) + 2^ ^-2X{x,ii{x)) i'{x) 2  z  6d V  X(x,Lt(x))(^'(x))  2 2  +  r  '  2  ,  + — X(x,//(x))//'(x) = 0. d cV  by implicit differentiation. We have shown that  = 0 as J ^ X ( x o , ^ ( x ) ) =  H'(XQ)  0  a  g^X(xo, LIO) = 0. Then, from the last relation,  _af  2 A'(x ,Mo) 0  ^X(x ,// ) 0  so that  M"(XQ) ^  0 if and only if  0  no) ^ 0.  J^X[XQ,  Now consider the graph of the function X(-,n) fixed //. For  LI —  that the function  LIQ  we have that  X(.,Lto)  : R  1  —• R  a  9  : x >—> X ( x , / x ) for  o2 -^X{XQ,LIQ)  - ^ X ( x o , ^ o ) — 0 and also  has a local extreme value at x =  1  XQ  ^ 0 so  . Also, X ( x o , / i o ) = 0  so that this local extreme value is zero. From the implicit function theorem, there is a C X(LIQ)  2  function £(//) defined for LI near no such that J ^ X ( x ( / / ) , / / ) = 0 and  = xo as we have assumed that -^2X(xo,no) ^ 0. We can see that the point  x(/i) must be a local extreme point for the graph of neighborhood of [xo,no) • Also, since X ( x o , M o ) local extreme value for the graph of X(-,no)  =  X(-,LI)  as j^X  ^ 0 in some  X ( X ( / / ) , M ) = 0 we know that the  is zero. Let us compute the derivative  of the extreme value as the parameter LI is changed, j^X (x(n), n) , at fi — no . d —  X(x(/z),/z)  d  = —A"(z(/i),/i)x'(/x) -14-  +  d  —X(X(/X),M).  But, at fi — fiQ , we have x(//o) = %Q , and also ^X(xo,/io) = 0 so that — dp  X{X{HO),HO)  =  ^-X(X ,HO) OH 0  ^  0,  so that the value of the local extreme point X(H) of X(-,/x) changes sign as H is varied through the bifurcation value HO • This shows us that the graphs of the X(-,fj.)  are qualitatively the same as what is shown in Fig. 3.  In Fig. 3. we see that as the parameter H is increased through HO the local extreme value of the graph of X(-, H) (which in this case is a local minimum) changes from negative to positive. This induces a change in the qualitative behaviour of our system near  XQ  . Because the graph of  X(-,H)  is concave up near  XQ  , then  when the local minimum of this graph is negative (H < HO) then there must be two distinct zeroes of the graph of X(-,/n), which will correspond to two distinct critical points of the vectorfield X(-,/x). As the parameter H approaches the bifurcation value HO these two zeroes approach each other, until, at the bifurcation 5  value HO the the two zeroes of the graph meet, and the graph has a double zero corresponding to a quadratic tangency. Above the bifurcation value HO the graph of X(-,H)  has no roots near XQ. It is worth noting that this qualitative analy-  sis depends only on the conditions we have given on the Taylor series of X at (XQ,HO)  5 which  state that X has a root at (io,Mo) which is non-degenerate in the  //-direction (i.e. j^X(xo,HO) (i.e.  ^X(X ,HO) 0  ^ 0) and quadratically tangent in the x-direction  = 0, ^X{xo,HO)  ^  0)-  Orbit Structure In the 1-dimensional case it is easy to obtain the phase diagram  of a vectorfield  from its graph. The critical points of the vectorfield are the zeroes of the graph of the function. A zero such that the derivative j^X  is positive is a source, and if  is negative then it is a sink. A double zero is neither a source nor a sink, but is instead the coalescence of a source and a sink as shown in the middle phase portrait -15-  -16-  1.2  of Fig. 3. The critical points of the vectorfield are joined by orbits between them, the direction of these orbits being determined by the sign of the function in between the two zeroes associated with the critical points. For adjacent pairs of non-degenerate critical points the orbits joining them will start in a source and end in a sink. From this viewpoint the bifurcation shown in Fig. 3 is caused by two critical points, one source and one sink, coalescing and annihilating one another. Generic Conditions for Vectorfields In the above analyses, we have prescribed various conditions on the Taylor series of the vectorfield X about a critical point (xo,Mo) • It seems reasonable to ask why we should choose these particular conditions instead of some others, or to ask if are "likely" to hold. Consider a vectorfield X on R  1  with a critical point xo which  is non-degenerate in the sense that ^ X ( x o ) ^ 0. We know that such a critical point will persist smoothly under small perturbations of the vectorfield X, indeed, if X(x, e) = X(x) + f f ( x ) , then we have already seen that for small values of t the vectorfield X has a critical point x(e) near xo where x(e) is a C  2  function. It is  easy to see that this new critical point x(e) will also be non-degenerate as  which will be non-zero for t sufficiently small. This shows that non-degenerate critical points persist and remain non-degenerate under small perturbations. On any compact subset of R  1  (or on a compact manifold such as S ), there will be 1  only a finite number of non-degenerate critical points, as non-degenerate critical points are separated from other critical points by finite distance. So, in the case of a vectorfield defined on a compact set or manifold, the property of a vectorfield having all of its critical points be non-degenerate will persist under suitably small perturbations of the vectorfield. In the analysis of the bifurcation above, the vectorfield X(.,H) -17-  had a degenerate  r  V  \  Figure  1.2.4-  Perturbing  a vectorfield  critical point XQ for one value fi  0  to  make  0  a regular  value  ^  °f the parameter, but this degenerate critical  point either vanished or became two distinct non-degenerate critical points if the value of the parameter ft was changed. Most critical points of vectorfields are nondegenerate in the sense that a vectorfield with some degenerate critical points can, through arbitrarily small perturbations, be made a vectorfield whose critical points are all non-degenerate. Indeed, Sard's theorem states that for a C to R  1  1  map from R  1  that the set of critical values of the map has measure zero and hence is  nowhere dense. If the vectorfield A' : R  1  -> R  1  has a degenerate critical point,  then 0 is not a regular value of X. But we can find an arbitrarily small e £ R  1  that is a regular value for A' (unless we have chosen X = 0, which is silly). Then the perturbed vectorfield X = X — e has zero as a regular value and all the critical points of A' are non-degenerate as an Fig. 4. Thus, we see that for such vectorfields on a compact space the property of -18-  1.2  having all critical points non-degenerate is "open" in the sense of persisting under perturbation and "dense" in that any vectorfield can be approximated by one with only non-degenerate critical points. This property is then said to be generic for vectorfields (on 1-dimensional compact manifolds). G e n e r i c Conditions for 1-Parameter Vectorfields For vectorfields depending on a parameter, we still expect that the zeroes of the function X : (x, n) H-> X(x,n)  will be generically non-degenerate, but now the  non-degeneracy means something different for the critical points of the various vectorfields to  X(.,LI)  from the non-parameterized case. Since X is now a map from  R , the derivative of X at a point 1  DX(x ,no) 0  (XQ,MO)  : R — • R : (v,w) 2  1  2  is linear map a  d  >—> —-X(x ,no)v  dx  R  0  +  d — X ( X  on  , L I  0  0  ) W .  Even if (io,Mo) is a regular point of X , there will be some direction (vo,wo) such that DX(xo,no)(vo,  tun) = 0. If (xo,no) is a zero of X , then there is a along which the value of X is zero. This  curve tangent to (vo,tuo) at (XQ,LLO)  follows from a corollary of the implicit function theorem. Thus, one could still have ^X(x ,no)  - 0 if (zo,Mo) is a regular point of X , but only if J^X(xo,no)  0  If a vectorfield X(x,fi) and  j^X(xo,no)  has a critical point (xo,no) such that  7^ 0.  no) — 0»  J^X(XQ,  ^ 0, then, does this type of critical point persist under pertur-  bation? Under the non-degeneracy condition we have assumed for this bifurcation, ^ j X ( i o , M o ) 7^ 0, we may apply the implicit function theorem to the map JX L  T  where X  : {x,n,t) 1 — » ( X ( z , / x ) , — X ( x , / / ) ) £  4  is the perturbed vectorfield  E  X ( x , / x ) = X(x,n) e  +  e£{x,n).  Indeed, we compute d —  J X (X,M,C) = ]  £  d ( — X(x,n)  d d + t — £{x,ii),-^X(x,ii) 2  -19-  d dx^^ ^ 2  +  e  X,fl  and  when (x,n,e) =  , this becomes  (XQ,PO,0)  £j'X {x ,p ,0)\ (  0  0  0  ^A'(z ,Mo)  \^A(x ,Mo),  sgi^xo./xo)  (  =  a\J M*»iW>#)) l  0  0  as g^A'(io»Mo) = 0. This is an invertable matrix as both of the terms  ^A(XQ,^O)  and ^A"(xo,/io) are nonzero. Thus, it follows from the implicit function theorem that there are points (XQ(C),Mo(0)  depending smoothly on e such that  A' (i (c),/io(0) = t  0  0  and  — X (io(e) Mo(e)) = 0. £  5  of the whole 1-  Thus, the degenerate critical point xo persists under perturbations parameter  family  of vectorfields  X(x,p).  Since a 1-parameter family of vectorfields  is the smallest family which may contain this kind of degenerate critical point in a persistent way (recall that it wasn't persistent for vectorfields with no parameters), we call this type of critical point a codimension-1 We  critical point.  may also use Sard's theorem to obtain an approximation result for 1-  parameter families of vectorfields. Consider the map JX l  : (x,p)  > (X(x,p),  Since we have assumed that X was at least C  2  ox  £-X[x,»)).  , then the map J X is a C 1  Sard's theorem gives us the existence of a regular value [to,t\) of J X, 1  1  map.  arbitrarily  close to (0,0). Then, if the range of X contains a neighborhood of 0, we have that for the perturbed vectorfield X(x,n)  = X{x,p) -20-  - t  0  ~  1.2 (0,0) is a regular value for the associated map  J'X.Then, if  J X(IQ,MO) — 1  0  5  or equivalently X(xo>Mo) = 0 and J^X(zo,/in) = 0, we know that the derivative map DJ X(xo,Mo) !  is surjection onto R a  D^Xizoifio)  2  . We compute  ^rX(z ,Mo)u + ^ X ( z , M o ) w 0  a-  w  \  0  I  92  ~  ^X[XO,LIO)W H-  g^X{x ,(j, )v 0  as ^ X ( x o , M o )  =  0  ft-  + g^A'(x ,A*o)w) 0  0. This map can only be a surjection onto R  non-degeneracy conditions f^X(x ,n ) 0  0  ^ 0 and f^X(xo,no)  Therefore, degenerate critical points of the vectorfield X(.,fx)  2  if both of the  ^ 0 are satisfied. must satisfy these  non-degeneracy conditions. This shows that this type of codimension-1 degenerate critical points are generic for 1-parameter families of vectorfields.  -21-  2.1  2.1  Tangent Bundle, Vectorfields  In the study of dynamical systems we regard vectorfields as differential equations whoseflowsdefine dynamical systems. In this chapter we are mostly concerned with the set of critical points of a given vectorfield and as such shall consider a vectorfield as a differentiable map between manifolds. While there is no difficulty in defining a dynamical system by its flow on a compact mainfold, it is not immediately clear how to define the associated vectorfield. Specifically, a vectorfield X on M is a smooth map from M to what manifold? Local Vectorfields In the case of a flow $ defined on an open subset U of R  n  , a vectorfield X  is given on U by the differential equation  |*(*,0 = *(*(*,0) or X{x)  =  j *{x,t)\ . t  Then, X is a smooth function X : U —» R  n  t=0  which we call a local vectorfield. Note  that the value of X at a point x depends only on the tangency of the integral curve *) {t) — ${ ,t) x  x  at f = 0. Indeed, one possible approach to extending the  notion of tangent vectors and vectorfields to manifolds that is adoped by many texts (Abraham and Marsden [1978], Abraham et. al. [1983], Chillingworth [1976]) is to define tangent vectors to a manifold by the equivalence class of curves on the -22-  manifold that are tangent at a point. We shall follow a slightly different approach here in order to motivate the important concept of a vector bundle.  Transformation  Rule for Tangent Vectors  F i r s t , we consider what happens to vectorfields under a change of coordinate system. We require that our vectorfield in the new coordinates has the same flow as the old vectorfield after changing coordinates. Precisely, let X : U —> R " be a local vectorfield and <p : U —• V be the difleomorphism  that gives our change of  coordinates. Then, we require that our new vectorfield X' defined on V has the flow ip o  where $ is the flow of A' on U. Thus  = Dp(*(x,0)-^*(*,0, =  D<p{*{x,t))-X{*{x,t)),  or equivalently X'{y) =  Dv{<p- {y))-X{<p- {v)). 1  1  T h i s is the transformation rule for vectorfields (indeed even vectors) under changes in coordinates. Now suppose we have a flow $ on a compact n-manifold M which has an atlas of charts {{U , <p )} . V i a the charts, the flow $ defines local flows $ a  a  a  on open  subsets of R by n  A s we have already seen, these local flows give rise to local vectorfields  X  Now  : 7L \<p [U ) n  a  Q  a  —> R . n  we will apply the "globalization process" which will "patch together" these  local objects X  Q  into global object which w i l l be a vectorfield on M. -23-  2.1 By our transformation rule for vectorfields, we know that  X  where <p fl = <p oip~ , x  a  of  a  p  = D<pp [<p p{x)) a  •  a  <pp = <pp otp~  X {<p p(x)), a  a  are the chart transition maps for our atlas  l  a  M.  Equivalence Relation  Consider the disjoint union  x <p (U ) x R " .  S = \J{a}  a  Q  a  O u r transformation rule for vectors motivates the following equivalence relation on S\ (a,x,i>) ~ iff  (0,y,w)  x = <p p{y) a  and  v = D<p p(<pp (x)) a  • w,  a  or equivalently x = <Pa(3{y)  and  v = D<p p(y) a  • w.  We see that this does define an equivalence relation from the chain rule. Indeed, reflexivity follows as X = <Pap{y)  y = <PBa[x)  and so v = D<p p(y) a  w = D<pp (x)  •w  a  •v  since D<p p{y) • D<pp (x) a  a  = D<p p{<pp {x)) a  = D(ip p a  -  D{id)  -24-  a  • D<pp {x)  o <PBa)(x) = id  a  2.1 by the chain rule. Transitivity of the relation also follows from the chain rule. We have <p = tp p o <pp , so that 1  1Q  a  = D<p p{<pp (x)) • D<pp (x)  D<p {x)  1  ia  <Pia — P-,[i  0  a  a  <PpQ- by the chain rule.  Tangent B u n d l e We define the tangent  If Ad is a C  bundle  TM  of the manifold M  TM  = S/ ~ .  by  manifold, then the tangent bundle has a C "  given by an atlas on TM  1  manifold structure  that is inherited from our atlas {{U ,<p )} a  a  on M  as fol-  lows: Letting [(a,x,v)] denote the equivalence class of (a,x,v) under the relation ~ , the charts of this atlas are (TU ,T<p ), a  TU  a  where  a  R ),  = {\{a,<p {p),v)} :peU ,ve  n  a  a  c  and <—> (x,w).  T<p : \(a,x,v)} a  The charts {TU ,Ttp ) a  of this atlas are called vector bundle  a  charts for TM. The  transition maps for this atlas are given by T<p  aB  : R \vp(Up) n  x R " ^ R \<p {U ) n  Q  a  x R  71  : (x,u) '—• (V? (X),Z)V?Q/?(X) • « ) . q/?  It is easy to verify that these transition maps are C  r  _  1  diffeomorphisms and that  the so-called cocycle condition T<Pai = T<p p o T<pp^ a  follows from the transitivity of our equivalence relation ~ . It is not at all suprising that the transition maps for this atlas on TM -25-  came directly from the definition  2.1 of the equivalence relation ~ when one considers that the manifold M can itself  Uc»{}  a x  be defined as the disjoint union  'PaiUa)  partitioned by the equivalence  relation given by the chart transition maps; i.e. (a,x) ~ (0,y)  <=> x = <p p{y) • a  This is essentially what the globalisation process i s — t h e creation of a global object from its component local representations in different, local coordinates, and noting that such a global object is denned by any set of local objects that is consistent with a transformation rule for the object in question.  Vector B u n d l e s The tangent bundle TM is the prototypical example of a vector bundle. That TM  is a vector bundle means:  1. TM has a local product structure given by the vector bundle charts as  = R \ip (U )  T<p (TU ) a  x R ,  n  a  a  n  a  so that TM is locally diffeomorphic to the product of a neighbourhood in M ( U ) and a linear vectorspace ( R ). n  a  2. The maps T<p  a  are "inherited" from the chart maps <p , in that the first a  component of the map T<p  is the chart map <p ; i.e.,  a  Q  T<p (\{a,<p {p),v)}) a  a  = [<p [p),v), a  or, for any tangent vector v € T M, (see below) we have v  P  T<p {v ) = {<£>a{p),something G R ) . n  a  v  3. The chart overlap maps in the atlas of vector bundle charts are linear isomorphisms on the second factor; i.e. for fixed x ,  T<PaB{Xit)  = {<p p(x),D<p (x) a  -26-  a0  • tl)  2.1  is a linear isomorphism in the variable £ £ R  from the vectorspace {x} x R"  n  into the vectorspace {<p p(x)} X R" . Also, the chart overlap map 'p fi is the a  a  first component of the overlap map T<p p as was hinted at in (2) above. a  We see that the vector bundle charts preserve the vector space structure of the set TM P  :v6R }.  = {!(<*,Pa(p),v]  n  This set is called the fiber of TM over p. Clearly T M is isomorphic to R . We n  V  may also define T M as V  T M=  TT-^P),  P  where 7r : TM —» M is the natural projection map of the tangent bundle given by  <f>Miv)\) = p. Items (2)—(3) above state that the vector bundle chart maps Ttp T<p \T M a  satisfy  a  = <p {p),  p  a  and that this restricted map is an isomorphism from T M and R . n  p  Vectorfields, Sections The set T M is also called the tangent space to M at p and can be thoughtof p  as the space of all tangent vectors to M which are based at p. A vectorfield X on M is a map which takes a point p £ M to a tangent vector to M at p; i.e., X{p) £ T M. Equivalently, a vectorfield is any smooth map X : M —* TM which p  is a section of the projection map TT ; i.e. that satisfies {noX)(p)=p,  Vp£M.  The space of all C -vectorfields on M , or sections of TM /sections of n is a linear space which we shall denote by X ( M ) . In section 2.4 we shall define a topology of r  -27-  2.1 X (M) r  that makes it a Banach space. Given that  will be able to establish properties of the evaluation  ev : X ( M ) x M -> TM r  X ( M ) is a Banach space, we r  map  : {X,p) >-> X(p)  E  TM  that we shall use in our analysis of the bifurcations of the critical points of vectorfields on M .  -28-  2.2  2.2  Jet Bundles  In our study of bifurcations of critical points of vectorfields on 1-dimensional manifolds in section 1.3, we looked at the effect of various assumptions about the Taylor series of a vectorfield X about one of its critical points p on the changes that could occur in the set of critical points of the vectorfield and also in the local dynamics of the vectorfield under perturbation. In generalising this approach to consider the case of vectorfields on compact manifolds, we realize that the Taylor series of a vectorfield depends on the coordinate system in which it is expressed. In order to make the most consistent use of the tools of differentia] theory, we shall consider another example of a vector bundle, the bundle of k-jets of vector fields on a compact manifold M . In the same way that a tangent vector v(p) G T M is p  a coordinate independent object that is represented by its local representatives in a coordinate independant way, the k-jet of a vectorfield X at a point p G M is in essence a coordinate independant notion of the k-th order Taylor polynomial of any local representative X  a  at the corresponding point <p (p) G R Q  n  . The bundle  of k-jets of vectorfields arises as the globalisation of local (coordinate) definition of the k-th order Taylor polynomial of a local vectorfield at a point. Transformation  R u l e for Taylor Polynomials  Recall that a vectorfield X G X (-M) is a smooth map r  (TU ,Tip ) Q  a  is a vectorbundle chart for TM , then X\TU  a  :U  a  -29-  —-  TV , a  X : M —• TM. If  2.2  and the local representative X T<p oXop- :  a  a  <p (U ) C R " -  1  Q  and X  is defined using the chart maps tp ,Ttp  Q  Q  T<p (TV )  a  a  = <p (U ) x  a  a  by  a  Q  R, n  is the second component of this map; i.e., Tp  o X o<p~ (x) =•  (x,X (x)).  ]  Q  For a point p G U , the local vectorfield X a  a  a  has a k-th order Taylor polynomial  at x = <p (p) , Q  P * X ( x , /i) = X (x) + rjx (i) -ZU- ^ £ > X ( x ) • (h, h) + ... + 7 ^ * - a ( * ) - ^ ^ ^ 0 • 2  a  a  Y  a  Q  k times  The coefficients of this polynomial map P X (x)  are  k  a  X {x),DX {x), a  ^D X (x),...,  ir>*A- (i).  