Dynamical Systems

5/16/2018 Dynamical Systems

1/74

Lectures on Dynamical SystemsChapter 5. Stable manifolds: introductionChapter 6. Stable manifolds: smoothnessChapter 7. Stable manifolds: dynamics

Chapter 8. Unstable manifoldsChapter 9. Invariant manifolds: miscellany

Oscar E. Lanford II IMathematics DepartmentETH-Ziirich

8092 Switzerland

September 2, 1997

This text is a slightly revised version of part of a set of lecture notes distributedto the students in my course on dynamical systems at the ETH Zurich in theWinter Semester of 1991-92. Ihave had a number of requests for copies ofthese notes, and this has encouraged me to think that it might be useful tomake them more generally available. The reader should be warned, however,that, since Iam working on a major rewriting of this set of notes, Idid notwant to take the time to polish this older version carefully (although there isa good deal which needs to be done.) Iwill be grateful for comments aboutthese notes and particularly grateful to have any errors called to my attention.

Oscar E. Lanford III ([email protected])


2/74

34


3/74

Chapter 5Stable manifolds:introduction5.1 IntroductionWe will consider here a mapping I, at least once continuously differentiable,defined on an open set in a Banach space E , A fixed point Z o for I will besaid to be hyperbolic if the linear operator D I ( z o ) is hyperbolic in the sensedefined in 2.4, i.e., if its spectrum is disjoint from the unit circle. We willfix our attention here on a particular hyperbolic fixed point z o o By Corollary2.4.2, there is a splitting of E as a direct sum of two closed linear subspaces [sand [u, each invariant under DI ( z o ) , such that the spectrum of the restrictionof DI( z o ) to [s is entirely inside the unit circle and that of the restriction to[u entirely outside. Furthermore, [s can be characterized by

[s= {z E [: Anz --+ 0 as n -+ oo.}In this and the following chapters, we are going to establish the existence

of a "non-linear version" of [s. Loosely formulated, the idea is as follows: LetWS denote the set of points z such that

In (z ) remains near Z o for all n 2 0 and -+ Z o as n -+ 00.So defined, WS is just a set, manifestly mapped into itself by I and manifestlycontaining z o , but which a priori might contain nothing else. We are going toshow, in fact, that

35


4/74

36 CHAPTER 5. STABLE MANIFOLDS: INTRODUCTIONWS is a smooth submanifold of E and the tangent space to WS at Zo is t.

This invariant manifold is known as the stable manifold for f at Xo . We willconcentrate, for the first part of the analysis, on the behavior in the immediatevicinity of the fixed point, and will construct only a little piece of invariantmanifold. In nice cases, it is possible to extend the small invariant manifold weconstruct here in a natural way to a global invariant manifold. Thus, what weare concerned with here is more properly called a local stable manifold. Thereis an analogous non-linear version of [u; it is called the unstable manifold andwill be denoted by WU . If f is invertible, the unstable manifold for f can beconstructed as the stable manifold for f-1. As we will see later on, however,unstable manifolds can also be constructed without assuming invertibility of f.

Stable and unstable manifolds are objects of fundamental importance inthe study of hyperbolic fixed points and in dynamical systems theory in gen-eral. Fortunately, they are very well-behaved objects, and in particular thereis no loss of differentiability in passing from f to the corresponding stable andunstable manifolds: If f is of class C r, 1 ~ r ~ 00, then WS and WU aresubmanifolds of class C r2. The theory of these objects, while not exceptionallydifficult, is of substantial depth, and a circumspect indirect approach seems tobe needed to make the proofs reasonably simple, especially if the objective isto obtain results with weakest possible hypotheses. In broad outline, we aregoing to proceed as follows:

- At the outset, we forget about the fact that WS is supposed to be theset of points whose orbits converge to Zo and concentrate on finding aninvariant manifold tangent to [S at z o o

- It is in fact convenient to elarge the framework a little and to look for aninvariant set with a bit less regularity that necessary for a submanifold.

- We convert the search for an invariant set to the search for a fixed pointof an auxiliary operator on an appropriate function space, and we showthat this operator is contractive and hence has a unique fixed point.

- With the fixed point-i.e., the invariant set-already in hand, we proceedto prove first that the invariant set is a C1manifold; then that it has atleast as much additional smoothness as f itself.

IThis means in particular that the dimension ofW8 is equal to that of 82The converse doesn't hold: A mapping of finite differentiability can easily have-by

accident, so to speak-Coo stable and unstable manifolds


5/74

5.2. PRELIMINARY REDUCTIONS 37- Finally, we show that our invariant manifold does indeed consist exactlyof those points whose orbits remain near the fixed point and converge toit.

5.2 Notation and preliminary reductionsOur basic notation and standing hypotheses are as follows: We are consideringa mapping f defined on an open set in a Banach space E . When nothing is saidto the contrary, it is always to be understood that f is at least continuouslydifferentiable. We investigate f in the neighborhood of a fixed point, whichwe will take-for notational simplicity-to be the origin in E . We will denoteD f (0) by A, and we assume that A is a hyperbolic operator. It follows that Ehas a representation as

E = e, E9s.;and that, in this representation, A has the form

(As 0)o Au 'where1. the spectral radius of As is < 1, and2. Au is invertible and the spectral radius of (Au)-l is < 1.

When referring to points of E = E s E9E u , we will generally use- x (with subscripts, accents, etc.) to refer to the Es co-ordinate,- y to refer to the Eu co-ordinate, and- z to refer to the pair (x , y) .

We choose norms on Es and Eu so that(5.1)

and we equip E with the normI I ( x , y ) 1 1 = m a x { ll x l l , I ly l l }

We now separate off the linear part of f near the origin by writing:f(x, y ) = (Asx + fs(x, y), AuY + fu(x, y)) , (5.2)


6/74

38 CHAPTER 5. STABLE MANIFOLDS: INTRODUCTIONwhere I s and I are continuously differentiable and vanish, together with theirfirst derivatives, at the origin.

We will refer to the I s and i as the nonlinear parts of f. A first veryimportant observation is that, because we want to prove a result which is localat the fixed point, we can without loss of generality assume that I s and i are"small". This goes as follows: By assumption, the I s and i are continuouslydifferentiable and vanish, together with their first derivatives, at (0,0). Hence,the norms of their first derivatives can be made as small as we like on the ball ofradius E about (0,0) by making E small. We can then replace I by z I-t il(EZ)to get a new mapping with the same linear parts as I but whose non-linearparts have first derivatives which are uniformly as small as we like on the unitball. That is: Working on the unit ball with the nonlinear parts small in theC1 sense is equivalent to working on a small ball for a general C1mapping f.We will refer to this reduction as "magnification".

We next want to give a concrete analytical formulation of the geometricalnotion of manifold tangent to E s at O. It both corresponds to our geometricalintuition and can easily be proved from the standard definitions in manifoldtheory that a submanifold containing 0 is tangent to E s there if and only ifa sufficiently small piece of this manifold around 0 can be represented as thegraph f(w) of a mapping w, defined on a neighborhood of 0 in E s and takingvalues in Eu, such that

w(O) = 0 and Dw(O ) = o . (5.3)(The first condition simply says that the origin lies on the submanifold; thesecond expresses tangency.) Furthermore, the smoothness ofthe submanifold-or rather, of the small piece of it so represented-is exactly the smoothnessof w. In our original-unmagnified-problem, what we are looking for is afunction, defined on a small neighborhood of 0 and satisfying (5.3), whosegraph is mapped into itself by I . After magnification, we can assume that w isdefined on the open unit ball. Thus, what we want to prove is something alongthe following lines:Preliminary formulation of desired result. Let As and A u be as above.Then for any pair of functions I s , i , defined and continuously differentiableon the open unit ball of EsffiEu, vanishing together with their first derivatives atthe origin and having sufficiently small C1 norm on the unit ball, the mapping

I: (x, y) f---t (Asx + Is(x, y), A u Y + lu(x, y))leaves invariant a set of the form I'(w), where w is a C1 function from the unitball of E; into Eu, vanishing together with its first derivative at the origin.


7/74

5.3. LIPSCHITZ FUNCTIONS 39With this preliminary formulation in mind, we now ask when the graph

of a function w is mapped into itself by f of the form (5.2). To answer thisquestion, we write out, in our detailed notation for t,he image under f of ageneral point of the graph of w:

f(x, w (x)) = (Asx + fs(x, w (x)) , A uw (x) + fu (x, w (x)) .This will again be in the graph of w if and only if

Auw ( x ) + fu (x, w (x) ) = w (A sx + fs(x, w (x)) ) , (5.4)i.e., if and only if

w(x ) = A : ; ; - l (w (A sx + fs(x, w (x) )) - fu (x, w (x))) . (5.5)Now the right-hand side of (5.5) can be taken as defining a (non-linear) operatoron functions w, i.e., (5.5) can be interpreted as the fixed-point equation w = Fu)where

(Fw)(x) = A : ; ; - l (w (A sx + fs(x, w (x))) - fu (x, w (x)) ) . (5.6)The final result, then, is:

The graph o f w is mapped in to itse lf by f if an d o n ly if w is a fixed po in t fo rF as de fin ed in {5.6}.(Included in the fixed-point condition is the requirement that the right-handside of (5.6) is defined on all of the domain of w , i.e., that A sx + fs(x, w (x)) isin the domain of w for all x in the domain of w .)

5.3 Lipschitz functionsThe idea now is to choose a function space on which the operator F is a con-traction. Although many choices of function space can be made to work undersufficiently strong hypotheses on t,some care is needed to get the sharpestpossible results. A space which turns out to work well is a space o f Lipschitzco ntin uou s ju nction s, equ ipped w ith a ju diciou s mo dification o f the su premu mnorm . The technical advantage here in working with with Lipschitz functionsis that

- for our estimates, a Lipschitz condition is almost as good as a bound onthe first derivative, while

- Lipschitz continuity, unlike differentiability, is well-behaved under uniform-or even pointwise-limits.


8/74

40 CHAPTER 5. STABLE MANIFOLDS: INTRODUCTIONWe need to introduce a little formalism. If (Xl, pd, (X2' P2) are metric

spaces, we say that a mapping I : Xl -+ X2 is Lipschitz continuous if thereexists a constant k such that

P2(f(x),/(xl)) ~ kp2(X,XI) for all x,xl EXl.The set of k's for which this inequality holds is closed; its infimum will bedenoted by Lip(f). If the range space is a normed vector space, then the set ofLipschitz continuous functions is a vector space and Lip(f) is almost a normon this vector space. Although it fails to be a norm because it vanishes onconstant functions, we will nevertheless refer to Lip(f) as the Lipschitz normof f.

