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J. Differential Equations 245 (2008) 692721www.elsevier.com/locate/jde

Transversality in scalar reactiondiffusion equationson a circle

Radoslaw Czaja a,b, Carlos Rocha b,

a Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland b Centro de Anlise Matemtica, Geometria e Sistemas Dinmicos, Departamento de Matemtica,

Instituto Superior Tcnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Received 1 August 2007; revised 17 January 2008

Available online 6 March 2008

Abstract

We prove that stable and unstable manifolds of hyperbolic periodic orbits for general scalar reaction

diffusion equations on a circle always intersect transversally. The argument also shows that for a periodicorbit there are no homoclinic connections. The main tool used in the proofs is Matanos zero number theorydealing with the Sturm nodal properties of the solutions. 2008 Elsevier Inc. All rights reserved.

MSC: primary 35B10; secondary 35B40, 35K57

Keywords: Periodic orbit; Heteroclinic orbit; Transversality; Global attractor; Zero number

1. Introduction

We consider the scalar reactiondiffusion equation of the form

u t = u xx + f(x,u,u x ) (1.1)

for one real variable u = u(t, x) on a circle x S 1 = R / 2 Z . In other words, we consider (1.1)together with periodic boundary conditions

Partially supported by the Fundao para a Cincia e a Tecnologia through the Program POCI 2010/FEDER and bythe Project PDCT/MAT/56476/2004.

* Corresponding author. E-mail addresses: [email protected] (R. Czaja), [email protected] (C. Rocha).

0022-0396/$ see front matter 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.jde.2008.01.018

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R. Czaja, C. Rocha / J. Differential Equations 245 (2008) 692721 693

u(t, 0) = u(t, 2 ), u x (t , 0) = u x (t , 2 )

and discuss (1.1) with initial condition

u( 0, x) = u 0(x), x S 1 . (1.2)

Below we use suitable assumptions on f so that the problem (1.1), (1.2) denes a global semiowin X = H 2 (S 1) , 34 < < 1, for which there exists a global attractor, i.e. a nonempty compactinvariant set attracting every bounded subset of X . The existence of global attractors and otherqualitative properties of the dynamical systems generated by reactiondiffusion equations undervarious boundary conditions have been extensively considered in the literature. For the interestedreader we mention the following excellent surveys [12,27,28].

It has been shown in [3,20,21] that time periodic solutions may appear in the description

of dynamics of (1.1). In case the function f does not explicitly depend on the x variable, i.e.f = f(u,u x ) , it was proved (see [3,9] for details) that the global attractor consists exclusively of equilibria, orbits of periodic solutions of the form

u(t,x) = v(x ct ), t R , x S 1 with some c = 0 (1.3)

(called rotating waves ) and heteroclinic orbits connecting the above-mentioned critical elements,when all are assumed hyperbolic. Moreover, necessary and sufcient conditions for the existenceof heteroclinic orbits between critical elements were established in [9]. However, as it followsfrom [31], in case of general x-dependent nonlinearities homoclinic orbits may belong to the

attractor as well and the periodic solutions do not have to be, in general, of the form (1.3). Thishappens due to the lack of S 1-equivariance, which was a crucial property used in [9] to excludehomoclinic connections.

One of the most important results concerning (1.1) is the PoincarBendixson type theoremproved by Fiedler and Mallet-Paret in [8, Theorem 1] (see also [22]). It states that if u0 X ,34 < < 1, then either its -limit set (u 0) consists in precisely one periodic orbit or (v 0) and (v 0) are subsets of the set of all equilibria for any v0 (u 0) .

In this paper we investigate closely the situation when a bounded orbit from the global at-tractor connects two hyperbolic periodic orbits. First, we exclude the existence of a homoclinicconnection for a hyperbolic periodic orbit (cf. [23]) in order to nally prove the main result of thispaper stating that the intersection of the global unstable manifold of a hyperbolic periodic orbit with the local stable manifold of another hyperbolic periodic orbit + is always transversal,i.e.

W u ( ) W sloc ( + ).

The paper is organized as follows. In Section 2 we formulate the abstract Cauchy problemfor (1.1)(1.2) and using the theory from [14] we solve the problem locally. Further we obtaina priori and subordination estimates, which ensure that the solutions exist globally in time. Thesemiow of global solutions constructed in this way is point dissipative and compact, thus has acompact global attractor. In Section 3 we examine the properties of the semiow and the evolu-tion system for the linearization around a given solution. Moreover, we recall the properties of the zero number of solutions of linear parabolic equations. Section 4 is devoted to the operatorcalled a period map for a periodic orbit. We describe its spectrum and decompose the phase space