2  a  a  a  These coefficients lie in the vector space P | ( R ) of symmetric k-th order polynon  mials on R ; n  (X (x),DX (x),..., a  ^D X (x))  G R" x L(R )  k  a  n  Q  x ... x  L (R ) k  n  s  = P (R ), fc  n  where L ^ ( R ) denotes the space of symmetric j-fold multilinear maps from R n  R  n  n  to  . PX  In order to discover the transformation rule for the Taylor polynomials  k  a  under changes of coordinates, recall that X/}{x) = D<pp {<p e(x)) • a  To obtain the coefficients of P Xp(x) k  a  X {tp p{x)). a  a  in terms of the coefficients of  we differentiate the transformation rule for vectors; LfiXgix)  = D* [DiPBai'PaBix))  -30-  •  X {<p {x)) a  aB  P X (<p p(x)), k  a  a  The composition D<pp • X a  is bilinear; Hence, we may apply Leibniz' Rule for  a  bilinear maps, Abraham and Robbin [1967], p.3, which states that  0<l<q  whence D  q  -I  Xa{<Pap{x))  0<l<q  In order to differentiate the terms D<pp (<p p(x)),X (<p p(x), a  Q  a  a  we employ the com-  posite function rule, Abraham and Robbin (1967], p. 3, which states that  1<]<S  where the o (i\,...,  ij)  s  \i\ = S  are constants obtained inductively in the proof of the result.  Then we have  D<Pp [<Pap(x)) a  1<J</ $=l  and D - {x {^{x))) q l  a  Y  Y  (p <p p{x),...  o _ {n ,...,n )D X {<p p{x))m  g  l  1  rn  a  ,D "><p p{x)),  ni  a  n  a  a  1 <m<q — l jn-j=9-— /  so that the full change of coordinate formula becomes: D X {x) q  p  =  E E  E E  0<l<ql<j<l\<m<q-lfl=l\n\  |  P  = q-l  •DJ ip (<p {x)) aP  •D X (<p p(x)) m  a  Q  V  • • • ,ij),o -i{ni,... q  ' (D *< {x),...,D >-<p 3(x))  •  +1  Pa  u  •  -31-  i  i  al  Pal}  (p ^ p{x),...,D >»<p p{x)). n  n  a  a  ,n ) m  This transformation rule expresses the derivative D X p ( x ) , for 0 < q < k , q  in terms of X {<p p(x))  , the old coordinate representation of X and its first q  a  a  derivatives D X ( t p p ( x ) , . . . , D X (<p p(x)),  at the point ip p{x)  g  a  a  a  Q  a  that corresponds  to x in the old coordinates <p . It is interesting to note that the transformation rule a  for the k-th derivative of a vectorfield depends on the first k-\-1 derivatives of the chart transition maps. This is because the trasition maps for vectorfields involve the derivative of the chart transition maps, so that in order to consider C the manifold M must be at least C  r  +  1  vectorfields,  T  .  Bundle of k-Jets Given the change of coordinate formulas for the derivatives of a vectorfield, we know how the Taylor series transforms under changes of coordinates. Unfortunately, the transformation rules are rather unwieldy, and as such we will not proceed as we did in defining the tangent bundle. Instead, we use the following definition;  Definition. Let X, We say that and  Y  Y  pG  X (M),  E  a  C M , where  (U ,<p ) is a cnart on a  a  X and Y have the same k-jet at p if the local representatives  have the same k-th order  a  U  Taylor  polynomial  if and only if P X {<p (p))  =  k  a  at a point  The k-jet of a vectorfield  a  at the point  M  X  .  a  <p {p) J-e., a  P Y {<p {p)). k  a  p is the eqivalence  Q  class of vectorfields  having  the same k-jet at p. From our observations about the transformation rule for the Taylor series of a vectorfield, it follows that this definition does not depend on the choice of chart {Ua,  <p ) • Indeed, if local representatives a  X  a  and Y  a  have identical k-th order  Taylor polynomials at a point <p {p) then in new coordinates tpp , the coefficients 5  a  of the k-th order Taylor polynomials of the new local representatives  Xp and Yp  at the point <pp{p) may be expressed in terms of the coefficients of the k-th order Taylor polynomials of X  a  and Y" and so are also equal. Q  -32-  2.2  T h e k-jet of the vectorfield X at the point p is denoted by j X(p).  T h e set of  k  all k-jets of vectorfields at a given point p forms a vector space which we denote by J {TM)  . Indeed, the vector space structure is given in the way we would expect,  k  w i t h A • j X(p) k  = J (XX)(p)  .  k  a  n  d  J X(p)  + j Y(p)  k  J {TM)  =  k  = j {X  k  + Y)(p).  k  Letting  U J (TM), k  r€M  we may define a vector bundle structure over M as follows: Given a chart for M , the associated vector bundle chart {J U , k  a  JU  =  k  a  where 7r* : J {TM) k  Then J U  J <p )  (U ,ip ) a  a  is defined by:  k  Q  *?{V ) a  —• M is the natural projection map given by n(J [TM)) k  = p.  is the set of k-jets of vectorfields at points p G U . We define the chart  k  a  Q  maps J <p k  a  for the bundle by J <p  : J {TM)\J U  k  k  a  :  j  k  — • R  k  X  Q  ( P ) r —  n  P [R )  x  k  n  (<p (p),P X (<p (p))). k  a  a  a  In other words, the chart map assigns to a k-jet j X{p)  the k-th order Taylor  k  polynomial of the local representative of the vectorfield X (or any vectorfield w i t h the k-jet j X{p) k  at p ) .  In order t o show that the charts (J U ,  J <p ) define a vector bundle structure  k  k  a  on J (TM) k  1. J {TM) k  Q  it suffices to note: has a local product structure given by the chart maps. T h i s is evident  as J <p (J U ) k  k  a  a  = R \<p (U ) n  Q  Q  x P|(R ). n  T h i s m a p is surjective, for if Qk € P * ( R ) is an arbitrary k-th order symmentric n  polynomial, we may define a vectorfield X' such that its local representative X'  a  has Qk for its k-th order Taylor polynomial at a point <p (p) . a  -33-  2.2  2. The induced chart transition maps J <p p k  a  J <p p k  a  : R \ipp(U ) n  p  x  = Jf  0  k  a  J <Pp k  >  l  p|(R») —» R > ( £ / ) Q  are linear isomorphisms of the space P (R ) k  n  x  a  for fixed x  E  P|(R"),  <pp{Up)  .  This follows  from looking at the transformation rule we obtained for the first k derivatives of a vectorfield under changes of coordinates. The maps we obtained were linear in the derivatives of the local representatives X (<p p(x)),..., a  D X (p p(x))) k  Q  n  These maps must also be isomorphisms as the map J <p p a  J <pp . k  a  As we have already noted, the chart transistion map 3 <p p  depends on the first  k  a  k+1  .  has an inverse given  k  by  a  derivatives of the chart transistion maps <p p of M . Thus, for a C a  M , we have that the bundle of k-jets of vectorfields is a C ~ ~ T  -34-  k  l  manifold  vector bundle.  S.l  3.1  The Banach Space  x (M). r  In the example of a bifurcation in Chapter 1, the vectorfield X depended on a scalar parameter /x and so was a map from R  x R  1  to R  1  (xn./zn) °^  bifurcation associated with a given critical point  geometry of the the set of critical points of X near (XQ,LI ).  1  ^  . We analysed the by looking at the  This analysis depended  0  heavily on the fact that X was a differentiate map, and as such we could apply several results from differential theory. In generalising this approach to bifurcation, we do not use a particular parameterised family; instead we think of a bifurcation as being associated with a particular vectorfield and wish to consider the possible changes in the set of critical points (or other parts of the dynamics) that occur when the system is perturbed.  The spaces B (<p (U ):R") T  n  n  Consider a finite collection of charts {{U , <p )}a=i that cover M. For a fixed a  a  a, we have the map that takes a vectorfield X € X (M) r  X :R \<p {U )  —+R ,  n  Q  Q  to its local representation  n  Q  defined through the vector bundle charts (TU ,T<p ) Q  by  a  X (v?a(p)) = second component of  T<p (X(p)),  Q  a  since we have T<p {X{p)) a  =  {<p {p),X {<p (p)). a  -35-  a  a  S.l  This map is a surjection  X (M)—>B {<p {U );R ), r  n  T  Q  a  where B (<p (U ); R ) is the space of C maps from the open subset (with compact r  n  Q  a  closure) pft([/a) of R" into R" which are bounded in the C-norm sup  We shall make use of the following well known result. Lemma. Let U c R " be an open set. Then the space B (U,R ) ||j  r  above is a Banach space.  Proof. Clearly B (U;R ) n  T  is a vector space under pointwise addition and scalar  multiplication of fuctions, and ||-|| is a norm on B (U;R ). r  n  r  B (U;R ) r  with the norm  n  T  We must show that  is complete in this norm.  n  Let {X } n  C  B (U;R ) T  n  be a Cauchy sequence in the norm ||-|j . Then, since the r  convergence is uniform, X ^X n  and for some continuous functions X,X  DX  -> X  q  n  ,...,X  on U. Clearly, the X,X  T  are all  q  uniformly bounded on U. It remains for us to show that D X(x) q  — X (x) for q  g — l,2,...,r and for x 6 U. Let us first show that DX(x) • v — X*(x) - v. This entails  lim  X{x  + tv) - X{x) - X (z) • tv t  i-»0  But X  n  = 0.  —• X so that this limit becomes lim lim  X (x-r n  tv) - X {x) - X (x) • tv n  t—>0 n—»oo  But, by the mean value theorem X {x + tv) -X {x)-X\x) n  n  - tv  \DX {x+ n  -36-  £v) -v-  X ( i ) •v\ J  = \DX {x  + £v) • v - X  (x + iv) • v + X ( x + iv) • v - X (x) • v\  < \DX {x  + iv) • v - X (x + iv) • v\ + | X (x + iv) - v - X  1  n  1  1  n  for some iin(0,t).  !  (x) • i;|  In the last expression the first term goes to zero in n uniformly  in / , and the second term goes to zero in t as X DX  l  = X . By induction, we see that D X 1  —X  q  9  is continuous. This proves that  1  for q — l , . . . , r and the lemma  is proved. Constructing  X {M) r  Now, consider the direct sum of these Banach spaces  *  =©?=iB>a(t7 );R ), B  Q  with the usual norm that makes X into a Banach space; ||X, © . . . © X \  N r  We shall show that X {M) r  = ||Xi|| + . . . + ||XjvIr. f  X ( M ) is isomorphic to a closed subspace of X , whence r  is itself a Banach space.  Clearly the map X (M)  —> X  r  is an injection (here the X  a  : X ^ ( X  U  . . . , X  N  )  are the local representatives of X in the a-th coor-  dinate chart on M.). The image of this map is the subspace of X defined by {{X!,...,X )e N  X :X  a  =  T<p oXg}, ag  where T<p p are the vector bundle chart transition maps for TM and the X Q  idi? x X , etc . . . In other words, the image of X (M) r  {i  Q  a  =  under this map is the set  of collections of local representatives that are consistent with the transformation rule for vectorfields. This is exactly what we have already seen in section 2.1. To see that this is really a subspace, it suffices to note that the transition maps are -37-  S.l linear in the Xp.  Also, this is a closed subspace. Indeed, if {X™,...,  Xpf  j is a  Cauchy sequence in X satisfying  X?(x)  lim X?= m—>oo  = D<p p(pp (x)) a  lim  •  a  Xf{<pp {x)), a  /;^(^ (x))-X^(^ (x)) 0  a  m—xx m—>oo  Thus,  X (A:/) C X r  has closed range and so  subspace of the Banach space X . T h e n  '  X {M) T  X ( M ) inherits a topology from this emr  bedding that makes it a Banach space. This topology on uniform C  C  r  is isomorphic to a closed  X (M) T  is the topology of  -convergence on compacta, which is the same as the topology of uniform  -convergence since M  is compact.  -38-  3.2  3.2  Differentiability of the Evaluation Map  In this section we shall show that the evaluation map for vectorfields,  ev : X (M) r  (X,p)  x M —>  —  TM,  X(p),  is a C — map and we will compute a formula for the derivatives of this map. T  In addition, we shall show that the derivative split-surjective;  Dev(X,p)  that is, it is surjective and its kernel  at the point  ker(Dev(X,p))  (X,p)  is  splits in the  Banach space T( , ){ X ( M ) x M). r  X P  The purpose of showing these properties of the evaluation map, is that we may then apply the results for differentiable maps that we will obtain in the next chapter, which will yield some results that are at the core of generic bifurcation theory. The results that obtain from the study of the evaluation map itself are primarily useful for the consideration of bifurcations that involve only the critical points of a vectorfield. However, it is possible to consider the more general relatives of the evaluation map in this same framework, and obtain similar results for periodic orbits and the like. The power of these so-called evaluation-transversality techniques is in reducing different kinds of bifurcation questions to questions about the geometry of certain submanifolds of X ( M ) x M. In chapter 5, we shall consider the example of r  the saddle-node bifurcation in detail in terms of this framework, as well as indicating how we might generalise the approach taken there so that it would include other - 3 9 -  3.2  .  bifurcations of critical elements and connections of critical elements. T h e Derivatives of ev Let us begin by formally differentiating the evaluation map. We have ev : X (M) x M -+TM r  ev:{X,p)^X{p)£T M. F  The derivative of this map will be a globalisation of the derivative of the map ev  a  induced by local coordinates <p , so it suffices to consider the derivative of this a  map. We have ev : X {M) x R \<p {U ) —• <p {U ) x R r  n  a  a  a  a  n  a  = T<p (TM\U ), a  a  where this is defined by ev = T<pc oevo ( ^ x ( M ) r  x  a  Now,  ev (X,x)  = X (x),  a  where X  a  Q  is the induced local representative of X in  local coordinates tp . In order to differentiate this map formally, we consider that Q  Dev {X,p) Q  • (t,v) = Dev {X,p)  • ( f , 0 ) + Dev {X,p)  a  a  • {0,v).  Each of these partial derivatives is easy to calculate. First, since for a fixed x we have that ev  a  is the linear functional X >—> X (x)  in the vectorfield X, we have  a  that Dev (X, x) • (£,0) = f a I )- For the other partial derivative, we notice that 1  a  Dev (X,x){0,v) a  ~ X {x + v) a  X {x) a  ~ DX (x) • v, a  so that we would guess that the derivative of ev  a  Dev {X,x) a  is  • ( £ , « ) = ^(x) + DX  That this is in fact the derivative is trivial to verify. We consider ev ((X,x) a  + (f,v)) - ev {X,x) = X (x + v) + f ( x + v) - -Xa(x) a  a  Q  = ta{x + v) + X {x + v)a  -40-  X (x). a  S.2 Now \Z {x + v)+ X {x a  a  + v) - X {x)  - U {x)  a  \t {x + v) - £ (x)| + \X {x + v)a  • v)\ <  a  X {x)  a  Q  + DX {x)  a  - DX (x)  a  • v\,  a  and the second term clearly goes to zero faster than ||f|| + |u| as X is differentiable. r  The first term goes to zero as f  is continuous. Furthermore, by the mean value  Q  theorem, we have that  \t {x-rv)-Z {4=\ Z°{c)-v\ D  a  a  for some c between x and x + v. So, since f  is bounded in the C -norm, we r  Q  have  and we know that this goes to zero like  (||cj + \v\) r  + |y|) . Thus the above function is Dev  Q  Now, let us consider the higher derivatives of ev . a  x) • ((£', v ), ( f , v )) 1  a  2  2  We consider the map  tlW+DXcW-v .  (A-,z)—>  2  from the inequality \ty\ <  by the definition of derivative.  2  Then, D ev (X,  2  1  is just the derivative of the above map in the  direction ( f , u ) . Again, we compute the partial derivatives of this map. We see 2  2  that derivative in (0,u ) direction = D£l(x) 2  • v + D X (x) 2  •  2  a  {v\v ), 2  and that derivative in (£ ,0) direction = D£,l(x)  • v, 1  2  since the the part of the map that depends on X is a linear functional in X. So, we would guess that the formula for D ev (x)  is given by  2  a  D ev (x) 2  a  •  (fV,fV) = D&(x)  • v + Dil{x)  -41-  2  • v + D X (x) 1  2  Q  •  (v\v ). 2  .3.2  Continuing in this fashion, we arrive at the following formula for the first r derivaev ,  tives of  a  •(e,v\...,e,v )=D X (x)-(v\...y) p  D ev (x) p  p  a  Q  ~J2D^H(x)-(v\...,i\...y),  (v ,...,  where the notation  v\ ..., v )  1  is used for the (p-l)-tuple that does not  r  v.  contain  l  We proceed to prove this formula by induction. Let us compute the ( p + l ) s t derivative of ev ( £  P  +  I  , t 7  p  +  1  a  by taking the derivative of the above formula in the direction  We have  ) .  D^ev (X,x)iey,...,e+\v^) a  x) • ( f  = D (D?ev (X, a  1  V,...,£*>")) • (£ , P+I  where 4 > denotes our formula for the p-th derivative, p  D^X (x)-(v\...,vn^f2D^U )-^\---^\---,v )-(e^,v^).  =  P  x  a  As before, we compute, f  p +  \x + u  |$(X + f  p + 1  p + 1  ) - *{X,x)\  ,x + v  p + 1  ^>(X,x + v )  ) - $ ( X , i + r; p + 1 )!-f -  p+1  <  $(X,x)|  and  < \(Dn (x) p  a  ..., v')\ + \(D»X (x  • (v\  (D^(  (X + u  Q  + r  a  p + 1  ) - D"- £ (x)) 1  A  p + 1  ) - D X (x))  -(v\...  p  a  ,r )|+ p  • ( V , . . . , v\ ..., v"),  i=i  Considering each of the terms in the second part above separately, we see that (£ X (x+i> p  a  p+1  )-.C> X (x)), and each of the terms {D -^ {x+v )-D^ p  p  p+1  £ {x))  1  a  Q  are approximated by their derivatives, D * X (x) -42F  1  a  •v  p + 1  a  a n d D £ (x)-v . p  p  a  A s long  3.2  as p + 1 < r, these derivatives w i l l exist as X and the f 's are all C . We know 1  r  that \p X {x  +v )  v  p+i  a  - L> A' (*) - D p  faster than f?.' |, and hence faster than p+1  tiability and the fact that A' is C . p  l  +  Q  - D^  „P+i)  goes to zero faster than sup.,. | D  ||f ||  X ( i ) • t; || - 0 p+1  a  + | i » | , by the definition of differenp+1  r  Also, each of the terms  r  \D - U{x  p + 1  a  P + 1  - D f (x) •  1  p  Q  £ (x)|| • ( v  p + 1  Q  | <  ||f |r a  t, + | p  ]  •k | p+1  by the mean value  theorem, since the norm here is uniform. This shows that the formula for the p-th derivative of the evaluation map is  D ev {X,x)iZ\v\...,Z y) p  =  v  a  D x (x) p  a  + J2  .(v\..., ) p  v  D  P  '  }  -(v\...y,.-.,v ). p  Since all of the derivatives in this expression exist and are continuous for p < r, we know that the local representative ev of the evaluation map ev is C  and hence  T  a  the evaluation map itself is a C  r  map from the Banach manifold  X {M) x M to r  TM.  S p l i t S u b j e c t i v i t y o f ev  The C  1  implicit function theorem from advanced calculus is usually stated for a  -function / : R  Dif{xo,yo)  m  —* R", such that the derivative in the  first  n-coordinates  has m a x i m a l rank n. Then there is a unique implicit function h such  that f{h(y),y)  — /(xo,!/o) for y near yo- To generalize this theorem to the case of  a function between Banach spaces, / : E —» F, we must replace the assumption of m a x i m a l rank w i t h an appropriate generalisation, namely that D/[XQ)  is surjective  at the point XQ. Additionally, we must assume that the kernel of Df(xo) E , that is, that there is a direct s u m of closed subspaces E = ker(Df(xo))  splits in © K'.  T h i s is necessary for the decomposition of E into a direct s u m of two components -43-  3.2  so that an implicit function can be expressed as a map from one component to the other. In chapter 4, we will look at the implicit function theorem and some of its global generalisations in the Banach space/Banach manifold setting with the intention of applying these results to the evaluation map. For this reason, the remainder of this section is devoted to showing that the derivative of the evaluation map is surjective and kernel-splitting. Consider the local representative ev  of the evaluation map, and its derivative,  a  Dev (X,x) Q  • (£,v)  clearly this map is surjective onto R  = £ (x) + DX (x)  • v.  Q  Q  as we may have an arbitrary value for f ( x ) .  n  Q  To show that the kernel of this map splits in X ( M ) x R , we consider the subspaces r  Kj = U :  t {x)  = 0} x  a  ker(DX (x)), a  K  2  = {(£,») : fa(x) # 0 and S {x) + DX {x)  K  3  = -U:£a(x)=0}xK',  K  4  = { £ : £a(x) / 0} x M M  Q  Q  (*)), The K, are all closed subspaces of  a  r  n  and it is easy to see that  X (M) T  and  -v = 0},  a  where K' is a complement of ker{DX [x)). X (M) x R  n  that ker(Dev (X,x)) a  xR  n  - Kj © K  = K i © K2,  2  © K  3  © K  4  and so is complemented. Thus tv  kernel-splitting submersion.  -44-  is a  4.1  4.1 The Implicit Function Theorem and Transversality  In this section we consider the implicit function theorem for smooth maps of Banach spaces and Banach submanifolds. Viewed geometrically, the implicit function theorem gives us conditions under which the inverse image, or pull back, of a point under a smooth map is locally a smooth submanifold of the domain. Introducing the definition of transversality of maps to submanifolds allows us to extend these results to the pull-backs of embedded submanifolds. In the next chapter, these results will be applied to the evaluation maps of Chapter 3 for the purpose of studying the relationships between the dependence of critical points of a vectorfield on perturbations of the vectorfield and the jets of the vectorfield at its critical points. Implicit Function T h e o r e m We state a version of the implicit function theorem for C — maps of Banach T  spaces. The statement and proof of this theorem is found in [Abraham et. al. [I983],p.l07]; However, the statement there omits a necessary condition for the uniqueness of the implicit function. T h e o r e m (implicit function theorem). Let U C E,V C F be open subsets of the Banach spaces E, F, and let f : U x V —> G space G. For some  (xo,yo)  E  U xV  assume that  be C , (r > l ) , into the Banach  /(xo»!