The Lipschitz norm is particularly well-behaved with respect to compositionof functions:Proposition 5.3.1 Let (XI,pd, (X2,P2), (X3,P3) be metric spaces, I: Xl -+X2 and 9 : X2 -+ X3 Lipschitz continuous mappings. Then go I is Lipschitzcontinuous and Lip(g 0 f)~Lip(g) Lip(f).Proof. For x, Xl in Xl,

P3(g(f(X)),g(f(XI))) ~ Lip(g)p2(f(x), I(xl)) ~ Lip(g)Lip(f)PI(x, z"),D

Once we have decided to look for the invariant set as the graph of a Lipschitzcontinuous function, it is no longer necessary to assume that I is continuouslydifferentiable. Instead of requiring that Is and I be continuously differentiableon the unit ball with uniformly small first derivatives, it suffices to assumeinstead that they are Lipschitz continuous with small Lipschitz norm. Thecondition that their derivatives vanish at the origin no longer makes sense inthis more general framework, but we retain the assumption that Is(O) = 0 andlu(O) = o .

We can now formulate our first existence and uniqueness result:Proposition 5.3.2 Let As, A~l be operators of norm < 1 on Banach spacesEs, Eu respectively, and let Is, I be Lipschitz-continuous mappings from theunit ball of E; E 9 Eu into Es, Eu respectively which vanish at the origin. If theLipschitz norms of Is, I are small enough, then there is a unique mapping Wsfrom the unit ball of E; into Eu, Lipschitz continuous with Lipschitz norm ~ 1,whose graph is mapped into itself by

I: (x, y) f---t (Asx + Is (x, y), AuY + lu(x, y))


9/74

5.3. LIPSCHITZ FUNCTIONS 41Proof. Let

XF:= {W : B; -+ Eu , w (O ) = 0 and Lip(w) ~ I}(where we have written B; for the open unit ball in Es. ) The idea is going tobe to show that F as defined in (5.6) maps XF contractively into itself. Forthe remainder of this proof, w will always denote an element of XF. We willfrequently use without comment the simple estimates

Ilw(x)11~ Lip(w)llxll ~ Ilxll < 1for all x E Bs.

To simplify the formulas arising in the course of the proof, it is convenientto introduce a streamlined notation. We rewrite the definition (5.6) of theoperator F as

(Fw)(x) = F(x, w (x), w (v(x, w (x)) )) (5.7)where

F(x, y , y l) = A~l [ y l - lu(x, y ) ] and v(x, y ) = A sx + Is(x, y ) . (5 .8)Note that-from the vanishing of Is , i , and w at the origin-

v ( O , O ) = 0 and F ( O , 0, 0) = o .When we consider functions of several variables, like F and v, we will writeLip, for the Lipschitz norm with respect to the i-th variable with the othervariables held fixed (i.e., in the differentiable case, the supremum norm of thei-th partial derivative.) From the definitions, we have

LiPl (F) < IIA~lll LiPl (fs) (arbitrarily small),LiP2(F) < IIA~l IILiPl(fu) (arbitrarily small),LiP3(F) IIA~lll, (5.9)Lip(v) < IIAsl1+ Lip(fs) IIAsl1+ E ) ,LiP2 (v ) < LiP2(fs) (arbitrarily small).

The last column in the above table-in parentheses-indicates how small thequantity in question can be made by making the Lipschitz norms of Is and I small.

We note further that, because of our choice of norm on Es f f i Eu , the mappingx I-t (x ,w(x)) has Lipschitz norm one, so

Lip(x I-t v(x,w (x))) ~ Lip(v).We will present the proof of the proposition as a seqence of steps, each a

simple computation, showing successively that


10/74

42 CHAPTER 5. STABLE MANIFOLDS: INTRODUCTION1. Fw is defined everywhere on B,2 . Fw (O ) = 3. F preserves the condition Lip(w) ~ 14. F is contractive

In the course of these computations we are led to impose a number of boundson F and V ; it follows easily from the estimates in the last column of (5.9) thateach of these bounds can be made to hold by taking the Lipschitz norms of I sand Iu small enough.Step 1. If

Lip(v) ~ 1then Fu: is defin ed on all o f B; for all w E XF.In terms of the notation introduced above, Fu) will be defined on all of B;provided

(5.10)

Ilv(x,w(x))11 < 1 when Ilxll< 1.Since v(O ,w (O )) v(O ,O ) = 0, this will follow if we can arrange that themapping x I-t v(x ,w(x) ) has Lipschitz norm ~ 1. We have already remarkedthat this Lipschitz norm is ~ Lip(v). D

From now on, we assume that Fu) is defined everywhere in the open unitball.Step 2. Fw(O ) = for all wE XF.This follows at once from w (O ) = 0, v(O, 0) = 0, and F(O, 0, 0) = 0. DStep 3. I f

(5.11)then

Lip(Fw ) ~ 1 fo r all w E XF.

From the definition (5.8) of F,Lip(Fw) < LiPl(F) + LiP2(F)Lip(w) + LiP3(F)Lip(w)Lip(v)

< LiPl(F) + LiP2(F) + LiP3(F)Lip(v).D


11/74

5.3. LIPSCHITZ FUNCTIONS 43Next comes the questions of contractivity of F. We do not prove contrac-tivity with respect to the Lipschitz norm, but with respect to a much weakerone. The norm we choose is

Ilwll :=sup Ilw(x)ll.#0 Ilxll (5.12)

It is easy to see that- Ilwll ~ Lip(w) if w ( O ) = O .- II II is a norm on the space of continuous mappings w : B8 -+ E u for which

Ilwll < 00, and this space is complete with respect to II II.- If Wn is a sequence in XF which converges with respect to II II, then thelimit is again in XF. (In fact: Pointwise convergence is enough.) Thus,

XF is complete with respect to II IIStep 4. I f

then F is contractive on XF with respect to II II.Let WI, W 2 be in XF and x E B8, X f : : - O . Then

F W I (X ) - F W 2 ( X ) = F ( X ,W I,W I( v d ) - F ( X ,W 2 ,W I( v d )+ F ( X ,W 2 ,W I( v d ) - F ( X ,W 2 ,W 2( v d )+ F ( x , W 2 , w 2 ( v d ) - F ( x , W 2 , W 2 ( V 2 ) )

where we have suppressed many arguments, using the convention that ui; with-out argument means W i ( X ) and that V i means V ( X , W i ( X ) ) . We then use theestimates:

and

to getIIFwI(x) - FW2(x)11

~ [Lip2(F) + LiP3(F) Lip(v) + LiP3(F) LiP2(v)lllwI - w21111xll


12/74

44 CHAPTER 5. STABLE MANIFOLDS: INTRODUCTIONDividing by I l x l l and taking the supremum over x, we see that : F is a contractionon XF provided

as asserted. DPutting all the pieces together: By the contraction mapping principle, : F

has one and only one fixed point in XF provided that (5.10), (5.11), (5.13) allhold, and, since

the estimates in (5.9) show that all three ofthese conditions hold ifthe Lipschitznorms of js and i are small enough. Thus, the proof of Proposition 5.3.2 iscomplete. D

5.4 GeneralizationsFor the proof of Proposition 5.3.2 we need only estimates (5.10), (5.11) and(5.13), and these can be made to hold provided

The condition I I A ~ l l l < 1 was never needed. This simple observation leads tothe following useful generalization: Let p ~ 1 be such that

a(Dj(zo)) n { I . x l = p} = 0 ,but such that

S_ :=a(Dj(zo)) n { I . x l < p} and S+ := a(D j(zo )) n { I . x l > p}are both non-empty. There is then an invariant direct sum decomposition ofE as E _ E9E+ , where E _ (respectively E+ ) is the spectral subspace for Dj(zo)corresponding to S_ (respectively S+.) In this representation for E , D j(zo)has block-diagonal form

and by proper choice of norm we can arrange that


13/74

5.4. GENERALIZATIONS 45The analysis of the preceding section applies in this more general situation

and proves the existence of a Lipschitz function, defined on an open e-ball aboutZo in E_ and taking values in E+, whose graph is invariant under f. The proofof the smoothness of the function-to be given in the next chapter-and ofthe vanishing of its derivative at Zo also extend to this more general situation.Thus we get an invariant manifold tangent to E_. Such an invariant manifoldis called a stro n g sta ble m an ifo ld; it can be characterized dynamically as theset of points whose orbits stay near to Zo and converge to it faster than p"(asym ptotically.)It is also useful-but a bit more difficult-to make a further extension to

allow for p > 1. The following situation is particularly important in practice:Suppose that the spectrum of D! ( zo ) can be split into two parts:

a (D !(z o) ) = s; U s;where

sup{loXl: oXE Scs} = 1 and inf{loXl: oXE Su } > 1.There is again a corresponding splitting E = E c s E9E u , etc., and we would liketo show that there is an invariant manifold tangent to E c s . Such an invariantmanifold is called a c en t er -s ta ble m a n if o ld .

The trouble in extending the above proofis in getting started, i.e., in provingthat F w is defined on the whole unit ball. The estimates leading to preservationof Lip(w) ~ 1 and to contractivity rely only on

(with a self-explanatory notation), and this bound can be made to hold withoutdifficulty. To get Fu) defined on the whole unit ball, we needed Lip(v) ~ 1,and for this the strict inequality

I I A s l 1 < 1is essential.

We can get around this difficulty-to some extent-by using the followingidea: We cut off the non-linear terms in ! away from the fixed point to geta mapping which is defined everywhere in E and globally close, in a Lipschitzsense, to a linear mapping. We then apply the analysis of the preceding sectionto prove the existence of a global invariant Lipschitz manifold W for the cut-offmapping. Because everything is defined everywhere, the problem of gettinga large enough domain for Fu) disappears. The cutting-off is done in such away as to make the cut-off mapping agree exactly with! on a neighborhood


14/74

46 CHAPTER 5. STABLE MANIFOLDS: INTRODUCTIONof the fixed point; then the manifold W which is globally invariant for the cut-off mapping is lo c ally in v a ria n t for I in the sense that, for U a small enoughneighborhood of the fixed point, if z E u c . W, and I(z) E U, then I(z) E W.

In more detail, the idea is as follows: We consider, in an obvious extensionof earlier notation, mappings I of the formI(x,y) = (Acsx + Ics(x,y) ,AuY + lu(x,y),

where Ic s and I are defined on the unit ball in E , vanish at the origin, andhave small Lipschitz norms.