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694 R. Czaja, C. Rocha / J. Differential Equations 245 (2008) 692721

according to the spectrum. We also recall the notions of local stable and unstable manifolds of ahyperbolic periodic orbit and list their properties. In Section 5 we analyze the local stable mani-fold of a hyperbolic periodic orbit and show that for any u 0 / from the local stable manifold

of there exists a such that u(t ; u 0) p(t ; a) tends exponentially to 0 as t and

z(u 0 a) i( ) + 1 + 1 + ( 1) i( )

2 , (1.4)

where i ( ) denotes the total algebraic multiplicity of eigenvalues of the period map for out-side the closed unit ball. Similarly, in Section 6 we investigate the global unstable manifold of a hyperbolic periodic orbit . We prove that for any u 0 / from the global unstable manifoldthere exists a such that u(t ; u 0) p(t ; a) tends exponentially to 0 as t and

z(u 0 a) i( ) 1 + 1 + ( 1) i( )

2 . (1.5)

In Section 7 we combine the estimates (1.4) and (1.5) and nd, in particular, that there is nohomoclinic connection for a hyperbolic periodic orbit. Finally, in Section 8 we follow the ideasfrom [6] and introduce ltrations of the phase space with respect to the asymptotic behavior of solutions for the linearized equation around an orbit connecting two hyperbolic periodic orbits.A proper choice of the spaces from the ltrations carefully combined with the corresponding zeronumber estimates for the functions from these spaces yields the transversality of the intersectionof the stable and unstable manifolds of two hyperbolic periodic orbits. The transversal intersec-tion of invariant manifolds of critical elements is one of the ingredients for genericity results (cf.e.g. KupkaSmale theorem) or structural stability theorems (cf. [13, Chapter 10], [24]) in thetheory of dynamical systems. In Section 9 we make some concluding remarks about structuralstability for the semiow generated by (1.1).

Under different boundary conditions many authors have considered problems of the same typeas discussed here. For separated boundary conditions, the results of Henry [15] and Angenent[1] on the transversality of the stable and unstable manifolds of stationary solutions constituteobligatory references. A problem of this type has also been considered by Chen, Chen and Halein [6] for nonautonomous time periodic equations with f = f( t ,x,u) under Dirichlet boundaryconditions. The effect of radial symmetry on the transversality of stable and unstable manifolds of

equilibria for problems dened on symmetric domains inR n

has been studied by Pol cik in [26].For special classes of ordinary differential equations on R n , Fusco and Oliva have considered thetransversality between stable and unstable manifolds of equilibria and periodic orbits (see [10,11]). Here we extend the results of [11] realizing the plans sketched by these authors for furtherpossible extensions.

2. Abstract setting of the problem and existence of the global attractor

Assume that f : S 1 R R R is a C 2 function satisfying the following conditions

there exist 0 < 2 and a continuous function k : [0, ) [0, ) such thatf(x,y,z) k(r) 1 + | z | , (x, y, z) S 1 [ r, r ] R for each r > 0, (2.1)

yf(x,y, 0) < 0, (x, y) S 1 R , |y | K for some K > 0. (2.2)

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In this paper we are going to use fractional Sobolev spaces of 2 -periodic functions H s (S 1) ,s > 0, and their properties (cf. [29, Appendix A]). Among others we will frequently use theSobolev embedding

H s S 1 C 1 S 1 if s > 32

.

We consider the operator A : L 2(S 1) H 2(S 1) L 2(S 1) given by

Au = u xx + u, u H 2 S 1 .

Since A is a positive denite self-adjoint operator, it is a positive sectorial operator. Henceforthwe consider fractional power spaces

X

= D A

, 0,

with norms uX = A uL 2(S 1 ) , u X (cf. [14, Section 1.4]). Note that X0 = L 2(S 1) ,

X 1 = H 2(S 1) and

X = L 2 S 1 , H 2 S 1 = H 2 S 1 , (0, 1)

(see [35, Section 1.18.10] and [32, Section 3.6.1]). Since H 2(S 1) is compactly embedded inL 2(S 1) , it follows that A has a compact resolvent.

We rewrite (1.1), (1.2) as an abstract Cauchy problem in X 0

u t + Au = F(u),u( 0) = u 0 ,

(2.3)

where F is the Nemycki operator corresponding to

F(u)(x) = f x,u(x),u x (x) + u(x), x S 1 .

For a xed ( 34 , 1) , F takes X into X 0 and is Lipschitz continuous on bounded subsets of X .

By the theory presented in [14] it follows that for each u0 X there exists a unique local

forward X

solution dened on a maximal interval of existence, i.e.u C [0, u0 ), X

C 1 (0, u0 ), X0 C (0, u0 ), X

1

and satises (2.3) on [0, u0 ) in X0 . Moreover, either u0 = or

u0 < and lim supt u0

u(t ; u 0) X = .