/o) = ton, and that  £>2/(xo,yo) F — » G -45-  4.1 is an isomorphism. a unique  UQ of x  Then, there exist neighborhoods  0  and WQ of u>o and  C' — map g : UQ X WQ —> V such that g{x ,w )  (t)  0  (ii)  = y  0  0  — w for ail x,w 6 UQ X WQ.  f(x,g(x,w))  The content of this theorem is essentially geometrical. It states that the part of the inverse image / (tuo) that passes through the point (xQ.yo) _1  gw { )  the graph of a C - f u n c t i o n pull-back  f~ (wo) 1  x  s{ , o)x w  =  {)  W(l  This means that near (xn,yo); the  of WQ is a submanifold of E x F. Furthermore, we can compute  the tangent space of this submanifold at (xo,yo) since f(x,g (x))  is locally given as  by implicit differentiation. Indeed,  = wo, then D\f[x ,yo)  + D f{x ,yo)  0  • Dig (x )  0  2  W()  =0  0  which implies Dig { o) Wll  since D2f(xo,yo) of the form  =  x  -(j?2/(xo,yo))  • (-Di/(zo.s/o))  is an isomorphism. Thus the tangent space 7(i0,y0) ^ / ( ^ o ) ^ is _ 1  D (x ) igwo  0  • f)  t  :^Ej}.  K e r n e l Splitting Submersions a n d Regular Values Often we are interested in the preimage/pull-back of a point p by a m a p / where / : t / c E — > F i s a E  C — map from an open set U in a Banach space r  into another Banach space F. We may reduce this to the case of the implicit  function theorem setting if / is locally a kernel splitting submersion. We see this in the following corollary; C o r o l l a r y . Let f : U C E —> F be C ,(r r  > 1), defined on the open set U. Assume  that for some un € U, we have /(UQ) = WQ and that Ei = ker[Df(uo))  splits in E. Then E = E i © E  -46-  2  D/(UQ)  is surjective and  a n d there exist  neighborhoods  4.1  Ui,U  in E i , E  2  submanifold f~ {p)  with U © U x  2  C U and such that / ( t u ) n (C7i © l/ ) is a _1  2  0  given by the graph of a C — function r  2  —» E j .  Furthermore  at UQ.  is tangent to ker(Df(uo))  l  g : E  2  Proof. Since E i = ker(Df(uo))  splits, then there exists a closed complement E  2  to E i in E , whence E = E ] © E . As Df (UQ) is surjective, then Z>/(u )|E 2  is an isomorphism from E  0  to F. Thus, the conditions of the implicit function  2  theorem are satisfied for the function f{x,y) D2f{xo,Vo)  2  = f(x + y)  on E] x E , since 2  where uo = XQ + j/o- We can then infer the existence  = Df(uo)\E2  of a unique C — f u n c t i o n g : E  2  —» E ] such that 0(3/0) = xo, and f{g{y),y)  =  wo for y in some neighborhood of t/o- Also, as shown previously, i>2ff(j/o) =  -  (p f{xo, t/o))  • Dif(x ,y ),  2  0  which is zero since Dif(x ,yo)  0  0  = D/{UQ)\EI  = 0  as E = A;er(Z?/(u )). 0  The generalization of this result to maps between Banach manifolds is immediate; C o r o l l a r y . Let Assume hood  f : M —> N  be a C -map of Banach manifolds T  that f is a kernel splitting submersion  Up of p such that f~ (q) 1  with f(p)  = q.  at p. Then there exists a neighbor-  is a C -submanifold T  tangent to ker{Df{p))  at  P-  Proof. Introducing local coordinates <p at p and ijjg at q gives a local represena  tative fa  that satisfies the hypotheses of the previous corollary.  A point q <G N is a regular value for a C -map f : M —> N if for each point P £ f~ {q), l  f is kernel-splitting submersion at p. It is evident from the previous a  corollary that the inverse image/pull- back of a regular value is a C -manifold. r  P u l l - B a c k s of Submanifolds. Transversality -47-  4.1  :  A point q of a manifold TV is a particular case of a submanifold of JV. We now consider the pull-back of a submanifold 5 of a Banach manifold N via a C -map. r  Recall that S is an embedded submanifold of the C-manifold TV if for each point q 6 5 there is a chart (Vp.ij'p) about q in the atlas of N that has the submanifold  property  <pp \Vp -» E, and <pp{Vp n 5) = E i x {0} C E, where E i is a subspace that splits in E . Then S inherits a manifold structure from TV with chart maps taking values in E i . If E i is a finite dimensional subspace of dimension k, then S is clearly an n-dimensional (sub)manifold. However, if Ei  has a closed complement E  submanifold  2  of finite dimension k, then we say that 5 is a  of codimension-k.  It is clear from the above that any submanifold of codimension-k can be locally expressed as S rtW = A (0) for some neighborhood W in TV where A : W —• R* -1  is a submersion, since we may take A to be the projection onto E  2  of %bp above.  This provides some motivation for the following definition. D e f i n i t i o n . Let S C TV be a codimension-k submanifold  and let f : M —» TV be  a C -map. We say that the map f is transverse to S at the point p 6 M if either 0) f{p) &  o  r  00 f{p) £  Df(p)  Df( )-T M P  The  notation  at the point  f ^ S p  p  + T  m  means that the map  p in the domain of f.  set W, we write f^wS, all points in its  is kernel-splitting and  or simply  If f  S  f  =  T N. f(p]  is transverse to the submanifold  S  is transverse to S at all points in some  f ^ S  to mean that f  domain.  We can now easily obtain the following result. -48-  is transverse to S at  4.1 T h e o r e m (pull-back v i a transversal maps). Let f : M —> N be a C of Banach manifolds, i  is compact. Furthermore, codimension  k in  -map  S C N be a C -submanifold, and assume that f <^ S. Then  is an immersed submanifold  f~ (S)  r  of M, and is an embedded submanifold  if S has finite codimension  k in N,  then f~ (S) 1  if S has  M.  Proof. First consider a small neighborhood V of a point q of 5. As we have noted, Sf)V  — A ( 0 ) for some surjection A : V —» R*. We show that 0 is a regular value _1  of XQ f. First, let p G ( A o / ) - ' ( 0 ) . Then p G /  ( 5 n K). Since f ^ S, then we  _ 1  have that Df(p)-T M-rT S p  Applying DX(f(p))  =  m  T N. m  to both sides,  DX(f(p)) • Df(p) • T M + DX(flp)) • T V  S  f[p)  D(Xof)( ).T M P  p  = DX(f(p)) • =  T N, f{p)  R, k  by the chain rule, and noting that A is a submersion and Tj/ }S is the kernel p  of DX(f(p)).  This shows that A o / is a submersion. Furthermore, we know that  (£>(A o /)(p)) (0) = {Df{p))_1  1  •T  S  f{p)  is a subspace of T M p  which is com-  plemented and whose complement is isomorphic to the complement of Tf^S Tf/p-jN.  in  Indeed, for any linear surjection A : E —> F, we have that the induced  map A : E / A ( F ' ) —» F/F' is an isomorphism. Thus A:er(D(A o f)(p)) has closed _1  complement so that A o / is kernel-splitting. Thus, A o / has 0 as a regular value and so (A o / ) ( 0 ) = / -1  _ 1  ( 5 n V) is an embedded submanifold of M.  Taking a  union of neighborhoods that cover S, we see that / ( 5 ) is a union of embed_ 1  ded submanifolds, which will be an immersed submanifold. In the case that 5 is compact, the above union can be made finite, so that f~ (S) 1  is still an embedded  submanifold. Also, if S has codimension-k, then we know that the complement of -49-  Tj^S  4.1 in Tf( )N  is isomorphic to R . By our observation, the complement of Tj,(/ (S) fc  p  -1  would be isomorphic to R*, so that /  _ 1  (5) is also a submanifold of codimension-k.  The following corollary to this result is essentially a direct extension of the implicit function theorem. Corollary. Lei, f : M -> N,S C N, f ^ S as above. If for p 6 f~ {S) l  thai  TM p  = Ep © E  such that the transversality  2  being direct when T M p  =  f{p)  TM  condition holds with the sum  is replaced by Ep, i.e.,  Df{p)-E],®T S  then in a neighborhood  U  p  T N f{p)  1  of p, we have that for any local coordinates  such that <p = {<p\<p ),<p{p) = (0,0) with D<p {p) 2  p  that the component  we have  of /  _ 1  {  (5) n U  •TM  <p : U —> p  = E j , we know  p  through p is the graph of a C -function from T  p  E j to E . 2  Proof. Consider the function F — A o / o ip" : Ep x E —• F' where A is as above. 1  2  Then D F{0,0) }  -T M P  = DF{0,0)  - El  = £>(Ao/)(p).Ej = £>A(/(p))-£>/(p).Ej. But this equals DX(f(p))  since Tj^S  • (Df(p)  = ker(DX(f(p)).  • El - T S) f[p)  = DX(f( )) P  •  T N f[p)  Since A is a submersion, we have that Z?iF(0,0) is  surjective. Also, we have that ker(DiF(0,0))  is trivial. Otherwise, there would be a v G Ep  with Df(p) • v £ fcer(Z?A(/(p))) = Tj^S,  which cannot happen as the sum in the  statement was direct. Thus, D\ 7^(0,0) is an isomorphism, and the result follows from the implicit function theorem. -50-  4.1 We will use this result in section 5.2 to obtain a parameterisation of the submanifold of critical points of the evaluation map at a point corresponding to a bifurcating critical point. We will use this to compute the relationship between qualitative properties resulting from the geometry of this manifold and the jets of the vectorfield at the critical point.  -51-  5.1  5.1  Critical Points of the Evaluation M a p  In this section we w i l l look at critical points of the evaluation m a p  ev : X (M) r  The evaluation m a p ev(X,p)  x M —•  TM.  = X(p) can be thought of as a parameterised  vector-  field on M where the parameter is the vectorfield A' € X ( M ) . We are primarily r  interested in the critical points of individual vectorfields and families of vectorfields in the study of bifurcations of critical p o i n t s — a n d these are related to the critical points of the evaluation map:  A point p is a critical point, for a vectorfield A' iff  (A", p) is a critical point for ev. However, the critical points of ev are especially useful for studying bifurcations as the local geometry of the set E Q of critcal points of ev depends on the relationship between changes in a vectorfield (perturbations) and changes in critical points. E x p l o i t i n g the properties of ev that were developed in the last chapter, we shall examine the relationship between the k-jets of a vectorfield A" at a critical point p and the local geometry of E Q at ( X , p). The advantage of this approach is that it allows us to take a particularly geometric view of parameterised  families of vectorfields. If X  is a family of C — r  M  vectorfields depending on a parameter /z, where the parameter is in some compact manifold A, possibly w i t h boundary, then the family (if it is at all reasonable) X ^ defines an embedding X {M).  A —>  r  -52-  5.1 X [M).  