We choose a Lipschitz-continuous real-valued function ' I j ; on E cs which isidentically equal to one on a neighborhood of 0 and identically zero for I l x l l 21 - E for some strictly positive E. Such a function can easily be constructed bycomposing the norm with a smooth function on the positive real axis which isone on a neighborhood of 0 and 0 to the right of 1 - E. (Note, however, thatby the remarks in 3.1, it may not be possible to take ' I j ; to be differentiable ifE cs is infinite dimensional.) Define

Ics(x,y)lu(x,y)f(x,y)

'I j; (x )/ cs ( x, y )'Ij;(x)lu(x,y)(Acsx + Ics(x, y ), A uY + lu (x, y ) )

Although I might not have been defined on all of E , we can evidentlyextend Ics, lu to vanish when I l x l l 2 1 and extend 1orrespondingly. Bymaking Lip(fcs) and2:-ip(fu~mall on the unit ball, we can guarantee that theLipschitz norms of Ic s and I taken o ver the set o f all (x,y) w ith I l y ll ~ I l x llw ith n o restriction at all o n x are as small as we like". Dropping the tildes, weare led to consider a global version of the invariant manifold problem, i.e., toconsider a mapping Iwhich is defined and uniformly near to linear in the senseof Lipschitz norm on-essentially-all of E cs E9E u and to look for a functionw defined and satisfying the condition Lip(w) ~ 1 on all of E cs whose graphis mapped into itself by I. Reexamination of the argument in the first fewparagraph ofthis section shows that it works perfectly in this situation, withoutrequiring Lip(v) ~ 1, provided that all the Lipschitz norms are understood tobe taken over the appropriate domains, e.g., the LiPi(F) is to be understood3We could have taken a cutoff function ' I j J which cuts off in both x and y, in which case

the condition I l y l l :S I l x l l would be unnecessary. Cutting off only in x has the advantagethat it may be easier to find smooth cutoff functions on t : c s than on t: (particularly if t : c s isfinite-dimensional and t: u infinite-dimensional.)


15/74

5.5. A MORE GENERAL SETUP 47as taken over the domain

{(x, y , y l) : Ilyll ~ Ilxll and Ily/ll ~ Ilxll with no restriction on z.]

5.5 A more general setupThus, to deal prove the existence of both stable and center-stable manifolds,we need two versions of the basic existence theorem, one which requires strictcontractivity of the "less expansive" part of D!(zo) and works on the unitball, and a second which does not require this strict contractivity and works onthe whole space. Once an appropriate cutoff of the non-linear terms has beenmade, the proofs in the two cases are essentially identical. It therefore seemsefficient to carry out, simultaneously, two versions of the argument, adapted tothe two situations. The idea of cutting off the nonlinear terms can evidently bemade to work in more general circumstances than that described above, andwe may as well carry out the proofs with this extra generality. We are thus ledto carry out the analysis which will follow-notably, the proof of smoothnessof the invariant manifolds-in the following somewhat abstract general setting:Notation and standing hypotheses: We consider two Banach spaces E_and E+, equipped with bounded operators A_ and A+ respectively. We denoteby E the direct sum E_ E 9 E+, and we equip E with the norm

II(x,y)11= max{llxll, Ilyll}We define

p : := SUp{IAI: A E a(A_)} and p+:= inf{IAI : A E a(A+) } . (5.14)Our fundamental hypothesis is that

p : < p+ ,so that the open annulus {A . : p : < IAI< p+ } is disjoint from the spectrum of

A:= (Ao- AO + ) on E.In particular, A+ is invertible. We will always assume that the norms on Eare chosen so that

IIA_IIIIA:;:lll < 1;we may later need to impose further restrictions on how the norm is chosen.We will consider in parallel the following two situations:


16/74

48 CHAPTER 5. STABLE MANIFOLDS: INTRODUCTION- (contractive case): p : < 1; B_ denotes the open unit ball in E_; and B

denotes the open unit ball in E. The norm on E_ is chosen so that

I I A _ I I < 1.- (non-contractive case): No assumption on p : ; B_ denotes E_; and B

denotes an open convex neighborhood of {(x, y) E E : I l y l l ~ I l x l l } .In both cases, we consider mappings of the form

f(x, y) = (A_x + f _(x, y), A+y + f+(x, y)), (5.15)with

andJ(O,O) = o .

Hlhen we speak of Lipschitz norms of function of x (respectively (x,y)), wemean Lipschitz norms over B_ (respectively B.)Summarizing-and extending slightly-the results of this chapter, we see thatwe have proved the following fundamental existence and uniqueness result:Proposition 5.5.1 If the Lip(J) are small enough, there is a unique

w : B_ -+ E+ with w(O) = 0 and Lip(w) ~ 1whose graph is mapped into itself by f. In terms of the notation (5.8), a set ofsufficient conditions for the existence and uniqueness of w is:

Lip(v) ~ 1LiPl (F) + LiP2 (F) + LiP3(F)Lip( v) ~ 1

LiP2(F) + LiP3(F) Lip(v) + LiP3(F) LiP2(v) < 1In the non-contractive case, (A) is not needed.

(A )(B)(C)

The reason for including the-not very illuminating-explicit sufficient condi-tions (ABC) in the statement ofthis proposition is so that they will be availablelater for comparision with the conditions needed to deduce smoothness of wfrom smoothness of f. Indeed, since the condition (ABC) (respectively (BC))are needed for our proof of the existence of the invariant set, it will be conve-nient to add them to our standing hypotheses:


17/74

5.5. A MORE GENERAL SETUP 49Standing hypotheses 2: From now on, we assume the above estimates (B)and (C) hold. In the contractive case, we assume further that (A) holds.Although it is efficient to consider the contractive and non-contractive casestogether, the reader should be aware that the final results are considerably lesssatisfactory in the non-contractive case:First: In the infinite-dimensional situation, there is the problem of whether thecut-off function ' I j ; can be taken to be smooth, or, more generally, whether wecan find any way of modifying the initial mapping outside a small neighborhoodof 0 to produce a smooth global mapping with uniformly small nonlinear part.Unless this can be done, there would seem to be no hope of proving smoothnessof the local invariant manifold.Second: We will see in the next chapter that, if the f are of class C r with firstderivatives vanishing at the origin, and if

then w is also c r . This latter condition is a consequence of p : < p+ in thecontractive case, but not in the strongly non-contractive case ip : > 1.) Ex-amples show that it is really necessary. Thus, in the strongly non-contractivecase, there is an intrinsic finite limit on the smoothness of the invariant mani-fold even for very smooth f 's; no such limitation exists in the contractive case.(For the borderline case of the center-stable manifold, i.e., p : = 1, there areno problems for any finite r, but we will give later an example of an analyticmapping with no C o o center-stable manifold.)Third: As we shall see, the invariant manifold is unique in the contractive casebut not-in general-in the non-contractive case. This may seem paradoxicalin view of the uniqueness statement in Proposition 5.5.1. The explanation forthe apparent paradox is that, to get from the original mapping to the f towhich Proposition 5.5.1 applies, we need, in the non-contractive case, to cutoff the non-linear terms. It is not surprising that the invariant manifold shoulddepend on the cutoff. What may be surprising is that changing the cutoffcan change the invariant manifold even very near to the fixed point, where thecutoff does nothing. That is: In the non-contractive case, local invariance isnot really a local property; the invariant manifold can contain orbits which,although starting out very near to the fixed point, eventually get well awayfrom it.


18/74

50 CHAPTER 5. STABLE MANIFOLDS: INTRODUCTION


19/74

Chapter 6Stable manifolds:smoothness6.1 Differentiability of w.We will consider here the general setup of 5.5. Proposition 5.5.1 then assures usof the existence-and uniqueness in an appropriate class-of a function whosegraph is mapped into itself by f. For purposes of this chapter, w, without anyfurther label, will always denote this function. We now add to our standinghypotheses thatStanding hypotheses 3: the J are continuously differentiable and theirderivatives vanish at the origin.Our objective here is to prove:Proposition 6.1.1 With our current standing hypotheses, w is continuouslydifferentiable on B_ and Dw(O) = o .We will later investigate to what extent further smoothness of f implies fur-ther smoothness of w. It is useful to observe that this result requires no new"smallness" conditions on the f beyond those needed to make the originalexistence proof (Proposition 5.5.1) work.Proof. We will use the notation of the preceding chapter: The function wsatisfies the equation

w(x) = F(x,w(x) ,w(v(x ,w(x) ) ) )51


20/74

52 CHAPTER 6. STABLE MANIFOLDS: SMOOTHNESSwhere

F(X,y,yl) = A+l [y l - f+ (x,y )] and v(x,y) = A_x + f-(x,y).With our stronger assumptions, F and v are continuously differentiable, andthere are simple expressions for their derivatives at the origin. We will generallysupress arguments when they are clear from the context. Thus, for example,we would normally write the right-hand side of the above functional equationfor w simply as F(x,w,w(v)) .

The general plan of attack is as follows:1. We differentiate formally the functional equation satisfied by w to get thefunctional equation

for its derivative. Although we do not yet know that w is differentiable, we doknow that, if it is differentiable, its derivative satisfies this equation.2. We show that the above equation "can be solved uniquely for Dw . " That is:We define an operator K acting on a space of mappings from E_ to (E_,E+)by

Ka(x) = D lF + D 2Fa + D 3Fa(V )[D l v + D 2va], (6 .1 )and show that this operator has a unique fixed point in an appropriate functionspace. Again, we denote this unique fixed point simply by a. Thus, withoutknowing that w is differentiable, we know that a is the only possible candidatefor its derivative.3. We then use the functional equations for wand a to verify that a does havethe defining property of the derivative of w, i.e., that

Ilw(x + 8 x) - w (x) - a(x)8 xll = o(118xll),and hence that w is indeed differentiable.

We will work in the space XK of continuous mappings a from E _ to (E _, E +)with a(O) = 0 and Ila(x)11 ~ 1for all x. (This last bound is chosen to corre-spond to the condition Lip(w) ~ 1.) We equip XK simply with the supremumnorm. From the definition (6.1) of K , and from v (O ,O ) = 0, w (O ) = 0, andDlF(O , 0, 0) = 0, it follows that Ka(O) = 0 if a(O) = o . Also, simply by takingthe norm of the defining equations, and using estimates like IIDlFl1 ~ Lip. (F),we see that IIKa(x)11 ~ 1for all x provided Ila(x)11 ~ 1for all x and provided

LiPl (F) + LiP2(F) + Lip3(F)Lip(v) ~ 1.


21/74

6 .1 . D I FF EREN TI A BI LI TY OF W. 53This latter inequality is exactly condition (B) in our standing hypotheses, andthis is not an accident; (B) was introduced in order to guarantee that : F pre-serves the condition Lip(w) ~ 1.In the above we used the estimate:

To prove this, let (xo, Y O ) be a point of B , and let 170 be an operator from E _ toE+ with norm ~ 1. Then x f-t v ex, y o + l7o (x - xo )) has Lipschitz norm not greaterthan that of v. Thus, the derivative of this mapping at zo has norm not greater thanLip(v), i.e.,

To find conditions which make K a contraction, we use the definition towrite

D 2F(al - (72 ) + D 3 F(al(v) - 1 72(v) )[D lv + D2a l ]+D 3Fa2D 2v(al - (72 ) .Taking norms and estimating in a completely straightforward way shows thatK is a contraction provided

and this is exactly condition (C), i.e., the bound needed to make : F contractive.We thus now let a denote the unique fixed point for K in XK. We want to

show that a(x) is the derivative of w at x for each point x. For this purpose,let

M .- 1 Ilw(x + 8 x) - w (x) - a(x)8 xll.- sup im sup II' II .x ox--+o uXIf follows from Lip(w) ~ 1and Ila(x)11~ 1that M ~ 2. The idea now will beto use the functional equations to show that

M ~ ",M for some", < 1.Since M is finite, this will imply M = 0, which is exactly the statement thata(x) is the derivative of w at x for all x.