Using assumption (2.2) and the maximum principle it follows that if for some R 0 we haveu 0L (S 1 ) K + R , then there exists a positive constant = (K,R) such that

u(t ; u 0) L (S 1 ) K + Re t , t [0, u0 ). (2.4)

This implies that each forward X solution is bounded in L (S 1) .

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Therefore the semiow {S(t) : t 0} is point dissipative in X . Note also that S(t) is a compactmap on X for each t > 0 by [7, Theorem 3.3.1], since A has a compact resolvent. Thus thesemiow {S(t) : t 0} has a global attractor A in X . We recall that A is then the union of all

bounded orbits.3. Properties of the semiow

Fix s R . Let u( ; s, ) be the global forward X solution of the problem

u t + Au = F (u), t > s,u(s) = .

(3.1)

Since f is C 2 , it follows from [14, Theorem 3.4.4, Corollary 3.4.6] that the function

(s, ) X (t, ) u(t ; s, ) X

is continuously differentiable. Moreover, for each xed t s the function

X u(t ; s, ) X

is also continuously differentiable and, for each X , its derivative in the -direction given by

w(t ; s, ) = D u(t ; s, ) X , t s,

is a unique global forward X solution of the linear variational problem

w t + Aw = D u F u(t ; s, ) w, t > s,w(s) = .

(3.2)

Taking into account the regularity of X solutions we see that (3.2) is the abstract equivalent of

w t = wxx + b(t,x)w + d(t,x)w x , t > s, x S 1 ,w(s,x) = (x), x S 1 ,

(3.3)

where

b(t,x) = f y x,u(t ; s, )(x),u x (t ; s, )(x) , t > s, x S 1 ,

d(t,x) = f z x,u(t ; s, )(x),u x (t ; s, )(x) , t > s, x S 1 .

We dene the evolution system

T(t ,s ) = w(t ; s, ), t s, X , (3.4)

where w(t ; s, ) is a unique global forward X solution of (3.2). Note that we have T (t , s ) =D u(t ; s, ) , so it follows that T(t , 0) = (D u0 S(t)u 0) . Moreover, for t > s the operatorT(t ,s ) L (X , X ) is compact in the Hilbert space X (see [14, Section 7.1]).

Below we prove the injectivity of the semiow {S(t) : t 0} and the injectivity of {T(t ,s ) : t s}.

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To show that the semiow is injective in X suppose that for some u1 , u 2 X and somet 0 > 0 we have

S(t 0)u

1 = S(t

0)u

2.

Dene v(t) = S(t)u 1 S(t)u 2 , t 0. Then we have

vt + Av = F S(t)u 1 F S(t)u 2 , t > 0,v( 0) = u 1 u 2 .

(3.5)

Moreover, we know that v(t 0) = 0. Note that A is a positive denite self-adjoint operator in theHilbert space X 0 = L 2(S 1) and X X

12 . Furthermore,

v C [0, ), X C 1 (0, ), X 0 C (0, ), X 1

and

F S(t)u 1 F S(t)u 2 X 0 L S(t)u 1 S(t)u 2 X12

= L v(t)X

12

, t [0, ),

where L is a constant depending on

supt [0, )

S(t)u i X < , i = 1, 2.

By [7, Proposition 7.1.1] (see also [34, Lemmas 6.1, 6.2]) we get

v(t) = 0, t [0, t 0].

In particular, we obtain u1 = u 2 . This proves the injectivity of the semiow.Suppose now that

T (t 0 , s 0) = 0

for some t 0 > s 0 and X . Dene w(t) = T (t + s0 , s 0) , t 0, and choose any T 0 > t 0 s0 .

Then we havew t + Aw = D u F u(t + s0; s0 , ) w, 0 < t T 0 ,w( 0) = .

(3.6)

Moreover, we know that w(t 0 s0) = 0. For t [0, T 0] we estimate

D u F u(t + s0; s0 , ) w(t) X 0 C1 w(t) X 0 + C2 w(t) X12

M w(t)X

12

,

where C1 and C2 depend on

sup(t,x) [0,T 0] S 1

f y x,u(t + s0; s0 , )(x),u x (t + s0; s0 , )(x) ,

sup(t,x) [0,T 0] S 1

f z x,u(t + s0; s0 , )(x),u x (t + s0; s0 , )(x) ,

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4. The period map

Consider a periodic orbit with period > 0 and choose a periodic point a . Thus

= p(t) : t [0, ) ,

where p : R X 1 is a periodic solution of (2.3) with p( 0) = a . We consider the linear varia-tional problem (3.2) around p and the corresponding evolution operators T (t , s ) , t s . In par-ticular, the operator T = T ( , 0) = D u0 S( )a is called a period map (cf. [14, Denition 7.2.1])and the function w(t) = T(t , 0) satises the linear nonautonomous equation

w t = wxx + b(t,x)w + d(t,x)w x , t > 0, x S 1 , (4.1)

with

b(t,x) = f y x,p(t)(x),p x (t)(x) , d(t, x) = f z x,p(t)(x),p x (t)(x) .