The image of this embedding will be a submanifold of (the Banach space)  T  Thus, a parameterised family of vectorfields can be regarded as a submanifold of X [M) T  or a smooth embedding A —» X (M). T  This geometric point of view makes  it much easier to see the mechanism behind certain bifurcations, and will provide us with a coherent approach to the whole study of bifurcation theory. In a later section, this geometric viewpoint, is used in conjunction with transversality theory to obtain genericity results for vectorfields and families of vectorfields. C r i t i c a l Points. Zero Section In order to define critical points for vectorfields on compact manifolds, we will use the definition of a critical point for the local representatives of such a vectorfield and then extend the definition in the obvious way. X G X [M)  D e f i n i t i o n . A point p is a critical point of the vectorfield (and hence for any) chart (U ,ip ) a  X  a  with p G U , the induced local  a  iff for some  T  a  representative  defined by  T<p oXo<pl\x) Q  = (x,X (x)) a  G R \<p (U ) x ir\ n  a  Q  has the corresponding point x = tp (p) as a critical point; i.e., the value Q  of the local representative  at the corresponding  X (<p (p)) Q  a  point is zero. Equivalently,  p is a  critical point iff  7 W X ( p ) ) = (<p {p),0). a  It is obvious that this definition is independant  of the choice of chart  (l/ ,£> ). Q  a  While this may seem like an awfully formal definition for such a straightforward concept, this definition does motivate us to define the zero section of the tangent bundle  TM.  The zero section OTM °f the tangent bundle TM is simply the set of all vectors in that are zero vectors in the sense that their local representatives are zero vectors. -53-  5.1  Thus OJM C TM.  It is easy to see that OTM is a submanifold of TM as  T<p oO \TU Q  TM  = R  x {0},  n  a  for any vector bundle chart map T<p . Clearly OTM has codimension n in a  The  zero section can also be thought of as a vectorfield on M:  TM.  For each point  p £ M , there is a zero vector 0TM{P) * P- Thus, the zero section is a section of a  the tangent bundle, and is also a vectorfield as the map OTM : M —» TM is smooth.  The  Manifold En  The set E n of critical points of the evaluation map is the pull back of the zero section by the evaluation map;  E  0  =  ev~ {0 ). l  TM  In section 3.2 it was shown that ev was a kernel-splitting submersion. Thus, applying the results of sect 4.1, E Q must be a codimension-n submanifold of X ( M ) x M r  as OTM is a codimension-n submanifold of TM. If {X,p) is a point in E n , then the tangent space T ^ ^ E n lationships between perturbations in X example, if  indicates the re-  and changes in the critical point p. For  £ T ^ ^ E o , then perturbing the vectorfield X in the direction f  will move the critical point p in the direction v; More precisely, we can say that for the 1-parameter family X  t  = X + ef, there is a corresponding 1-parameter family  of critical points p(t) for small e such that  p(0) = p, X (p(c)) = 0 {p{e)), and p'(0) = v. £  TM  In order to compute the tangent space T ^ p ) E n , recall that since  ev(X,p) = -54-  OTM(P)  for all (X, p) G E o , then differentiating along E o gives us that T(x,r>) {^i>) =  T 0 {v)  ev  p  TM  for all (£,i>) £ 7(jy ,)Eo. Putting this expression into coordinate form, i?  T Po.°T(x )ev(£,v)  = T <p o T p O ( T < ^ ; ( t ; ) ) , 7  2<  lP  1  a  rM  (  a  or T  (X,x) o.{Z, a) ev  = T [T<p  v  z  cO p- )(v ), l  a  TM  a  = (x,0,f ,0), Q  as Tifc oOr^(p) = (i/? (p),0). Recall that the local representative of the evaluation Q  map tv  a  was defined for a chart (U ,ip ) a  on M by  a  eu (X,x) = T<p o eu(X,(p~ (x)), 1  a  Q  or eu (X,x) =  x <p ) o tv o ( ^  T{id r  a  x  a  (M)  r x  (M)  x  1  so that Tet; = T £> o Tew o (J'(^> ) 2  a  a  _1  a  where <p = (*^x (M) ^a)- Finally we have r  x  a  eu (A ,x) = (x, A' (x)) r  a  a  so that Tet> (X ,x, f , v ) = | i , X o ( i ) , t ) . f ( i ) + M ( i ) - t ; j , a  a  a  a  a  a  a  a  where x = <p (p), and the subscripted quantities are the local representatives in a  the ip  a  coordinates. This shows that {Me  T T, {XtP)  0  iff £ (x) + £>X (x) -v Q  a  a  = 0.  This tells us some things about the geometry of E o at (X,p). First, in the case where the linearisation DX  a  is non-singular, we see that v -55-  a  can be expressed  f ( x ) of the vectorfield X  in terms of the perturbation  at the critical point  Q  a  — <Pa(p)- This means that an arbitrary perturbation  f of the vectorfield  A'  will cause the critical point to move in the local coordinates in the direction v  =  x  Q  — (DX (x) a  • £a{ )) x  to first order. In the case where the linearisation DX (x)  non-trivial kernel, then for vo 6 KerDX(p),  we have ( f o ^ o ) € ^ ( A » ^ ° ' for any  — 0 (t.e.f(p) = OTM(P)-) For other perturbations £  perturbation £ such that £(p) of X  have several directions v in which critical points may  we may  has  Q  move, or no  directions. We shall see what the interpretations of these results are in the following analysis.  Non-Degenerate Critical Points A critical point p of a vectorfield A' is non-degenerate if X ^¥ 0TM-  This  p  means that •T  DX{p)  M + T ( 0M  P  0tm  p)  =  T  T  0TM{P)  In terms of local coordinates, this means that DX (x)  is a submersion, as  a  T  o ( )°TM TM  P  TM. T <p lphao 2  a  = (x,0,R ,0). Indeed, n  :  DX(p)  T  P  M  ->  T  x  {  p  )  T M ,  whence  =  T <p oDX(p)o(Tip )~ (x,v ) 2  1  Q  Q  a  T ( T i p  a  o  X o IP' )  • {x,v )  1  Q  = (x,X (x),v ,DX [x) Q  which is surjective at (x,X (x))  if DX {x)  Q  We  may  a  - Va)  Q  is surjective.  a  apply the implicit function theorem in the case of a non-degenerate  critical point. Consider ev : X  T  but in the case where DX {x)  —> T M . We  know that ev  is surjective, we have that  a  Dev(X,p)  ( M )x M  T  p  M® T  0 t m [ p )  -56-  0  t m  =  T  TM,  0TM[P)  is surjective,  5.1  so that we may apply the extension to the implicit function theorem in section 4.1 to conclude that there is a unique implicit function 4> : X  T  ( M ) —> M such that  et;(A\*(*))eO™. For vectorfields X sufficiently near X in X (M).  In fact  T  e v ( X , * ( X ) )  so that 4>(X) 6 M  =  0  T  M  ( < S > ( X ) ) ,  is a critical point for the vectorfield A^. As we have already  seen in Sect. 1.3, this shows that the critical point $(X)  is a smooth function of  the perturbation A of A and is a locally unique critical point in a neighbourhood of the point p. Degnerate Critical Points  A critical point p £ M of the vectorfield X £ X ( M ) is a degenerate critical r  point if the derivative of X at p has non-trivial kernel;i.e„ the map  D X ( p )  : T  M  r  vanishes on a non-trivial subspace of T this means for the linearisation  ->  T  x  (  p  T M  )  M . In local coordinates, we can see what  V  of X (x),  DX (x)  where x  a  Q  —  <p (x). Q  We have  that T i p a c X i p '  (x,A' (x)),  : x i—•  1  Q  so that T <p oTXoT<p~ (x,v)  = {x,X {x),v,DX (x)  l  2  a  Since  T < p  a  ( T  p  M )  we see the that  = {x} xR", T<p (ker(DX(p)) a  a  and  T  2  < p  a  { T  X  • v).  a  (  p  ) T M )  =  (x, A ( x ) ) x R  = {x} x ker(DX (x)), a  Q  n  x R",  then  which is what we would  expect. In the next section we shall see how may apply the extension to the implicit function theorem in section 4.1 to a non- degenerate critical point in order to obtain a well known bifurcation result in a very geometric way. -57-  5.2  5.2  A n Example—The Saddle-Node Again  In the previous section we saw what the main distinction between non-degenerate and degenerate critical points of vectorfields. For a non-degenerate critical point p of a vectorfield A', the critical point varies smoothly under small perturbations of A. This is due to the implicit function theorem; More precisely, we can express the critical point p as a smooth function p(X), —*  p : X (M)\N T  X  with p(A') = p, defined in a neighbourhood N  x  M, of X £ X ( M ) . This is in turn r  due to the local geometry of Eo, specifically that the tangent space J p ^ E o  15  n  °t  "vertically tangent" in any direction. However, in the case of a degenerate critical point p, we have already seen that T^ )^o is tangent to {X} XlP  that are in  xM  along directions  KerDX(p).  In order to consider the analysis of degenerate critical points, let us first consider the simplest case in which Eo is tangent to {A} x M  along only one direction  v° £ TpM. This corresponds to a critical point p for which the linearisation has 1-dimensional kernel spanned by v°. quadratically tangent to {A} x M  DX(p)  Furthermore, let us assume that Eo is  in this direction. The graph of Eo is shown in  Figure 5.2.1. From the graph of this function, we can see what kind of qualitative change will occur when the vectorfield passes-through A. On one side of the graph, there is no -58-  .5.2  Figure  5.2.1  -59-  5.2  critical point near p, then as we cross the fold, the critical point p appears and divides into two separate points that grow apart at quadratic rates. This is exactly what happened with the saddle-node example in section 1.2. We need to consider a parametcrisation of En near (X,p);  i.e., we need to  express Do as the graph of a smooth function. In order to find suitable variables for such a parameterisation, let us try and find a subspace S C T^x ){ X {A1) T  lP  x Ad)  such that Dev(X,p)  •S© T  0 T M { p )  0  T M  =  T  TAd.  0TM[p)  Then, the subspace S will satisfy the conditions of the generalised implicit function theorem of section 4.1 and E Q will be locally diffeomorphic to the graph of a function OQ : S' —> S, where S complements S in T(x ){X  r  1  lP  {Ad) x M). Invoking  local coordinates, we have that T V a  o T(TM)  0  =  TT<p ((TM) ) a  = T(R  n  = R  0  x {0})  x {0} x R  n  n  x {0}  and T <p [T 2  a  0TM{p)  TAd)  = > ( p ) } x {0} x R" x R \ a  Since Tev (X,x, a  £,v ) a  = {x,X {x),v , a  a  £ (x) + DX {x) a  • v)  a  a  from before, we can see what vectors would comprise a suitable subspace S. Indeed, we need to find a set of (f ,i>) such that ( (x) + DX (x) a  the range of DX (x) Q  (0, DX(p)-v)  a  •v  a  will span R . Since n  is n — 1 dimensional, we can choose vectors of the form  to span an n —1 dimensional subspace of 5. Finally, adding a vector  of the form (f,0) where £ (x) complements the range of DX (x) a  a  n-dimensional subspace satisfying our requirements. -60-  would give an  5.2  Now, let 7r' be a projection onto a complement of range v'  be a vectorfield such that  subspace X" of X T(M)  7= 0. Clearly  ^'(v'^x))  v'  DXa(x)  in R . Let n  defines a 1-dimensional  which is complemented by the subspace X' = {ee X'(Af)|7r'(e (i))=0}. a  Then the subspaces S = X  v  x span(v'(p))  and S' = X' x ker(DX(p))  are as  previously required. We can see that near to (X,p),  the submanifold Eo will admit a parameteri-  sation o0  : X' x ker(DX(p))  — • E . 