To shorten the formulas, we introduce the abbreviations:8 w = w (x + 8 x) - w (x)

8 v = v( x + 8x,w + 8 w ) - v(x,w )


22/74

54 CHAPTER 6. STABLE MANIFOLDS: SMOOTHNESSFrom the functional equation for w, and the assumed differentiability of F,

8 w = D 1F8 x + D 2F8 w + D3F[W(V + 8 v) - w (v)] + o(8x).We then multiply 8 x by the expression given by the functional equation fora(x) and subtract from the above equation; this gives

8w - a8 x = D 2F [8 w - a8 x] + D3F[W(V + 8 v) - w (v) - a(v)8 v]+D 3Fa(V ) [8 v - D 1v8 x - D 2V 8w ] (6.2)+D 3Fa(v)D 2V [8 w - a8 x] + o(8x).

By the differentiability of v,

Also, 1 1 8 v ll ~ L i p (v ) 11 8 x ll ,and hence

1 I l w ( v + 8 v) - w (v) - a(v)8 vll L ( )Ml r : : : ~ P 1 1 8 x l l ~ ip v .Taking norms of both sides of (6.2), dividing by the norm of 8 x, letting 8 x

tend to 0, and inserting the preceding two estimates gives

so w is differentiable, with derivative a, provided

and this is our condition (C). Thus, the proof of Proposition 6.1.1 is complete.D

6.2 Holder continuity of first derivativesProposition 6.2.1 Assume, in addition to our standing hypotheses, that

- the f are of class C1+a with 0 < 0:< 1(C[1+0:]).


23/74

6.2. HOLDER CONTINUITY OF FIRST DERIVATIVES 55Then w is of class C1+a. A similar assertion holds for a = 1; we assume thatthe D f are Lipschitz continuous and prove that Dw is Lipschitz continuous.The inequality (O[l+aj) follows from our standing hypotheses (0) and (A)in the contractive case. In the non-contractive case, it can be made to hold-first by choosing the norms properly and then by making the Lip(f smallenough-if

Proof. We recall that a mapping 9 between metric spaces:

is said to be a-Holder continuous if there is a constant k such thatd2(g(x) ,g (x ') ) ~ kdl(x,x')a for all x,x' in Xl.

We will let Lip(a) (g ) denote the smallest such constant k, and we will refer toLip(a) (.) as the a-Holder norm.

We will present the proof of this proposition as a general lemma. This bothsimplifies the notation and permits us to avoid having to repeat essentiallythe same argument when we investigate higher derivatives. We showed abovethat Dw is the unique fixed point of a contractive operator K an appropriatefunction space. We will write the operator as

(Ka) (x) = G (x , a (x ), a (v (x))) (6.3)where

(with the usual rule for filling in supressed arguments) andv(x) = v(x ,w(x) ) . (6.5)

The variable x takes values in B_ , the variables s, s' in the closed unit ball in (E _, E +). With our assumptions, G is a-Holder continuous in its first argu-ment (uniformly in its second and third arguments.) It is in fact quadratic-hence, Coo-in its second and third arguments, but all we will use here isLipschitz continuity in these arguments. The space on which K is contractivecan be taken to be the set of all countinuous mappings from B_ into the closedunit ball of (E_ ,E+ ) , equipped with the supremum norm. (In the above, weimposed also the condition a ( O ) = 0, but this condition was not needed in theproof of contractivity.) v is a mapping of class CIfrom B_ into itself.

For later use, we consider the following setup:


24/74

56 CHAPTER 6. STABLE MANIFOLDS: SMOOTHNESS_ S denotes a closed ball in a Banach space z._ G denotes a continuous mapping from B_ x S x S to S which is Lipschitz

with respect to its second and third variables._ v denotes a Lipschitz mapping from B_ to itself.

We assume that(6.6)

from which it follows readily that the operator K defined by(Ka)(x) = G(x, a(x), a(v(x)))

maps the space of all continuous functions from B_ to S-equipped with thesupremum norm-contractively into itself.

Lemma 6.2.2 Let S, G, v be as above, and assume further that G is a-Holdercontinuous with respect to its first variable, uniformly with respect to the othervariables, and that

(6.7)Then the unique fixed point of K is a-Holder continuous.Proof of Proposition 6.2.1 from Lemma 6.2.2. The assumption on thea-Holder continuity of G with respect to the first variable means that

L (al(G)- {IIG(xI,s,s/)-G(x2,s,s/)II. _ _ j _ B I S }IPI .-sup I I I I .XI-r--X2E _,s,s EXl - X2 ais finite. If we take G, v as defined in (6.4) and (6.5), it is easy to see thatLip~al (G ) is finite and that

Lip3(G) ~ Lip3(F)Lip(v)Lip(v) ~ Lip(v).

Hence, (6.7) follows from


25/74

6.2. HOLDER CONTINUITY OF FIRST DERIVATIVES 57i.e., from our assumption (C[l+a]). Thus, a-Holder continuity of Dw follows.That (C[l+a]) follows from (C) if Lip(v) ~ 1 is obvious from inspection of theformulas. That (C[l +a]) can be made to hold if

(p_)1+a < p+follows easily from the table of estimates (5.9) DProof of Lemma 6.2.2. The idea will be as follows: We will look for a boundon Lip(a) (a ) which is preserved by K, i.e., for a number B such that

a: B_ -+S and Lip(a) (a) ~ B implies Lip(a) (Ka) ~ B.Once we have found such a B, it is easy to finish the proof of the Holdercontinuity of Dw: The set of mappings

a: B_ -+S with Lip(a) (a) ~ Bis closed and hence complete with respect to the supremum norm. It is mappedinto itself by K , and K is contractive in the supremum norm. Hence, K hasa fixed point in this set. But K has only one fixed point, so the unique fixedpoint must satisfy

Lip(a) (a) ~ B < 00.To show that such a constant B exists, we let a be any a-Holder continuous

mapping from B_ to S, and we estimate the a-Holder norm of Ko . For anyXl f : : - X2 , we have

G(XI , aI, a( ih)) - G (X 2, a2 , a(v2 ))G (X I,a l,a(v d) - G (X 2,al,a (v d)

+G(X 2,al,a(v d) - G (x2 ,a2 ,a(vd)+G(X 2, a2 , a(vd) - G (X 2, a2 , a(v2 ))'

Here we have written subscripts 1 and 2 to indicate objects to be evaluatedat X = Xl and X = X2 respectively. We make the following straightforwardestimates:

IIG (X I,a l,a(vd) - G (x2 ,al,a(vd)11 ~ Lip~a)(G)llxI - x211a.IIG (x2 ,al,a(vd) - G (x2 ,a2 ,a(vd)11 ~ LiP2(G)Lip(a) (a)llxI - x211a

IIG (X 2, a2 ,a(v d) - G (x2 ,a 2, a(v2))11 ~ LiP3(G)lla(vd - a(v2)11~ LiP3(G)Lip(a) (a)llvI - v211a~ LiP3(G)Lip(a) (a ) (Lip(v))'" IlxI - x211a


26/74

58 CHAPTER 6. STABLE MANIFOLDS: SMOOTHNESSInserting into the above formula for Ka(xd - Ka(x2) gives

IIKa(xd - Ka(x2)11 ~ IlxI - x211'"xx (LiP~"')(G ) + [Lip2(G) + Lip3(G)(Lip(v))"'] Lip("') (a))

Dividing by IlxI - x211and taking the supremum over Xl f : : - X2 gives anestimate of the form

whereA := Lip~"')(G) and K,:= LiP2(G) + Lip3(G)(Lip(v))"'.

By (6.6), K, < 1, so, if we takeB :=A /(l - K ,) ,

we get

as desired D

6.3 Higher derivativesProposition 6.3.1 Assume, in addition to our standing hypotheses, that

- the f are of class Cr, 1< r < 00,(C[r]).

Then w is of class c r . In the contractive case, if the f are of class Coo , thenw is of class Coo .Proof. The assertion for 1 < r < 2, and a partial result for r = 2, has alreadybeen proved in Proposition 6.2.1. The complete proof will go by induction onthe integer part of r, and part of the induction step has already been proved(Lemma 6.2.2). The careful and complete formulation of the induction argu-ment is a little heavy, and it is therefore perhaps useful to say at the outsetwhat the idea is, and to explain where the condition (C[r]) comes from. Forthis purpose, let us consider only integer r. Suppose that the f are of classc r and that, by the induction step, we already know that w is of class Cr-l.


27/74

6.3. HIGHER DERIVATIVES 59The idea is to imitate the proof of differentiability of w. Differentiating thefunctional equation for w r times (formally) gives an equation of the form

D2FD Tw + D 3FDTw (v) (Dvr+D 3FDw ( v ) D 2VDTW + lower order terms,

where the "lower order terms" are ones not involving D Tw and hence are undercontrol by the induction hypothesis. (D v is used here as an abbreviation for thefirst derivative of x I-t v (x , w ( x) ). ) We can regard this formula as an equationto be solved for D Tw in terms of the already-known lower order derivatives.In contrast to the situation with the first derivative, right-hand side of thisequation is affine (i.e., linear plus constant), and the equation will have aunique solution if the linear operator on the right has norm < 1. This will bethe case if

and this is exactly (C[r]). Thus, if (C[r]) holds, we can compute what D Twmust be before we know that it exists. We can then show, by roughly the samemethod as for the first derivative, that the solution to this equation is indeedthe r-th derivative of w.

Although the proof of this last point is similar in spirit for r = 1 and r > 1,the details in the two cases are sufficiently different so that it does not seemto be practical to invent hypotheses general enough to encompass them both.What we will in fact do is to repeat the argument of 6.1 with the appropriatechanges. We will do this is the general framework introduced in the precedingsection which is adapted to formalizing the induction argument. The technicalresult needed is as follows:Lemma 6.3.2 Let S, G, v be as in Lemma 6.2.2, and assume in addition tothe hypotheses of that lemma that

- G E C1, and- LiP2(G) + Lip3(G)Lip(v) < 1.

Then a is of class C1 and Do satisfies(6.8)

With the original choices for G, etc., this lemma shows that w is of class C2if the J are and if (C[2]) is satisfied. The key to the induction on r is theobservation that the equation for Do has the same general form as the equation


28/74

60 CHAPTER 6. STABLE MANIFOLDS: SMOOTHNESSfor a and so the lemmas can be made to apply successively to D 2w, D 3W, etc.We will first give the proof of the lemma, then explain the induction argument.Proof of Lemma 6.3.2. Differentiating the equation for a, we see that thederivative of a, if it exists, must satisfy (6.8). We define a linear operatorK on the space X of all bounded continuous functions from B_ to (_,2)(Reminder: 2 denotes the Banach space in which the values of a lie) by

(KT)(X) = D 2G T(X ) + D 3 G T(V (X )) D v(x),so the equation for Do reads

D a = D IG + K(Da) .We equip the space iwith the supremum norm; then

Call the quantity on the right n, By assumption r;, < 1, and it follows easilythat the mapping

T f---t D IG + KTmaps the ball S of radius B :=LiPI(G)/(l - r;,) in i contractively into itself.Thus, this mapping has a unique fixed point, and we will from now on let Tdenote this fixed point.