Since T is a bounded compact operator in the Hilbert innite-dimensional space X , thespectrum (T ) of T consists of 0 and a countable number of eigenvalues converging to 0. Eachof these eigenvalues is called a characteristic multiplier and has a nite algebraic multiplicity.

Moreover, if we choose p( ) instead of a and linearize around the periodic solutionp( + ) , then the evolution operators are T ( + t , + s) , so the period map is equal toT ( + , ) = D u

0S( )p( ) . Thus, by [14, Lemma 7.2.2], the spectrum of T = D u

0S()a

does not depend on the choice of the periodic point a , but the eigenfunctions do dependon a . Observe also that 1 is always a characteristic multiplier with the corresponding eigenfunc-tion p t (0) X 1 (cf. [14, Lemma 8.2.2]). If 1 is a simple eigenvalue of T unique on the unitcircle, we say that is a hyperbolic periodic orbit.

We put the multipliers in a sequence { j }j 0 such that they appear according to their alge-braic multiplicity and are ordered by | j + 1 | | j | . It was shown in [3] that for all j 0 wehave | 2j + 1 | < | 2j | . In other words, denoting by E j ( ) the real generalized eigenspace of { 2j 1 , 2j } for j 1 and by E 0( ) the real eigenspace corresponding to the isolated eigen-value 0 , we know that dim E 0( ) = 1 and dim E j ( ) = 2, j 1. Moreover, [3, Theorem 2.2]

yields that any nonzero E j ( ) , j 0, has only simple zeros and z( ) = 2j .Now we consider three projections connected with the decomposition of the spectrum of T

P = 12 i (I T )

1 d, {s,c,u },

where , {s,c,u }, is a closed regular curve surrounding in mathematically positive senseand separating from the rest of the spectrum of T the following subsets of the spectrum of T

s = (T

): | | < 1 ,

c = (T

): | | = 1 ,

u = (T

): | | > 1 ,

respectively. Note that dim P u X , called the Morse index i( ) , is nite and equals the totalalgebraic multiplicity of multipliers outside the closed unit ball. Similarly dim P c X is nite andequals the total algebraic multiplicity of multipliers on the unit circle.

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Observe that X = P u X P c X P s X and P X , {s,c,u }, are positively invariantsubspaces of T and

(T |P X ) = .

Furthermore, the eigenvectors of T belong to X 1 , so P X X 1 , {c, u }. Moreover, T mapsbijectively P u X onto P u X and P c X onto P c X .

Assume that is a hyperbolic periodic orbit. Consequently, we have P c X = span{p t (0)}.We consider the Poincar map P a for the semiow {S(t) } corresponding to the cross sectiona + P u X + P s X (see [14, Section 8.4], [30, Section 4.1]). Then the spectrum of the tangentmap to P a at a is equal to (T ) \ {1} and hence a is a hyperbolic xed point of P a . Therefore is hyperbolic in the sense of [30].

Since a hyperbolic periodic orbit is a normally hyperbolic manifold for {S(t) } (see [30,Remark 14.3(c)]), it follows from [30, Theorem 14.2, Remark 14.3] (see also [30, Section 6.3])that the local stable manifold of in a small neighborhood U of dened by

W sloc ( ) = u 0 X : S(t)u 0 U, t 0

is a C 1 submanifold of X with codim W sloc ( ) = i( ) , whereas the local unstable manifold of in U dened by

W uloc ( ) = u 0 X : {u s }s 0 S(t)u s = u t s , 0 t s and u s U, s 0

is a C 1 submanifold of X with dim W uloc

( ) = i( ) + 1.Moreover, W sloc ( ) is brated by local strong stable manifolds at each a

W sloc ( ) =a

W ssloc (a)

and W uloc ( ) by local strong unstable manifolds at each a

W uloc ( ) =a

W suloc (a),

where, for sufciently small > 0, we have the following characterizations with certain , > 0

W ssloc (a) = u 0 X : S(t)u 0 S(t)a X < for t 0 and

limt

e t S(t)u 0 S(t)a X = 0 ,

W suloc (a) = u 0 X : {u t }t 0 u t S(t)

1a X < for t 0,

S(r)u s = u r s for 0 r s and limt et u t S(t) 1a X = 0 .

From [30, Section 15.2] (see also [4]) it follows that for each a , W ssloc (a) is a C1 submanifold

of X tangent at a to P s X and W suloc (a) is a C1 submanifold of X tangent at a to P u X .