0  We can use this parameterisation for the computation of certain quantities; Specifically, we are interested in obtaining conditions on the k-jets of X  at p that  are equivalent to the non-degeneracy conditions of the quadradic tangency of Eo to {X}  xM  in the direction v°. In terms of the parameterisation OQ, this non-  degeneracy condition is 7r'o£» o^(0,0) ^ 0 22  where O Q is the vectorfield part of o"o, and the £>  = 0TM{P) f°  ra i i  means differentiating twice  DX(p).)  along the second component (along the kernel of Recalling that ev(X,p)  22  y (A,p) in Eo, we have the parame-  terisation ev(ol(X',v),o 20(X',v))  for  in X ' x ker(DX(p))  {X',v)  0TM(ol(X',v)),  =  where  O-Q  and  are the vectorfield and manifold  o\  components of o"o, respectively. We may take tangents of both sides along {0} x ker(DX(p)) Tev  using the composite function rule: o (ol  x o-^)(0,0,0,«;) =  Tev  o (Tol(0,0,0,w)  x  =  T(0 o^)(0,0,0,«;)  =  T0 oTo (0,0,0,w).  rM  2  TM  -61-  0  TCT^(0,0,0,U;))  Taking another tangent along ker(DX(p)) T~tv o ((T o 2  by the composite function rule, we have  x T 2ol){0,0,0,^,0,0,0^))  ] 0  = T0M  ° T ^ (0,0,0,™, 0,0,0, u).  2  2  T  Now, we introduce local coordinate maps so that we can see what this equation means in terms of the Taylor series of X T ev : T X (M) 2  2  at the point p . So,  a  a  x TM  — • T M,  2  r  3  whence T ip  o T ev o  3  2  a  (T (id r 2  X  o  x (Pa))^  {M)  x (T ^ oT a ))(0,0,0,u;,0,0,0, )  (r 2al  = rVa  1  2  2  2  a  o TO 2  (rV )  O  TM  U  _ 1  a  o r <p 2  a  o  r a^(o,o,o,u;,o,o,o,u). 2  But T <p oT ev 3  o  2  Q  T (id r^ 2  x <£> )  x  We may compute an expression for T ev  a  a  = T ev . 2  a  from  2  ev {X,x)  _1  a  = (i,X (i)), a  T e u ( X x , f,t') = (x, A' (x),t;, f ( x ) + £>A (x) • u), a  Q  a  a  whence T ei; (A,x, £, v, n, u, f, w) 2  0  = ^x, A ( x ) , t>, f ( x ) + DA' (i) • v,u, r] (x) + DX (x) a  a  a  a  a  • u,w,  £>£ (x) • u + Dr) {x) • v + P A ( x ) • (u,r) + ? (x) + M ( i ) • w^j. 2  a  a  Q  Q  Now, we can compute expressions for the quantities £,v,r),u,c,w  a  from our  parameterisation OQ. We have ( a x a ) ( 0 , 0 ) = (X,p), 2  0  whence T{o0  x o^) (0,0,0, a) = (X, p, -62-  (0,0) • a, 7J ^(0,0) • a), 2  5.2 and, differentiating once more along kerDX(p), T 2{ol  we have  x ol)(0,0,0, a, 0,0,0,6) = {X,p,D2o0  • a,D2o 2  • a,D2ol  • b,D ol  • b,D  2  • (a,b),D22ol  o l0  22  •  (a,6)),  so that £ — D OQ(0,0)  •a  v =  D2ol-a  r, = D2ol(0,0)  •b  u=  D2ol-b  • (a,b)  v =  2  D22ol{0,0)  ? =  • (a.b).  D22o 2  Now, from the quadratic tangency conditions,the parameterisation  D2ol(0,0)  — 0, whence £,r/ above are zero. Also, D2ol(0,0)  will be along  Substituting this into the eighth (last) component of T ev . 2  a  D X {x) a  Q  a  22  by equating this with the last component of T <p oT0TMa  *' -D X {x) 2  a  a  -w = 0,  Applying the projection  2  7r' onto the complement of range{DX(p)),  ker(DX(p)).  we have that  • (u,u) + T<p o D o"o(0,0) • (a, 6) + DX {x)  2  OQ satisfies  we have that  • (u,v) = -T<p o  (7r'.£> ^(0,0) • (a,6))).  a  22  Recalling the non-degeneracy condition for the quadratic tangency, that 7r'-D ^(0,0) ^ 0 , 22  we have that the equivalent condition in terms of the jets of X at p will be TT'- D X{p) 2  for t; e  • (v,v) ^ 0  ker[DX{p)).  The Manifold S i The "fold" in Eo appears to be a submanifold of X (M) x M of codimensionT  n + 1, as it appears to be of codimension-1 in Eo- Knowing that this fold is the set of vectorfield-point pairs (X,p) such that X satisfies our saddle-node conditions -63-  5.2  at p-that  X(p)  — 0, the derivative DX(p)  has rank n — 1, and the second  derivative satisfies the quadratic non-degeneracy condition we derived above. It is easy to show that this defines a codimension-n + 1 submanifold when we consider that this is the pull-back of a codimension- n-\- 1 submanifold of the bundle of 2-jets of vectorfields. Consider the submanifold of J (TM) l  X(p)  and DX(p).  defined by our conditions on  In natural vectorbundle coordinates induced by a chart <p , a  these conditions will become X {x) a  = 0  and DX (x) Q  has rank n — 1 .  The set of n x n matrices that has rank n — 1 is a codimension-1 submanifold of L(R ) n  by the implicit function theorem. Indeed, consider A £ L ( R ) n  having rank  n — 1. Then, there is a neighborhood of A such that all matrices have rank at least n — 1. The determinant map det : L ( R " ) —> R is a submersion, and the set of rank n — 1 matrices is de< (0) restricted to the set of matrices of rank at least n — 1, -1  which is an open set in L ( R ) . Thus, the set of rank n-1 matrices is a submanifold n  of codimension equal to the codimension of {0} in R . 1  The quadratic non-degeneracy condition is open in that it persists under small perturbations in the 2-jet, so that our saddle-node conditions do define a codimension2 submanifold of J (TM). 2  The manifold E i indicated in Figure 3.1 is the pull-back  of this submanifold by the evaluation map for 2-jets, and so is also a codimensionn -f 1 submanifold. The important observation to make about E i is that locally it will project to a codimension-1 submanifold Ei of see that T y E i  X (M). r  For a vectorfield X € E i , we  is complemented by the direction X" shown in figure 3.1. This  direction is the direction in which v' • £ changes for perturbing vectorfields -64-  f.  -5.2  Geometrically we see what causes the saddle-node bifurcation. If we cross through the submanifold E i transversely, then we are crossing the fold in the graph of Eo, and this will cause the appearance of pair of critical points associated with the fold. For a one parameter vectorfield X\ crossing E i at the parameter value Ao, the • Xx){p) i 0- This motivates  condition that we cross E i transversely is that  1  the saddle-node bifurcation theorem, which 1 have taken from Guckenheimer and Holmes [1983], Theorem 3.4.1. T h e o r e m . Let x = f^(x) rameter  be a differential equation depending on the single pa-  p,. When p = / Z Q , assume that there is a an equilibrium  p for which the  following hypotheses are satisfied: ( SNl)  DJ x  vector  has a simple eigenvalue 0 with right eigenvector v and left eigen-  )lu  w.  ( SN2)  wD f{p,n )^0. tl  Q  ( SN3) w(D ffj, (p)  '  2  0  R  n  xR  ii) v  v  0. Then there is a smooth curve of equilibria in  passing through (p,/io), tangent to the hyperplane  one side of fio there are no equilibria  of /  of /xo there are two distinct equilibria  of  M  R  n  x {MO}- For p on  near p, while for p on the other side near p  We can see that the conditions (SNl) and (SN3) are the same as our conditions that ker(DX(p)) v 6 ker(DX(p)).  is 1-dimensional and that n' • D X (p) 2  a  • (v,v) ^ 0 for  If we think of a one-parameter family of vectorfields as a one-  dimensional arc in X (M),  then the condition (SN2) is really a requirement that  T  the arc Xx crosses through the point X £ X (M) r  "fold" in figure 5.2.1 onto X (M). r  transversal to projection of the  This projected fold is a codimension-1 subman-  ifold E i C X ( M ) , which we shall discuss in the next section. r  -65-  5.3  5.3  Concluding Remarks.  We have seen an example of how me may study the local geometry of an evaluation map to get a result for saddle-node bifurcations of critical points of vectorfields. However, it remains to be seen how this approach might be extended to obtain other results from generic bifurcation theory, or even what the connection between the material in the preceding sections is related to generic bifurcations. In this section, I will make some comments (of a somewhat speculative nature) on how this approach can be extended to include the other types of bifurcations that are encountered in generic bifurcation theory and also on the connection between this approach and that of generic bifurcation theory.  Transversality a n d Genericity We have the definition of transversality of a map / : M —• N to a submanifold S of N. We have considered transversality for two particular kinds of maps; vectorfields, which are maps M —* TM, and parameterised vectorfields, which are maps from a parameter space into the space X ( M ) of all is the r  transversality of these maps to various submanifolds that give rise to genericity results for vectorfields and parameterised vectorfields by the following well-known results. T h e o r e m (Openness of Transversal M a p s ) . Let M, N, S be C -Banach manT  ifolds, and f : M —> N be transverse to the closed submanifold  -66-  S C N. Then there  5.3 is a neighborhood  Uf of f in C (M,N)  such that g G Uj implies g <^ S. Thus,  r  the set of maps transverse to S is open in  The  C [M,N). k  statement and proof of this result is found in Palis and de M e l o (p. 24), or  A b r a h a m , Marsden and R a t i u (p. 179). Of course, the statement belies the definition of the topology of the space C {M, TV).  In the case where M is compact, this is  T  the C -compact-open topology. We saw an example of this in topology section 3.1 for the case of vectorfields on a compact manifold, where the topology on the space X (M) r  of sections was defined. A s in most of the results in section 4.1, the proof  relies on locally replacing transversality of maps with the equivalent surjectivity conditions. Specifically, consider  p G f~ (S). l  Then, there exists neighborhoods  Up C M , V^( ) C N, and a submersion A : V/(p) —> F ' such that f(U ) C Vf( ), p  and  p  SnV^p) = A ( 0 ) . Then there is a neighborhood Wf of / in - 1  p  C (M,N) r  such  that g(Up) C ^/(p) for all g G Wf. We have that g ^ f u S if and only if A o g is f)  a submersion on U . Finally, we note that the evaluation map that is defined near p  (/,p) G X (M,N) T  x M by  (g,q)-^D(Xog)(q)  is continuous, so that there are open neighborhoods W'j C Wj.U'^ C U  v  such  that Z?(A o g') is surjective as the set of linear surjections is open. This means that for all g in the C-neighborhood  f~ (S) i  is compact, we can cover  /  _  the intersection of the corresponding  W'j of /, we have g^u S. p  1  (5)  Finally, since  w i t h a finite number of the U' , and so p  W'j is a C -neighborhood of / of functions T  transverse to S. One consequence of the above proof is that we may generalize the above result and claim that the set of maps transverse t o a given submanifold is open in any space for which the evaluation m a p considered above is continuous. For example, the above theorem is not directly applicable in the case of vectorfields, as -67-  X (M) r  5.S  is not a space of the form C ( M , TV), but we still have that the set of vectorfields r  that are transverse to a given submanifold of TM is open in X (M).  .  r  There is a corresponding result concerning the density of transversal maps. First, recall Sard's theorem for a map / : M —-> N of finite-dimensional  manifolds, which  states that the set of regular values of the map is dense in TV. The C°° -version of this theorem is found in most texts on advanced calculus, differential geometry or introductory differential topology. The Sard-Smale theorem is a generalization of this result where the map / : M —> TV is now a C -Fredholm map and M, N are k  Banach manifolds. In this case, if / is sufficiently smooth, then the set of regular values is again dense. This extension of Sard's theorem is the subject of Appendix E. of Abraham et. al. [l983]and also covered in section 16 of Abraham and Robbin [1967]. Given an evaluation map, we may use the following lemma. L e m m a . Let F : A x M is a submanifold and n\ = n\\S  —» TV be transverse to S C TV. We know that S —  of A x M.  Let  : A x M —> A be the natural projection  Proof. If F\ is not transverse to 5, then for some (A,p) £ A x M we have DF(X,p)  T ( )S F  = DF(X,p)  XiP  Obviously T M + T ^S P  1  map  : S —+ A. Then, Fx = F(X, •) is transverse to S if and only if A is  a regular value of 7TA-  But,  F~ (S)  {X  •T M + T P  • T( ,p)S, X  -J> r  (A)P)  F  M  N  1>  T  F  M  N .  so that  ( A x M), so that  2I>7r (A,p)-r (AxM), A  -68-  ( M  5.S so that Dn (X,p) A  is not surjective, whence A is not a regular value of n^. Con-  versely, if A is not a regular value of 7 ^ , then for some p € M , we have D* {\,p)-T SJ>T A, A  {XtP)  x  which implies DTT {X, )-(T M A  P  + T  P  S)  J>T A,  {X<P)  X  so that T, S XtP)  Letting [v ,0) x  T MJ>T A.  +  X  p  be in T A x {0} but not in T( )S DF(\,p)  • (v ,0) ? DF(\,p)  p  +  p  • T M + Tp( }S XP  p  p  • (T M +  x  = DF(\,p)-T M whence DF(\,p)  + T M, we know that  XP  X  T s) {x>p)  T  F[XIP)  S,  so that F is not transverse to S.  Tp^ ^N, x  In the case where the above manifolds are finite dimensional, the projection map 7TJV  satisfies Sard's theorem, so that the set of A which are regular values of n  A  dense in A. Therefore the set of A for which F  x  are  S is dense in A. However, the  interesting application of this lemma is in the case where F is an evaluation map of the form evj : J(M,7V) x M ^ TV, where M  is a finite-dimensional manifold, and 7(M,N)  is a Banach space of  functions from M to TV. One example of this kind of map we have already seen is the evaluation map ev : X (M) T  x M —> TM for vectorfields. Another example  was the map (g,x) —> D(X og) that was used in the proof of the openness theorem for transversal maps above (the range of this map is actually a linear map bundle, and the proof used the fact that the set of surjections in in this bundle is openthis is done carefully in Abraham and Robbin [1967], section 18.) The set of all one-parameter families of vectorfields can also be considered in this way. A oneparameter vectorfield X , A G [0,1] can be considered as a map A —* X (M) r  x  -69-  in  5.3  the Banach space C ([0, l], X (M)). Then there is an evaluation map associated r  r  with parameterised vectorfields Ev : C([0,l],X (M)) x [0,1] —» X ( M ) r  r  (X,A)-I(A,.)GX'(M). In order to apply the above lemma to the evaluation map ev or Ev, we must verify that ev and Ev are transverse to any submanifolds their respective ranges, TM  and X ( M ) . We have already verified that ev was a submersion in section r  3.2. Similarly, it is easy to see that Ev is a submersion, for DEv(X\, Y\ -t- jxX\ (i  ()  An) • {Y\, v\) =  ~v\, and since we can choose anything we want for Y\ ,  the derivative  0  is surjective. Then, given a submanifold S of the range ( TM or X ( M ) ) , we need r  to verify that the induced projections n : (X (M) x M)|5 -» X (M), f  r  0  Tf, : (C ([0,l],X'(M)) x [0,1])|5 -> C ([0, lj, X (M)) r  r  r  are Fredholm maps, where S is the pull back of S by the evaluation map ev or Ev.  This requires that (i) both the kernel and range of  £?7r,  split, (ii) the kernel  is finite-dimensional and (iii) the range has finite codimension. If the submanifold S has finite codimension, then S has (the same) finite codimension, and it is not difficult to verify that (i)—(iii) hold. Then the Sard-Smale theorem holds for the 7r,, and we have the same density result as above. Specifically, for any submanifold S (remember that TM isfinitedimensional) of TM, the set of vectorfields transverse to 5 is dense in X (M). T  For one-parameter vectorfields, we have that for any  submanifold S C X ( M ) of finite-codimension, the set of one-parameter families r  X\  that are transverse to 5 is dense in C([0, l], X ( M ) ) . f  Thus, for a compact submanifold S, we have that the set of maps transverse to S is both open and dense. If S is paracompact, then the resulting set of maps transverse to S will be the countable intersection of open-dense sets, which is called a residual or generic set. This is where the connection between transversality and generic properties arises. We consider an example. -70-  5.3  Go is a generic property. A vectorfield A' G X ( M ) is said to have property Go if all critical points of X r  are non-degenerate in the sense of section 5.1. Since X has non-degenerate critical points if and only if X ^ OTM-, >d therefore the set of vectorfields satisfying Go ar  is open and dense (since OTM is compact.)  Generic 1-Parameter "Vectorfields In order to find generic properties of 1-parameter vectorfields, we must look for 1-parameter submanifolds of X ( M ) . In the last section, I hinted that one r  could project the "fold" that gave rise to the saddle-node bifurcation and get a codimension-1 submanifold E j C X ( M ) . If we consider the evaluation map for r  2-jets of vectorfields, then we may pull-back the submanifold of J (TM)  that  2  corresponds to saddle-node critical points. In natural vectorbundle coordinates on J (TM) 2  associated with a chart {U ,<p ), a  {0} x {A G L(R ) n  in R  n  a  : rank(A) — n. — 1} x {non-degenerate quadratic forms}  x L ( R ) x L ( R " ) . Since {0} C R n  we have that this becomes  2  maps is codimension-1 in L(R ) n  n  is codimension-n, the set of rank- n - 1  (since it is the pull-back of 0 G R  by the map  A H-V det(A)), and the set of non-degenerate bilinear forms is open in L ( R ) , then 2  the submanifold of saddle-node critical points is of codimension n + 1 in We may pull this back to a submanifold E i C ( X (M) T  n  J (TM). 2  x M) of codimension n + 1.  If this manifold projects to a codimension-1 submanifold E i C X ( M ) , then we r  will have a genericity theorem for one-parameter families of vectorfields. The openness result for transversal maps tells us that if a 1-parameter family has a saddle-node bifurcation (which crosses E] transversely), then nearby (in the sense of G ([0, l], X ( M ) ) ) 1-parameter families will also have a saddle-node bifurcation. r  r  The qualitative change associated with the saddle-node bifurcation occurs when the  -71-  5.3  manifold b\ is crossed transversely—we recall that on one side there are no critical points in a neighborhood of the bifurcating critical point, while on the other side the bifurcating critical point splits into two. E x t e n s i o n to O t h e r Bifurcations. The result that Go is a generic property for vectorfields is the first part of the Kupka-Smale theorem (in Abraham and Marsden [1978], chapter 7 or Abraham and Robbin (1967J), which gives several generic properties for vectorfields. Each of the generic properties ( GQ )—( G 3 ) is associated with some kind of non-degeneracy of critical points, periodic orbits, or the intersection of the stable and unstable manifolds of a pair of critical elements. In the literature on generic bifurcation theory (Sotomayor [1973a], Sotomayor [1973b], Sotomayor [1974]), codimension-1 bifurcations are examined that correspond to failure of each one of the non-degeneracy conditions/generic properties in the Kupka-Smale theorem, and submanifolds of X {M) T  are constructed that give rise to a corresponding genericity result for de-  generate (bifurcating) equilibria of 1-parameter families of vectorfields. The content of Abraham and Robbin [1967], is a modernised proof of the KupkaSmale theorem that relies on evaluation-transversality techniques like that used above to show that Go was a generic property in X ( M ) . In order to consider f  the corresponding codimension-1 bifurcations, it is necessary to extend all of the evaluation transversality results used in Abraham and Robbin [l967]to account for higher orderr terms. For example, the property Go involves non-degeneracy or the derivative, whereas it is necessary to have non-degeneracy of the second derivative for the saddle-node bifurcation. If this were done (and I believe it is possible), then the results from the literature on generic bifurcations could be reproduced along the lines of what I have done in this thesis, which would be more geometrically intuitive, and hence of some pedagogical value.  -72-  Bibliography  A b r a h a m , R.H., and Marsden, J.E. [ 19781. Foundations jamin/Cummings: Reading, M A .  of Mechanics. Ben-  A b r a h a m , R.H., Marsden, J.E.. and R a t i u , T . [1983]. Manifolds, Tensor Analysis and Applications. Addison-Wesley: Reading, M A . A b r a h a m , R.H., and R o b b i n , J . [1967]. Transversal Mappings and Flows. Benjamin: Reading, M A . C h i l l i n g w o r t h , D.R.J. [1976]. Differential  Topology with a View to  Applications.  P i t m a n : London. Guckenheimer, J., and Holmes, P.J. [1983]. Nonlinear Oscillations, Systems, and Bifurcations  Dynamical  of Vector Fields. Springer-Verlag: New York.  P a l i s , J . a n d de M e l o , W. [1982]. Geometric  Theory of Dynamical  Systems.  Springer-Verlag: New York, Heidelberg, Berlin. Sotomayor, J. [1974]. Generic one-parameter families of vector fields, Publ. Math. I.H.E.S.  No 43 (1974), 5—46.  Sotomayor, J . [1973a]. Structural Stability and Bifurcation Theory. In Dynamical Systems, M.M. Peixoto (ed.), pp. 549—560. Acedemic Press: New York. Sotomayor J. [1973b]. Generic Bifurcations of D y n a m i c a l Systems. In Dynamical Systems, M.M. Peixoto (ed.), pp. 5 6 1 — 5 8 2 . Acedemic Press: New York.  -73-  


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