To show that T is the derivative of a, we introduce, as before,M .- 1 I l a ( x + 8 x) - a(x) - T 8 x l l.- s~p im sup 1 1 8 x l l .

By Lemma 6.2.2, with 0: = 1, we see that a is Lipschitz continuous, and fromthis it follows that M is finite. We are going to prove

M ~ r;,M ,with r;, as above, and this will finish the proof.

To prove this estimate, we fix x and defineSo :=a( x + 8 x) - a(x),

and8 v :=v(x + 8 x) - v(x)

Then8 a G (x + 8 x, a + S a, a (v + 8 v)) - G (x, a, a(v ))

G I 8 x + G 28 a + G3(a(v + 8 v) - a(v) ) + o ( 1 1 8 x l l ) ,


29/74

6.3. HIGHER DERIVATIVES 61where we have used Lipschitz continuity to see that

8 v and a( v + 8v ) - a(v ) are both O(118xll).Multiplying the functional equation for T by 8x and subtracting gives

8a - T8x = G2[8a - T8x]+G3[a(v + 8 v ) - a(v ) - T (v )8 v + o(118xll).

Since118vll~ Lip(v)118xll,

we have1 Ila(v + 8v ) - a(v ) - T (v )8v ll L (-)M1 r : : : ~ P 118xll ~ ip v .

Combining the various estimates, we get

as desired. DLemma 6.3.3 Let S, G, v be as in Lemma 6.2.2, let 1 < r < 00, and assumein addition that G and v are of class Cr and that

(6.9)Then the unique fixed point a is of class cr.Proof. In the proof of Lemma 6.2.2 we saw that a is differentiable and thatT = Do satisfies the equation

T (X ) = G(X,T(X) ,T (V(X) ) ) ,where

G(X , t, t') :=DIG + D2Gt + D3Gt' Dti ,and where the variables t, t' take values in a sufficiently large ball S in theBanach space i := (_,2). If G and v are of class C r, then G is of classCr-I. It also follows immediately from the formula that


30/74

62 CHAPTER 6. STABLE MANIFOLDS: SMOOTHNESSPutting these considerations together we see that, if G, v, S fulfill the originalassumptions-in particular that

-and if in addition G, v are of class Cr, r > 1, and satisfy (6.9):

then G, v, S also fulfill the original assumptions, G, v are of class Cr-1, andsatisfy

which is just (6.9) with r replaced by r - 1. From these observations andLemmas 6.2.2 and 6.3.2, it is easy to see by induction that Do is of class Cr-1,i.e., that a is of class c r . DProof of Proposition 6.3.1. As already noted, Lemma 6.3.2 with the originalchoice ofG, etc., completes the prooffor r = 2, so we need only consider r > 2,and we know that a = D w satisfies the standard equation with

andv(x) = v(x,w(x) ) .

Although it follows immediately from the smoothness assumption of the propo-sition that F(x,y,y ' ) and v(x,y) are of class C r, the results of the precedingparagraph do not quite apply because w appears in the expressions for G andv-e.g., D IF is an abbreviation for (DIF) (x ,w(x) ,w(v(x ,w(x) ) ) ) - so we can-not directly conclude that G, v are of class Cr-1. To get around this we needto make another small induction argument: We write the r of the propositionas n + 0: with n an integer and 0 < 0: ~ 1 and argue by induction on n. Aswe have already remarked, the assertion is true for n = 1. We thus assume theproposition for n and prove it for n + 1. By the induction hypothesis, w is ofclass Cr-\ from this it follows at once that G and v are of class Cr-1, thenfrom the preceding paragraph that D w is of class Cr-1, i.e., that w is of classc r .

This completes the proof of the assertions for finite r. The assertion forr = 00 applies only in the contractive case, and our standing assumptions theninclude the bound Lip(v) ~ 1. Thus, if (C) holds, (C[r]) holds for all r > 1, sow is of class C r for all r, i.e. is of class Coo . D


31/74

6.4. SOME EXAMPLES 636.4 Some exampIes6.4.1 Non-existence of smooth generalized stable mani-

foldsWe will give here an example to show that the condition p " : _ < p + appearing inProposition 6.3.1 for the existence of a C T generalized stable manifold (in thecase where p : > 1) is, in at least one instance, sharp. Let . x > 1, and let

The origin is a fixed point, and

D f ( O , O ) = (~ ~ 2 ) .We split ]R 2 as the direct sum of the x and y axes, i.e., we take the x-axis forE _ and the y-axis for E + . The theory applies with p : = . x and p + = . x 2 Since

p " : _ = .x T < p + = . x 2 if and only if r < 2,our general theory asserts that there is, for each r < 2, a locally invariantmanifold of class C T tangent to the x axis. We will show that there is no suchmanifold of class C 2

A manifold tangent to the x axis at (0,0) is, locally near (0,0), the graph ofa function y = w(x) . The condition for invariance is easy to work out directly:The image under f of a point (x,w(x) ) of the graph is ( .x X , .x 2 W ( X ) + x2 ); thisis again in the graph if and only if

hence, the graph is locally invariant if and only if this equation holds for allsufficiently small x. It is nearly trivial to see that no function which is twice-differentiable on a neighborhood of can satisfy (* ) on a-possibly smaller-neighborhood of 0: Assuming that w is twice differentiable and satisfies (*),we can differentiate twice to get

then put x equal to to get the contradictory equation


32/74

64 CHAPTER 6. STABLE MANIFOLDS: SMOOTHNESS

6.4.2 Non-uniqueness of the center-stable manifoldConsider the (uncoupled) pair of differential equations

dydt = (log 2 )y .The general solution is easy to write down; it has the form

x(t) = 1x~ , y(t) = 2tyo- Xoand is defined for (-00 < t < 1/ xo ) if Xo > 0 and for (1/ Xo < t < 00) if Xo < O.The time-one solution mapping

xf : (x,y) I---t (-,2y,)I-xis defined and analytic in a neighborhood of the origin, and has the origin as afixed point. The derivative of f at the origin is

so the center-stable subspace is the x-axis and the unstable subspace the y-axis.Consider now a solution curve for the differential equation with Xo > o . Att -+ -00, y(t) goes (exponentially) to 0 and x(t) goes to zero like -l/t. Hence,the solution curve is asymptotic to the origin and has there a tangency ofinfinite order with the x-axis. The part of the solution curve near the origin isevidently locally invariant for the time-one map f. The union of such a solutioncurve with the negative x-axis (including the origin) is thus a locally-invariantC o o manifold tangent to the x-axis at the origin, i.e., is a center-stable manifold.Since there are continuously many pairwise disjoint solution curves, there arecontinuously many center-stable manifolds. It may nevertheless be remarkedthat there is only one of these-the x-axis-is an analytic manifold. This istrue in general: In the C o o center-stable case, all the derivatives of w at theorigin are uniquely determined by the functional equation for w in terms of thederivatives of f at the origin. Hence, there can be no more than one analyticcenter-stable manifold. Unfortunately, as the following example shows, theremay be none at all.6.4.3 An analytic mapping with no C o o center-stable man-

ifold.We start by generalizing the example given above of a mapping with no C2invariant manifold tangent to E _. Consider a one-parameter family of mappings


33/74

6.4. SOME EXAMPLES 65of the form

f>.(x,y) = (Ax,2y+h(x)) ,where h is analytic in a neighborhood of 0, vanishes to second order at 0, andis n o t a polynomial. Applying an idea due to Ruelle and Takens, we regardthe whole family of two-dimensional mappings as a single three-dimensionalmapping by treating the parameter as a third coordinate with trivial evolution,and we look at center manifolds of this augmented system 1. Specifically, weconsider the mapping

F: (A , x , y ) f---t (A , A X, 2y + h(x)).Each point of the form (A , 0, 0) is a fixed point for this mapping, but we wantto look in particular at the fixed point with A = 1. The derivative of F at thisfixed point is

o ~ nThus, the center-stable subspace is the (A , x) plane and the unstable subspaceis the y-axis. A center-stable manifold is a locally invariant manifold of theform { y = W(A,X) }Using the form of F we see-as in subsection 6.4.1-that the local invariancecondition can be expressed as the requirement that the equation

W (A , A X ) = 2W(A , x ) + h(x),hold in a neighborhood of (1,0). We can now generalize the argument ofsubsection 6.4.1 to show:Let A n den o te 21 /n . If (*) holds in a n eighborhood o f (A n ' 0), an d if M n) (0) - I - 0,then W can n o t be en at (A n ' 0).

The proof is very simple: If W were en , we could differentiate (* ) n timeswith respect to X to get

1We will make more serious use of this idea when we discuss bifurcation theory. Thetechnical point is that an invariant manifold for the augmented system gives a parametrizedfamily of invariant manifolds for the mappings depending on a parameter, and the device ofRuelle and Takens permits us to deduce smoothness of the invariant manifold with respectto the parameter from results about individual mappings.


34/74

66 CHAPTER 6. STABLE MANIFOLDS: SMOOTHNESSUsing (An ) n = 2 and setting x = 0 leads to the contradictory equation

By assumption, h(x) is not a polynomial, i.e., there is an infinite sequencenj going to infinity such that h(n;) (0) f : : - 0 for all j. The corresponding A n ; 'sconverge to 1, so the conclusion isThere is no Coo function w satisfying (*) in a neighborhood of (1,0).(This example is modelled on one due to S. J. van Strien, "Center manifoldsare not C O O " , Math. Z. 166 143-145 (1979).)


35/74

Chapter 7Stable manifolds: dynamicsWe initially introduced the stable manifold for a hyperbolic fixed point Z o as theset of points z with forward orbit ( f n ( x ) : n = 0,1, ... ) remaining near Z o forall n and converging to Z o as n -+ 00. We then constructed the stable manifoldusing primarily the requirement of invariance. We now want to show thatthe manifold we constructed really can be characterized as suggested above.This characterization has two aspects: We need to show both that orbits onthe stable manifold do converge to the fixed point and that orbits not onthe stable manifold do not. In both cases, there are generalizations to otherkinds of generalized stable manifolds. We will work in the following generalframework: The main estimates will be made in our usual magnified-and, inthe non-contractive case, cut-off and globalized-context, i.e., working on theunit ball or the whole of E_ and imposing smallness conditions on the non-linearterms. Although this is not really necessary, we will assume here that the fare continuously differentiable. Then, if for example we have established thatsome orbit converges to the fixed point, and we want to make sharp estimateson the asymptotic rate of convergence, we can assume-without changing themap, just by further magnification-that the non-linear terms are as small aswe like.