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5. Local stable manifold of a hyperbolic periodic orbit

In this section we consider a hyperbolic parabolic orbit with period > 0 and show that

for any u 0 W s

loc ( ) \ there exists a such that u(t ; u 0) p(t ; a) tends exponentially to0 as t and

z(u 0 a) i( ) + 1 + 1 + ( 1) i( )

2 . (5.1)

Choose u0 W sloc ( ) \ and let a be such that u0 W ssloc (a) . We consider the corre-

sponding solutions u(t) = S(t)u 0 and p(t) = S(t)a of (2.3). Let v(t) = u(t) p(t) , t 0, andnote that v satises the nonautonomous linear equation 1

vt = vxx + b(t,x)v + d(t,x)v x , t > 0, x S 1 , (5.2)

where

b(t,x) =

1

0

f y x, u(t)(x) + (1 )p(t)(x), u x (t)(x) + (1 )p x (t)(x) d ,

d(t ,x) =

1

0

f z x, u(t)(x) + (1 )p(t)(x), u x (t)(x) + (1 )p x (t)(x) d .

We also have with some > 0

limt

e t v(t) X = 0. (5.3)

We consider the sequence v(n ) = u(n ) a , n N . Note that u(n ) W ssloc (a) for all n N .Changing the norms to the equivalent ones, if necessary, but keeping the notation, we observethat

W ssloc (a) = u = a + h P s (u a) + P s (u a) : u BX (a, ) , (5.4)

where h : BP s X (0, ) BP u X P c X (0, ) is a C1 function such that h( 0) = 0 and h (0) = 0.

Let

= max | j | : | j | < 1 .

Then { j : | j | = } is a spectral set for T and we denote the corresponding projection in X

by P . If i( ) = 2N 1, then 2N 1 = 1 and 2N form a spectral set and thus P X is theone-dimensional space spanned by the eigenfunction corresponding to 2N , so P X E N ( )and

z( ) = 2N = i( ) + 1 for P X \ {0}. (5.5)

1 v(t) = 0 for all t 0, since u 0 / .

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If i ( ) = 2N , then 2N = 1 and P X is either E N + 1( ) or the one-dimensional space spannedby the eigenfunction corresponding to 2N + 1 . In both cases we have P X E N + 1( ) and

z( ) = 2N + 2 = i( ) + 2 for P X

\ {0}. (5.6)

It can be shown that for each a the set

W f sloc (a) = u 0 X : S(t)u 0 S(t)a X < for t 0 and

limt

e t S(t)u 0 S(t)a X = 0

for a certain = ( ) > is a C 1 submanifold of X , tangent at a to (P s P )X . We callW f sloc (a) the local fast stable manifold at a .

We are going to show that if u0 W ssloc (a) \ W f sloc (a) , then there exists a sequence t k

such that the normalized vectors u(t k ; u 0) p(t k ; a) tend to some P X \ {0}. Consequently,the zero number estimates for elements from P X \ {0} given in (5.5) and (5.6) will lead to (5.1)for u0 W ssloc (a) \ W

f sloc (a) . We will also show that (5.1) for u0 W

f sloc (a) \ {a } follows from the

previous case and the fact that W f sloc (a) is a submanifold of W ssloc (a) with codimension 1 or 2

within W ssloc (a) .Following [5, Lemma 2.2], we begin by proving that for u0 W ssloc (a) \ W

f sloc (a) the

(P s P )X -coordinate of v(n ) tends faster to zero than its P X -coordinate.

Lemma 5.1. For u0 W ssloc (a) \ W f sloc (a) we have

(P s P)v(n )X

Pv(n )X 0 as n . (5.7)

Proof. Note that

W f sloc (a) = u = a + g (P s P)(u a) + (P s P)(u a), u BX (a, ) ,

where g : B (P s P )X (0, ) B (P u + P + P c )X (0, ) is C1 and g( 0) = 0, g (0) = 0, is a subset

of W ssloc (a) . Taking into account (5.4) and setting y = P (u a) and z = (P s P)(u a) foru W ssloc (a) , we see that

W f sloc (a) = u = a + h(y + z) + y + z W ssloc (a) : y = Pg(z), z B (P s P )X (0, ) .

This means that in the coordinates (y,z) for W ssloc (a) the manifold W f sloc (a) is a graph of the

function y = Pg(z) .Consider rst the behavior of the sequence {v(n )} for u0 W ssloc (a) . Denote by T ( t , s ) :

X X , t s 0, the linear evolution operator corresponding to (5.2). We know that

U n = T (n + 1) , n T 0 as n (5.8)

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in the operator norm of L (X , X ) (see (4.1), (5.2)). Indeed, v(t) = T ( t , n ) , t [n , (n +1)], with X , satises

vt = v

xx + b(t,x)v + d(t,x)v

x, t n , (n + 1) , x S 1 , v(n ) = .

We change the variables vn (s) = v(s + n) , s [0, ]. Then vn satises

vns = vnxx + b(s + n , x) v

n + d(s + n , x) vnx , s (0, ], x S 1 ,

vn (0) = .(5.9)

Moreover, for w(s) = T (s, 0) , s [0, ], from (4.1) we have

w s = wxx + b(s,x)w + d(s,x)w x , s (0, ], x S 1 , w( 0) = .