For our purposes here we can forget most of the intricate notation of thepreceding two chapters. We will consider mappings of the standard form:

f(x, y) = (A_x + f _(x, y) , A+y + f+ (x, y));we take the J to be C1 and vanishing together with their first derivatives atthe origin; and we will always assume that the norms are chosen so that

67


36/74

68 CHAPTER 7 . STABLE MANIFOLDS: DYNAMICSWe further assume that there is a C1 function w, vanishing together with itsfirst derivative at the origin, with Lip(w) ~ 1, whose graph is mapped into itselfby I ; the main content of the two preceding chapters was to find conditionswhich guarantee the existence of such a w.7.1 Convergence to the fixed pointWe define

r _ = IIA_II + Lip(f_).Proposition 7.1.1 Assume that r _ < 1. Then for any z in the graph of w,

Proof. Because of the way we chose the norm on E_ E 9 E+, and becauseIlw(x)11 ~ Lip(w)llxll ~ Ilxll,

we haveII(x,w(x))11 = Ilxll

The x component of I (x ,w(x) ) isA _x + I - (x ,w(x) ) ,

so-by the invariance of the graph of w-for any z = ( x, w ( x) ) on the graph ofw

11/(z)11 IIA_x + 1-(x,w(x) )11< IIA_II + Lip(f-)llxllr_llzll

The assertion of the proposition follows at once by iterating this estimate. DWe can sharpen this result a little.

Proposition 7.1.2 Under the same hypotheses, for any z in the graph of w,. 1lim sup -log(llr(z)ll) ~ logp_.n--+oo n


37/74

7.2. EXPANDING WEDGES 69Proof. The preceding proposition says

. 1hmsup-Iog(llr(z)ll) ~logr_.n--+oo nOnce we know that the orbit of z converges to 0, we know that, for any E > 0,the orbit is eventually in a ball where Lip(f_) < E. Thus,

. 1lim sup -log(llr(z)ll) ~ log IIA_IIn--+oo nThe left-hand side of this inequality is unchanged if the norm is replaced byany equivalent norm, so we can minimize the right-hand side over all norms;this gives the desired estimate. D

7.2 Expanding wedgesThe second part of the argument is to see what happens to orbits which arenot on the invariant manifold. The situation here depends on the size of p.i.:

- If p+ > 1, then orbits not on the invariant manifold cannot stay near thefixed point for all time.

- If p+ ~ 1, orbits not on the invariant manifold may nevertheless remainnear the fixed point for all time and even converge to it, but they convergeless rapidly than orbits on the invariant manifold.

To prove these assertions, we use a very simple version of a set of ideasidea which have extensive ramifications in the analysis of the very importantphenomenon of exponential separation of orbits. As a first introduction to theidea, we consider for concreteness the standard hyperbolic case, and we assumethat the norms are chosen so that

The first inequality means that As is contractive; the second that A u is expan-sive, as

IIAuYl12 (IIA;;-lll)-1Ilyll for all y.Consider a pair of points Zl = (Xl, yd and Z 2 = ( X 2 ' Y 2 ) , and let z~ = (x~,yDdenote !(Zi). If the non-linear terms were not present at all, and assumingthat Yl f : : - Y 2 , we would have


38/74

70 CHAPTER 7 . STABLE MANIFOLDS: DYNAMICSIterating, we find that the separation between the orbits of Zl and Z2 growsexponentially.

The non-linear terms will generally spoil the simplicity of the above argu-ment. For example, it can easily happen, even with very small non-linear terms,that YI f : : - Y2 but y~ = y~. This can however only happen if Xl -X2 is much largerthan YI - Y2 As we shall shortly see, if-for example-IIYI - Y211> IlxI - x211,it then follows that IIY~- Y~II> IIYI- Y211,provided that the non-linear termsare small enough. Not only that, but it also follows that IIY~-Y~II > Ilx~-llx~11so that the argument can be iterated. The result is that the orbits of Zl andZ2 separate exponentially as long as they both remain near enough to the fixedpoint so that the non-linear terms remain small enough. Hence, they cannotboth remain near the fixed point forever.

We now proceed to formalize this idea. It is useful to do this in the generalcontext rather than just the original one of a hyperbolic fixed point. We let

If the Lip(J) are small enough, then r+ > r _; in the standard hyperbolicsituation we can similarly arrange r+ > 1> r _, etc. We also say that theseparation Z2 - Zl between the two points Zl, Z2 is predominantly expansive ifIIY2- YIII > IIx2 - xliiLemma 7.2.1 If r+ > r _, and if the separation between Zl and Z2 is predom-inantly expansive, then

- the separation between !(zd and !(Z2) is predominantly expansive.If, conversely, the separation between !(zd and !(Z2) is not predominantlyexpansive, then

- the separation between Zl and Z2 is not predominantly expansive

Proof. Write, as above, z~ = ( x~,YD for !(Zi), and assume first that theseparation between Zl and Z2 is predominantly expansive. Then, because wedefined

II(x,y)11 = max{llxll, Ilyll},we have


39/74

7.2. EXPANDING WEDGES 71Then

IIY~- Y~II IIA_(Y2 - yd + f+ ( Z 2 ) - f+(zdll> IIA_(Y2 - Ydll- Lip(f+llz2 - zlll> ((IIA:;:III)-1 - Lip(f+) IIz2- zlll

r+llz2 - zlll,i.e.,

Similarly-but without assuming the the separation between Zl and Z 2 is pre-dominantly expanding-we have

Ilx~ - x~11 < IIA_llllx2 - xIII + Lip(f-)llz2 - zlll< (IIA_II + Lip(f_)) IIz2 - zlllr-llz2 - zlll,

i.e.From (* ) it follows that

Ilz~- z~112 r+llz2 - zlllFrom (* ) and (t) together, and using also r., > r: , we see that IIY~- Y~II>Ilx~ - x~ll, i.e., that the separation between z~ and z~ is also predominantlyexpansive.

It remains to deal with the assertions for the case where the separationbetween f(zd and f(Z2) is not predominantly expansive. From the precedingresults, it follows at once that the separation between Zl and Z 2 cannot be pre-dominantly expansive. Furthermore, from the definition of "not predominantlyexpansive" , Ilz~ - z~II= Ilx~ - x~ II ,so the last assertion follows from (j). DProposition 7.2.2 Assume that r+ > r_, and let z in the unit ball of E butnot on the graph of w. Then

1. ifr+ > 1, there is an n such that r(z) is not in the unit ball of E ,2. ifr+ ~ 1, then either there is an n such that r(z) is not in the unit ballof E, or 1lim inf - log IIr (z ) II2 log r -t-"n--+oo n


40/74

72 CHAPTER 7 . STABLE MANIFOLDS: DYNAMICSProof. Write z = (x , y), and let z = (x , w (x) ) . Then, manifestly, the separa-tion between z and z is predominantly expansive. By the preceding proposition,the separation between jn(z) and jn(z) remains predominantly expansive, and

Now assume that r., > 1, so that the separation between r(z) and r(z)grows exponentially as long as both remain in B. If jn(z) remains in the unitball-as will be the case if r _ < I-the exponential growth of the separationimplies immediately that r(z) cannot remain in the unit ball for all n.We can avoid the assumption that r C i) remains in the unit ball as follows: Assumethat r C z ) remains in the unit ball for all n. Write C X n , Y n ) for r C z ) and C X n , Y n ) forr C i ) Using the invariance of the graph of w and Lipf w ) ~ 1, we get

C : j : )On the other hand, the proof of?? shows that

I lx n - x n l l ~ a l lY n - Y n l l where a:= r_/r+ < 1.Inserting the assumption that I IC x n , Y n )1 1 ~ 1for all n and using C : j : )give the bound

l lY n I I ~ a l lY n I I + 2,which implies a bound on I I Y n l l , which in turn-using C : j : )again-implies a bound onI l r C i ) l l , which contradicts the assumed boundedness of r C z ) and the exponentialseparation of r C i ) from r C z ) .Thus, the proof in the case r+ > 1 is complete. If r+ ~ 1, and if the orbit of zremains in the unit ball for all n, then

whereas, by Proposition ??,

Since r _ < r+,

DAs in the preceding section, we can sharpen the above: If z is not in the

graph of w, and if r(z) remains in the unit ball for all n, then1liminf -log I l r (z)11 210gp+.n--+oo n


41/74

7.3. REFORMULATION OF RESULTS 73We omit the proof. As a consequence: If p : < 1, then the graph of w consistsexactly of all those z in the unit ball whose orbits remain in the unit ball forall n and for which . 1lim sup -log Ilr(z)11 ~ logp_;n--+oo nwe can replace the limit superior on the left by limit inferior, and we canreplace p : by any number between p : and p+ . Anyone of these variantson the characterization of the points of w gives a very satisfactory un iquenessassertion for w .

7.3 Reformulation of resultsWe now want to re-express the main results obtained so far in a way whichis less closely tied to specific choices of coordinates, norms, and so on. Weconsider a mapping t,of class C r with r 2 1, defined on an open set U in aBanach space E , and we let Zo denote a fixed point of f.We let p be a positivereal number such that the spectrum of D f (zo ) does not intersect the circle{A . : IAI= p} but such that it does intersect both the inside and the outside ofthis circle. We define

p : SUp{IAI: AE a(Df(zo) ) , IAI< p}p+ inf{IAI : A E a(Df(zo) ) , IAI> pl

In the usual way, the splitting of the spectrum of Df (zo) into the parts insideand outside the circle {IAI= p} gives a direct sum splitting of E as E_ E9E+.We will correspondingly represent Zo as (xo , Y o) . We will use the phrase localCr man ifo ld tan gen t to E_ at Zo to refer to a set which can be represented asthe graph of a C r function w, defined on an open neighborhood Vw of Xo inE_, taking values in E+, with w(xo) = Y o and Dw(xo ) = o . We will also usethe following terminology: A set X is said to be locally in varian t for f at Zo ifthere is a neighborhood W of Zo such that

z E X nW implies f(z) EX.i.e., such that f(X nW) c X , and we say that two sets Xl and X 2 coinciden ear Zo if there is a neighborhood W of Zo such that


42/74

74 CHAPTER 7. STABLE MANIFOLDS: DYNAMICSTheorem 7.3.1 (Generalized stable manifold theorem) Let the notationbe as above. If p : 2 1assume

i. r < 00ii. (p-t < p+iii. There is a real-valued Cr function ' I j ; on E_ identically equal to one on a

neighborhood of 0 and vanishing outside the unit ball.Then:1. There exists a cr manifold Wl~-j tangent at Zo to E_ and locally invariantthere for f2. If, further, p : < 1, then:

_ Wl~-j can be taken-by judicious choice of Vw-to be invariant in theliteral sense, i.e., mapped into itself by f. (Such a choice is however notassumed in the following assertions.)

_ Wl~-j is "locally unique" in the sense that any two locally invariant C1manifolds tangent to E_ at Zo coincide near Zo .

_ if W is any sufficiently small open neighborhood of zo , then Wl~-j coin-cides near Zo with the set of all z E W whose forward orbits remain inW for all n and converge to Zo sufficiently rapidly that

. 1lim sup -log(llr(z) - zoll) ~ log(p_).n3. If p+ > 1 (whether or not p : < 1), we have the following partial uniquenessresult: If Wl~-j is any C1 locally invariant manifold tangent to E_ at zo , thereis an open neighborhood W of Zo such that any z such that r(z) E W for alln = 0,1, ... must be in Wl~-j.4 . If p: < 1< p+ (hyperbolic case), then for any sufficiently small neighbor-hood W of zo , Wl~-j coincides near Zo with the set of z such that r(z) is inW for all n = 0,1, ....


43/74

Chapter 8Generalized unstablemanifolds8.1 IntroductionGiven a splitting E _ E9E + as in the preceding sections, we want to investigateinvariant manifolds tangent to E + . We will call these ge ne ra lize d u n sta ble m an -i folds. The most important case is that where p : ~ 1 < p+ ; a correspondinginvariant manifolds are called an uns table manifold. Similarly, if p : > 1, wespeak of a s tr on g u n sta ble manifold, and, if p+ = 1, of a center-uns table mani-fold. If f is invertible, we can deduce the existence and properties of generalizedunstable manifolds by applying our results about stable manifolds to f-1. Itturns out, however, that a theory of unstable manifolds completely parallel tothat for stable manifolds can be developed w itho ut assu min g the in vertibility o ff. This is a useful thing to do, since in applications to partial differential equa-tions one expects that derivatives of the mappings encountered will often becompact operators and hence not invertible but with finite-dimensional E + 's, sorestricting to a generalized unstable manifold may give a means of extractingthe essential finite-dimensional part from such infinite-dimensional dynamicalsystems.

As before, we want to study mappings of the formf (x, y ) = (A_x + f _ (x, y ) , A + y + f+ (x, y )) ,

with f small in Lipschitz norm and vanishing at the origin, and we seekinvariant sets which are graphs of mappings w from E + to E _ with Lipschitznorm no greater than one.

75


44/74

76 CHAPTER 8. GENERALIZED UNSTABLE MANIFOLDSThe first question we have to address is: In what sense do we want to

require that a small piece of manifold W be invariant? There are two plausiblecandidates:

fW:J Wand f-1W C W,which are not equivalent in the non-invertible case. For orientation, considerthe case of a linear f with non-trivial null space. In this case, it is clear thatf E+ :J E+, whereas any W containing 0 and invariant in the second sense mustcontain the null space of f. The lesson we draw from this is that we shouldlook for a function w whose graph W satisfies

fW:JWThe general plan is now much as before: We are looking for a function

whose graph is invariant in this sense. We convert the condition of invarianceof the graph of w to a functional equation of the form

w = Fui.We then prove that F is contractive in an appropriate functions space (providedthat Lip(J) are small enough.) With the unique candidate for an invariantmanifold thus in hand, we proceed to analyze its smoothness and-in nicecases-to give it a dynamical characterization. In spite of the similarity ofthe general plan to that used for investigating the generalized stable manifold,there are significant differences in detail, and, unfortunately, the analysis in thepresent case turns out to be noticeably more complicated than for generalizedstable manifolds.

8.2 The Lipschitz invariant manifoldOur first task is to convert invariance of the graph into a functional equationfor w. That is: We want to express in terms of w the condition that everypoint z = (w (y) , y) of the graph of w can be written as f (w(y), y ). Using thedetailed form for f we find the pair of conditions

(8.1)and

w(y ) = A_w(y ) + f -(w(Y), y ). (8.2)We will handle these equations by showing first that, if f_ is small enough,(8.1) can be solved uniquely for y . Let us accept for the moment that this can


45/74

8.2. THE LIPSCHITZ INVARIANT MANIFOLD 77be done, and let us denote the solution by

y = v(y , w ).In contrast to the analogous function encountered in the proof of the existenceof generalized stable manifolds, v here depends of the whole function w andnot just on its value at one point.

Inserting the solution into (8.2) gives a functional equation, in the form ofa fixed-point problem, for w:

w = Fui,where

(Fw)(y )=F(w(v(y ,w) ) , v (y ,w) ) with F(x,y)=A _x+ f_(x,y) . (8 .3 )Thus: Assuming solvability of (8.1) for all relevant y, the graph W of w satisfiesfW :J W if and only if w = Fur.

Before turning to the detailed estimates, let us anticipate the fact that, asfor generalized stable manifolds, we will need to carry out two parallel versionsof the argument. What distinguishes the good from not-so-good cases here iswhether or not p+ > 1. We will use the following notation: B_ will denote-inboth cases-the closed unit ball in E_. Then either

- Expansive case. p+ > 1, in which case we choose the norm so thatIlk;lll < 1, and we let B+ denote the closed unit ball in E + , or

- Non-expansive case. p+ ~ 1, in which case-from the general hypoth-esis p : < p+ -we have p : < 1, so we choose the norm so that IIA_II< 1,and we let B+ denote E+ .

In both cases, B will denote B_ x B+ . The functions f are defined andLipschitz continuous on B ; Lipschitz norms mean norms over B . We alwaysassume f (O) = 0 (and, when we assume differentiability, we also assumeDf (O ) = 0.) In the non-expansive case, we require that

J(x, y) = 0 for Ilyll> 1.It then follows that

11J(x,y)11~ Lip(J) for all relevant x, y.The functions w which we consider are mappings from B+ to the unit ball ofE_ with

Lip(w) ~ 1 andw(O) = O.


46/74

78 CHAPTER 8. GENERALIZED UNSTABLE MANIFOLDSIn the expansive case, the requirement that

Ilw(y)11~ 1 for all yfollows from the preceding assumptions; in the non-expansive case, it must beimposed as a separate condition, and we have to check that it is preserved bythe operator F. From the formula (8.3) for F, it follows that IIFw(y)11 ~ 1 forall y provided that Ilw(y)11~ 1for all y and

IIA_II + Lip(f_) ~ 1, (8 .4)and we accordingly add this estimate to our list of standing hypotheses (in thenon-expansive case.)

The solvability of (8.1) is an easy application of the contraction mappingprinciple: We rewrite the equation in question as

(8.5)i.e., as a fixed-point problem for the unknown y . The right-hand side of (8.5) isa Lipschitz-continuous function of y with Lipschitz constant not greater thanIIA+l IILip(f+) (provided that Lip(w) ~ 1.) Thus, if

(8.6)the right-hand side is contractive. To apply the contraction mapping principle,we need to specify a domain in which contractivity holds and which is mappedinto itself. In the expansive case, we take as domain the closed unit ball of E+ .A straightforward estimate shows that, if

(Ae)the right-hand side of (8.5) maps the closed unit ball into itself for all relevant yand w. By good fortune, the condition (Ae) already includes the contractivitycondition (8.6) so:If (Ae) holds, then, for any given wand y as above, there is exactly one y inthe unit ball such that (8.1) holds. As already indicated, we denote this y byv(y, w ).In the non-expansive case p+ ~ 1, we need only assume (8.6) (with Lipschitznorm now taken over all of B_ x E+, ) and we can then solve for arbitrary y ; thesolution, of course, need not lie in the unit ball. On the other hand, because


47/74

8.2. THE LIPSCHITZ INVARIANT MANIFOLD 79we want to restrict the x variable to the unit ball, we need to impose (8.4).Thus, our first quantitative condition in the non-expansive case is

IIA:;:lllLip(f_) < 1 and IIA_II + Lip(f_) ~ 1, (Ane)Conditions (Ae) respectively (Ane) suffice to guarantee the solvability of

(8.1) and hence the definition of v(y , w ) for y in the closed unit ball respectivelyall of E+ .

We next want to estimate the Lipschitz norm of v with respect to y. Bydefinition, the mapping y I-t v(y, w ) is the inverse of the mapping

Y f---t A+y + f+(w(y) ,y) ,so an estimate on the Lipschitz norm of v means a lower bound on the extentto which this latter mapping expands distances. Thus, let Yl, Y2 be two pointsof B+, i.e., either the closed unit ball in E+ (expansive case) or all of E+ (non-expansive case), and let Xl, X2 two points of the closed unit ball B_ of E_ suchthatThen

II{A+Yl + f+ (xl,yd} - {A+Y2 + f+(X2,Y2)}112 IIA+ (Yl - Y2)11- Lip(f+)IIYl - Y2112 (1IA:;:lll-l - Lip(f+)) IIYl - Y211

We now let L(v) be the smallest constant such that

whenever the Xi and Yi are as above; the computation we have just done showsthatL(v) < IIA:;:lll - IIA-111+ O(Li (I ) )- 1-IIA:;:11ILip(f+) - + p + .

The rule of thumb is thus that we can make L(v ) almost as small as 1/ p.i., bychoice of norm and smallness conditions on Lip(f+). In the interest of keepingthe formulas simple, it is useful to note that-by an easy but not-quite-obviouscalculation-our standing assumption (Ae) in the expansive case implies thatL(v) ~ 1.

It is evident from the definitions that, for any w with Lip(w) ~ 1,Lip(y I-t v(y,w )) ~ L(v).


48/74

80 CHAPTER 8. GENERALIZED UNSTABLE MANIFOLDSWe note for later use that, if the f are continuously differentiable and if A isa linear operator from E+ to E_ with IIAII~ 1, then

(8.7)this can be seen by applying the definition of L(v) with YI and Y2 very closetogether.

It is now an easy matter to estimate the Lipschitz norm of Fu) in terms ofL(v): The construction of Fw(y ) can be factorized as

Y I-t y:= v(y,w ) I-t z:= (w (y ),y) I-t F(z).The first factor has Lipschitz norm ~ L(v), and the second Lipschitz normunity. Hence

Lip(Fw) ~ L(v) Lip(F).We thus impose as a second quantitative condition

Lip(F) L(v) ~ 1, (B)and it then follows that Lip(Fw) ~ 1for all w.

From the definition (8.3) of F,Lip(F) ~ IIA_II + Lip[j",.},

so we can make Lip(F) almost as small as p_. Since, as already noted, wecan make L(v) almost as small as 1/p+, condition (B) can be satisfied byfirst choosing the norms properly and then taking Lip(f) small enough. It isstraightforward to check that F preserves the condition w(O) = o .