Dene zn (s) = vn (s) w(s) , s [0, ], and note that it satises

zns = znxx + b(s,x)z

n + d(s,x)z nx + b(s + n , x) b(s,x) vn + d (s + n , x) d(s,x) vnx ,

with zn (0) = 0. If we denote by G(t, ) , 0 t , the evolution operator in X0 forzs = zxx + b(s,x)z + d(s,x)z x , then we obtain (see [18, (6.1.18), (6.1.19)])

G(t, ) X C X , 0 t , X ,

G(t, ) X

C

(t ) X0, 0 < t , X

0

,

and

zn (s) =

s

0

G(s, )h n ( ) d , 0 s ,

where h n ( ) = (b( + n) b( )) vn + (d( + n) d( )) vnx . Thus we get

zn (s) X C

s

0

1(s )

h n ( ) X 0 d . (5.10)

Moreover, from the variation of constants formula and (5.9) we obtain

vn (s) X c X + c

s

0

1(s )

b( )vn + d( )vnx

+b( + n) b( ) v

n

+d( + n) d( ) v

nx X 0 d .

Note that by the regularity of f we have

b( , x) M, d( , x) M, [0, ], x S 1 ,

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and

b( + n , x) b( , x) M, d( + n , x) d( , x) M,

for any [0, ], x S 1 and n N . Therefore

vn (s) X c X + 2Mc c

s

0

1(s )

vn ( ) X d , s [0, ].

From a Volterra type inequality we obtain for L = L(c, c ,M, ) > 0

vn (s) X L X , s [0, ]. (5.11)

Fix > 0 and let n0 N be such that for n n0 we have for any [0, ], x S 1

b( + n , x) b( , x) 0. Choose 0 < < 0 such that ( ) < . Suppose that for every n n0 = n 0( )

we have ||| zn ||| (P s P )X

||| yn ||| P X

. Then by (5.19) we get

( )

< ||| zn ||| (P s P )X

||| yn ||| P X ( + )n n0

||| zn0 ||| (P s P )X ||| yn0 ||| P X

, n n 0 ,

which is a contradiction. Therefore, there exists n1 n0 such that ||| zn1 ||| (P s P )X < ||| yn1 ||| P X .Hence from (5.18) it follows that for every n n 1 we have

||| zn ||| (P s P )X

||| yn ||| P X < .

Since > 0 was chosen arbitrarily, this shows that

znX

yn Pg(z n )X 0 as n . (5.20)

Since g(0) = 0 and g (0) = 0, we know that yn = 0 for all sufciently large n and

zn

X

ynX z

n

X

yn Pg(z n )X 1 + Pg(z

n)

X

ynX ,

i.e. we have

znX

ynX 1

Pg(z n )X

yn Pg(z n )X znX

yn Pg(z n )X .

This shows that

znX

ynX 0 as n ,

or in other words (5.7), which completes the proof.

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We compute

v(n )

v(n )X =

h(P s v(n ))

v(n )X +

P s v(n )

v(n )X . (5.21)

Observe that 2

limn

h(P s v(n ))v(n )X

= limn

h(P s v(n ))P s v(n )X

P s v(n )X

v(n )X = 0, (5.22)

since v(n ) 0 as n and h( 0) = 0, h (0) = 0.Let u0 W ssloc (a) \ W

f sloc (a) and note that for n large enough

P s v(n )v(n )X

= (P s P)v(n )Pv(n )X

Pv(n )X

v(n )X + Pv(n )v(n ) X

. (5.23)

Since P X is nite-dimensional and the sequence Pv(n )v(n )X

is bounded there, we can nd asubsequence {t n k } {n : n N } and P X

\ {0} such that

limk

Pv(t nk )v(t n k )X

= (5.24)

and, by (5.21), (5.22), (5.7) and (5.23), we obtain

limk

v(t nk )v(t n k )X

= limk

S(t n k )u 0 aS(t nk )u 0 a X

= . (5.25)

Since Lemma 3.2 applies to (5.2), we have for u0 W ssloc (a) \ W f sloc (a) and k large enough

z(u 0 a) = z v( 0) z v(t n k ) = z(). (5.26)

Let now u0 W f sloc (a) \ { a }. Since Lemma 3.2 applies to (5.2) and the zero number is boundedfrom below, there exists n N large enough so that v(n ) = u(n ) a has only simple zeros.Note that u(n ) W f sloc (a) and choose u W

ssloc (a) \ W

f sloc (a) such that

z v(n ) = z u(n ) a = z( u a).

Therefore, by the above considerations there exists P X \ {0} such that

z(u 0 a) z v(n ) = z( u a) z( ).