As in the investigation of generalized stable manifolds, we will prove con-tractivity of F with respect to the norm

Ilwll = sup Ilw(Y)II.y Ilyll

We consider a pair of mappings WI, W2, and we write(FWI - FW2) (y ) = F(WI(vd,vd - F(W2(vd,vd

+F(W2(vd,vd - F(W2(V 2),V2 ),where we have abbreviated v(y , Wi) by Vi, i= 1,2. Now

IIF(WI(vd,vd - F(W2(vd,vdll < LiPI(F)llwI - w21111vIII< LiPI (F)llwI - w21IL(v)llyll,


49/74

8.2. THE LIPSCHITZ INVARIANT MANIFOLD 81and

IIF(W2(vd,vd - F(W2(V2),V2)11 ~ Lip(F)llvI - v211To make effective use of this last bound, we need to estimate IlvI -v211 carefully.Recalling the definition of v(y, w ) as the solution of (8.1), we write

or {A+VI + f+(w I(vd,vd} - {A+V2 + f+(WI(V2),V2)}= f+ (WI(V 2),V 2) - f+ (W2(V 2),V 2).Taking norms and using the definition of L(v) gives

< II{A+VI + f+(w I(vd,vd} - {A+V2 + f+(WI(V2),V2)} II11f+(WI(V2),V2) - f+(W2(V2),V2)11

< LiPI(f+) IlwI(V2) - W2(V2)11< LiPI(f+) IlwI - w21111v211< LiPI(f+)llwI-W21IL(v)llyll,(where, to get the last expression, we used the estimate

Lip(y I-t V(y,W2)) ~ L(v).)Simplifying the above:

Combining the above estimates givesII (:FWI - :FW2) (y)11 ~ IlwI - w21111yllx

x (LiPI (F)L(v) + LiPI (f+)Lip(F)(L(v))2)Thus, if we impose

(LiPI (F) + LiPI (f+)Lip(F)L(v))L(v) < 1, (C )it follows that :F is contractive and hence that it has exactly one fixed pointin the space of w's under consideration. Since we can make LiPI (F) almost assmall as p :., L(v) almost as small as l/p+, and LiPI(f+) as small as we like,condition (C) can be satisfied.

Summarizing what we have shown so far:


50/74

82 CHAPTER 8. GENERALIZED UNSTABLE MANIFOLDSI f (A e) (in the expan s ive case) o r (A ne) (in the n on -expan sive case) , (B ) , an d(O) ho ld, the n there is a u n iqu e w : B _ -+ B+ w ith Lip(w ) ~ 1 an d w (O ) = 0,w ho se graph W satisf ies fW :J W.From now on,- We will assume (Ae) respectively (Ane), (B), and (C).- w will denote the unique fixed point of F.

8.3 DifferentiabilityWe now assume that the f are of class C1 with derivatives vanishing at 0,and we want to show that w is also of class C1with derivative vanishing at O.Again, this goes roughly as before: We first find a functional equation whichmust be satisfied by the derivative of w if this derivative exists; we then showthat this equation has a unique solution (subject to some bounds); and we thenshow that the unique solution is indeed the derivative of w.If w is differentiable than so-by the inverse function theorem-is v (as afunction of y, with w fixed), and

(Dv ) ( y ) = [A + + D d+ (w (v ) ,v )D w (v ) +D 2 f+(w(v ) ,V ) ] - 1 ;then differentiating the equation for w gives

Dw ( y ) = [ D 1FDw ( v ) + D 2 F] D v (y ),where suppressed arguments are now to be understood to be (w(v(y) ) , v (y) ) ,and where we are now regarding v as a function of the single variable y. Hence,if w is differentiable, then, writing a for D w , we have

We will denote the right-hand side of this equation as (J(a)(x), and we wantto show that J ( is a contraction on the space of continuous mappings y I-t a(y)

- from B+ to the closed unit ball in (E + , E _ )- with a(O) = 0,

equipped with the supremum norm. From D1F(0 ,0 ) = 0 and v (O ) 0 itfollows that J( preserves the condition a(O) = O. From (8.7),


51/74

8 .3 . D I FF ER E NT IA B ILI TY 83Also, by the argument in the fine-print section of 6.1,

Hence K preserves the condition I la ( y )1 1 ~ 1 for all y providedLip(F)Lip(v) ~ 1,

and this is condition (B) again.To find conditions which make K contractive, we choose al and a2 and

write(K ad(y ) - (K a2)(Y ) = D IF [al (v ) - a2(v)] [A + + DIl+al (v ) + D21+]-1+ [D IF + D2Fa2(V)] [A + + Dd+al(v ) + D 21+]-1 Dd+ x

x [a2(v) - al(v) ] [A + + DI l+al(v ) + D21+]-I,where we have used the standard operator identity

tr:' - V-I = U-I (V - U)V-I.Using the bound (* ) repeatedly, we get

which shows that K is contractive provided that condition (C) holds.Once again we let a denote the unique fixed point for K ; we write

M - 1 I lw ( y + 8 y) - w (y) - a(y)8 Y II .- sup im sup I I ' I I 'y oy--+O u ywe observe that M ~ 2; and we want to use the functional equation to showthat M ~ ",M for some", < 1. We will use the abbreviations

8 v :=v(y + 8 y , w ) - v (y , w ) ;8 w :=w (v(y , w ) + 8 v) - w (v(y , w ) ) .

(The latter, while a little unnatural, is convenient because w (v(y , w )) occursfrequently on the right-hand sides of the functional equations whereas w(y )does not.) Note also that suppressed arguments in E+ are to be taken to bev(y ,w) and in E_ are to be taken to be w(v(y ,w) ) .

Writing the equations satisfied by v(y , w ) and by v( y + 8 y, w ), subtracting,and using the differentiability of 1+ , gives

8 y A +8 v + DI l+8w + D2 1+8 v + o(8y )[A + + DI l+a + D2 1+ ] 8 v + D d+ [8 w - a8 v] + o(8y ) .


52/74

84 CHAPTER 8. GENERALIZED UNSTABLE MANIFOLDSSimilarly, writing the functional equation for w and using the differentiabilityof the f , we get

w (y + 8 y) - w (y ) = D 1F8 w + D 2F8 v + o(8y)= [D 1Fa + D2F ] 8 v + D 1F [8w - a8 v] + o(8y)= [D 1Fa + D2F ] [A + + D1 f+a + D 2f+ ] -1 8 y + D 1F [8 w - a8 v]- [D 1Fa + D2F ] [A + + Dd+a + D 2 f+ ]-1 D d+ [8w - a8 v] + o(8y) .

The first term on the right of this last equation can be recognized as a(y)8y ,so we get

w (y + 8 y) - w (y ) - a(y )8 y =(D 1F - (D 1Fa + D2F ) (A + + D d+ a + D 2f+ )-1 D 1 f+ ) (8 w - a8 v)+o(8y) .

Taking norms, dividing by 1 1 8 y l l , letting 8 y tend to zero, and using

we get

1 . I l w ( y + 8 y) - w (y ) - a(y )8 Y II1r::~~P 1 1 8 Y I I~ (LiP1(F) + LiP(F)LiP(v)LiP1(f+)) Lip(v )M.

The factor multiplying M on the right is < 1by (C), so M = 0, so w isdifferentiable with derivative a.

The analysis of Holder continuity of the first derivative and of higher dif-ferentiability goes in almost exactly the same way as for generalized stablemanifolds. We can write the equation satisfied by a = Dw ( y ) in the form

a(y) = G( y, a (v (y ) )) ,where

G (y , Sf) = (D1F + D 2Fs') (A + + D1 f+S ' + D 2f+ ) -1 .Lemmas 6.2.2, 6.3.2, and 6.3.3 apply here-with a G depending only on two ofthe original three variables-so the analysis given for the stable manifold canbe applied directly. The outcome can be summed up as follows:


53/74

8.4. EXPANDING AND CONTRACTING WEDGES 85

Proposition 8.3.1 Let the J be of class C r, 1~r ~ 00, with D J(O) = o .Assume (A), (B), (C) and also, if Lip(v) > 1,

(LiPI (F) + Lip(F)LipI (f+)Lip(v)) Lip(vr < 1.(from which it follows that r < 00.) Then F has a unique fixed point w withw(O) = 0 and Lip(w) ~ 1. This w is of class C r, and its first derivative vanishesat the origin.

8.4 Expanding and contracting wedgesWe now want to apply the expanding and contracting wedges idea of 7.2.We have defined the separation between two points Zi = (Xi, Y i) to be predomi-nantly expansive if I l x I - x 2 1 1 < I I Y I - Y 2 1 1 . If the separation is not predominantlyexpansive, we will say that it is predominantly contractive (in spite of the factthat the strict inequality in the definition of "predominantly expansive" makesthe two definitions slightly unsymmetric.) Also, by a backward orbit for a pointZ under a (not necessarily invertible) mapping t,we will mean a sequence Z-l,Z-2, ... ,Z-n with f(z-d = Z and f(Z-j) = Z-j+1 for j= 2,3, ... ,n. Since weare not assuming invertibility, there may be no backward orbit at all, or theremay be more than one.Proposition 8.4.1 Let the hypotheses and notation be as in 7.2, but assumein addition that- r., > 1- There is a function w from the unit ball in E+ into E_ with w(O) = 0 and

Lip( w) ~ 1whose graph is mapped onto itself by f.- f are continuously differentiable with derivatives vanishing at (0,0).

Then any Z in the graph of w admits a unique infinite backward orbit (z-n) inthe graph of w. This backward orbit converges to (0,0) fast enough so that

. 1lim sup -logllznll ~ -log(p+)n n

Conversely, a point Z in the unit ball but not in the graph of w does notadmit an infinite backward orbit in the unit ball such that

. 1lim sup - log I lz n l l ~ -log(r .,').n n


54/74

86 CHAPTER 8. GENERALIZED UNSTABLE MANIFOLDSWe now drop the assumption that r+ > 1and assume instead that

r _ < 1.Then if z is a point of the unit ball such that the Jj(z) up to j= n are all inthe unit ball, the distance from r (z ) to the graph of w is not greater than 2r~.From the last assertion it follows in particular that any orbit which stays inthe unit ball for all n converges to the graph of w, and any orbit which startsvery near to fixed point-and so stays in the unit ball for a long time-butwhich nevertheless eventually escapes from the unit ball must escape "along"the graph of w.Proof. Under our hypotheses, J is injective on the graph W ofwand JW :J W.Hence, there is a uniquely determined sequence of successive preimages Z-n inW, and these are easily seen to converge to the origin at least as fast as r+ n .Improving this rate of convergence to the one indicated is done by the samearguments as those used to prove proposition 7.1.2.

Suppose that z is not in the graph of w but that it admits successive preim-ages Z-l, Z-2, ... ,Ln, all in the unit ball. Write z = (x,y); put Zl = (w(y) ,y) ;and write Z'_j for the successive preimages of z' in the graph. By construction,the separation between z and z' is predominantly contractive, so - applyingrepeatedly the last assertion of lemma 7.2.1 - the separations between all theZj and z j are predominantly contractive and

Since the left-hand side of this inequality is non-zero and independent of n,I lz n - z ~ 1 1 cannot go to zero faster than (r _)-n. But we already know thatI Iz~ I I goes to zero at least as fast as (r +)-n, so I IZnI I cannot go to zero fasterthan (r _)-n.

We now turn to the last assertion. If In(z) is in the graph of w, there isnothing to prove. Otherwise, the argument of the preceding paragraph withz of that paragraph replaced by r(z) shows that there is a zIon the graphof w whose separation from z is predominantly contractive and such that thedistance from r(z) to the graph of w is not greater than (r _ ) n llz - z/ll. Sincez' = ( z " , y l) is on the graph of w, I l x / l l ~ 1, and since z is in the unit ball a

Dynamical Systems

Documents

Transcript of Dynamical Systems