Recalling (5.5) and (5.6), we summarize our considerations in the following

2 P s v(n ) = 0 for any n N , because otherwise v(n ) = h(P s v(n )) + P s v(n ) would be 0.

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Theorem 5.2. For any u0 W sloc ( ) \ there exist a and > 0 such that

limt

e t S(t)u 0 S(t)a X = 0

and, for 2N = z(p t (0; a)) ,

z(u 0 a)i( ) + 1 = 2N if i ( ) = 2N 1,i( ) + 2 = 2N + 2 if i ( ) = 2N.

(5.27)

6. Global unstable manifold of a hyperbolic periodic orbit

Following [14, Theorem 6.1.9], we prove a general result concerning the extension of sub-manifolds.

Lemma 6.1. Let S(t) : X X , t 0 , be a semiow, which admits a compact global at-tractor A in X . Assume that is a subset of A , V is an open subset of an m-dimensionalclosed linear subspace E of X and k : V is a homeomorphism (with endowed withthe induced topology from X ) and its inverse h = k 1 : V belongs to C 1(V,X ) withD v h(v) L (E,X ) injective for any v V . Moreover, let the semiow S(t) : X X , t 0 ,be injective, belonging to C 1(X , X ) and let D w S(t)(w) L (X , X ) be injective for anyt 0 and w . Then each set S(t) is a C 1 submanifold of X with dimension m.

Proof. Dene f t : V X by

f t

(v) = S(t)h(v), v V .

Since S(t) | A is a homeomorphism of A onto A , we infer that S(t) | is a homeomorphism of onto S(t) (both equipped with the induced topology from X ). Thus f t is a homeomorphismof V onto S(t) , f t C 1(V,X ) and for any v V we have

D v f t (v) = D w S(t) h(v) D v h(v) L E, X is injective .

Moreover, (D v f t (v))E is an m-dimensional closed linear subspace of X , so it has a closedcomplement in X . Thus by [30, Corollary B.3.4] f t is an injective C 1 immersion (at any pointv V ). Since f t is a homeomorphism of V onto S(t) with the induced topology from X ,it follows from [30, Proposition B.4.3] that S(t) is a C 1 submanifold of X with dimen-sion m .

In our problem we dene the global unstable manifold of a hyperbolic periodic orbit by

W u ( ) =t 0

S(t)W uloc ( ).

Using Lemma 6.1 we infer that this invariant subset of the global attractor A is the union of C 1 submanifolds of X . Moreover, we have

W u ( ) =t 0 a

S(t)W suloc (a),

where again by Lemma 6.1 each S(t)W suloc (a) is a C1 submanifold of X .

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Again observe that 4

limn

h(P u v( n))

v( n)X = lim

n

h(P u v( n))

P u v( n)X

P u v( n)X

v( n)X = 0. (6.5)

Since

P u v( n)v( n)X

, n N ,

is a bounded sequence in a nite-dimensional subspace of X , there exist a subsequence {t n k }{ n : n N } and P u X \ {0} such that

limk

P u v(t n k )v(t n k )X

= (6.6)

and by (6.4) and (6.5), we obtain

limk

v(t n k )v(t n k )X

= . (6.7)

If i ( ) = 2N 1, then 2N 1 = 1, 2N = z( p t (0; a)) = z(p t (0; a)) and

z( ) 2N 2 = i( ) 1 for P u X \ {0}, (6.8)

whereas if i ( ) = 2N , then 2N = 1, 2N = z( p t (0; a)) = z(p t (0; a)) and

z( ) 2N = i( ) for P u X \ {0}. (6.9)

Using Lemma 3.2, we have for u0 W u ( ) \ and k large enough

z(u 0 a) z( u 0 a) = z v( 0) z v(t nk ) = z(). (6.10)

Summarizing the above considerations we obtain

Theorem 6.2. For any u0 W u ( ) \ there exist a and > 0 such that

limt

e t u( t ; u 0) p( t ; a) X = 0

and, for 2N = z(p t (0; a)) ,

z(u 0 a) i( ) 1 = 2N 2 if i ( ) = 2N 1,i( ) = 2N if i ( ) = 2N.(6.11)

4 P u v( n) = 0 for any n N , because otherwise v( n) = h(P u v( n)) + P u v( n) would be 0.

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7. Exclusion of a homoclinic connection for a hyperbolic periodic orbit

In this section we will consider two (not necessarily distinct) hyperbolic periodic orbits

and +

with periods

> 0 and +

> 0, respectively. We also assume that there exists a pointu 0 W u ( ) W sloc (

+ ) \ ( + ).

Note that if = + , then u0 is a homoclinic point and the corresponding orbit is a homoclinicconnection for the periodic orbit.

Consequently, to u0 / + there corresponds an X solution u( ; u 0) of (2.3), which isdened for all t R , its orbit is bounded in X (thus belongs to the global attractor A ) and thereexist initial data a together with periodic solutions p (; a ) of (2.3) such that

limt u(t ; u 0) p

(t ; a

) = 0 (7.1)

in X (cf. [14, Theorem 8.2.3]). We also dene N N so that 2N = z(p t (0; a )) .

In order to combine the estimates (5.27) and (6.11) we observe in the following two lemmasthat there exists some neighborhood of {p + (t ; a + ) p (t ; a ): t R } in X consisting of anite number of balls such that in a sufciently bigger neighborhood the zero number of functionsis constant.

Lemma 7.1. If = + and a = a + , then

z p + (t ; a + ) p (t ; a ) = const, t R .

Therefore there exists a nite cover ni = 1 BX (p+ (t i ; a + ) p (t i ; a ), i ) of the set

{p + (t ; a + ) p (t ; a ): t R } in X such that the zero number is constant inni = 1 BX (p

+ (t i ; a + ) p (t i ; a ), 2 i ).

Proof. Suppose that v(t) = p + (t ; a + ) p (t ; a ) , t R , has a multiple zero at t = t 0 . Since vis periodic with period = + = , then by Lemma 3.2 we have

z v(t 0) = z v(t 0 ) > z v(t 0) ,

which is a contradiction.The second claim follows from the compactness of {p + (t ; a + ) p (t ; a ): t R } in X (as

a continuous image of a compact interval), since to each point v of this set there corresponds aball BX (v, 2v ) in which the zero number is constant and we can choose a nite subcover from

v BX (v, v ) .

Lemma 7.2. If = + , then we have

z p+

(t ; a+

) p

(s ; a

) = const, s, t R

.

Therefore there exists a nite cover ni = 1 BX (p+ (t i ; a + ) p (s i ; a ), i ) of the set

{p + (t ; a + ) p (s ; a ): s, t R } in X such that the zero number is constant inni = 1 BX (p

+ (t i ; a + ) p (s i ; a ), 2 i ) .

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Proof. Fix R and suppose that v(t) = p + (t ; a + ) p (t + ; a ) , t R , has a multiple zeroat t = t 0 . We consider two cases. If

+

is rational, say + =

km for some k, m N , then with

= m+ = k we have for any t R

v(t + ) = p + (t + m+ ; a + ) p (t + k + ; a ) = p + (t ; a + ) p (t + ; a ) = v(t),

i.e. v is periodic with period . Then we have

z v(t 0) = z v(t 0 ) > z v(t 0) ,

which is a contradiction.Consider now

+

irrational. Note that R t z(v(t)) R as a monotone function has atmost a countable number of points of discontinuity. Therefore we choose t 1 t 0 + < t 0 such

that v(t 1) does not have multiple zeros and observe that

z v(t 1) > z v(t 0) .

To obtain a contradiction it sufces to nd t 2 > t 0 such that z(v(t 2)) = z(v(t 1)) . Below we showthat

> 0t 2>t 0 p+ (t 2; a + ), p (t 2 + ; a ) p + (t 1; a + ), p (t 1 + ; a ) X X < ,

which implies that v(t 1) v(t 2)C 1 (S 1) < and thus z(v(t 1)) = z(v(t 2)) for sufciently small > 0. Fix > 0. Since p + and p are continuous and periodic, the uniform continuity impliesthat there exists > 0 such that for any points (t ,s), ( t , s) R 2 we know that |(t,s) ( t , s) | < implies

p + (t ; a + ), p (s + ; a ) p + ( t ; a + ), p (s + ; a ) X X < .

Choose n0 N such that

n0< and set s0 = t 0 t 1+ 1. Denoting by [ ] the oor function,

consider the n0 + 1 real positive numbers

[s0] + 1 +

, 2 [s0] + 1

+

, . . . , ( n 0 + 1) [s0] + 1

+

and their fractional parts, which are pairwise different, since +

is irrational. At least two of the

numbers, say m1 +

> m 2+ , have to have fractional parts closer than

1n0

. Therefore, we knowthat m1 m2 [s0] + 1 > s 0 and

(m 1 m2)+

= l + r,

where l N {0} and 0 < | r | < 1n0 .Hence the distance between the points (t 1 + (m 1 m2)+ , t 1 + (m 1 m2)+ ) and (t 1 +

(m 1 m2)+ , t 1 + l ) is less than

n0< . In conclusion, for t 2 = t 1 + (m 1 m2)+ > t 0 we

have

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p + (t 2; a + ), p (t 2 + ; a ) p + (t 1; a + ), p (t 1 + ; a ) X X

= p + (t 2; a + ), p (t 2 + ; a )

p+

t 1 + (m 1 m2)+

; a+

, p

(t 1 + l

+ ; a

) X X < .

This ends the proof of the rst assertion.The second claim follows from the compactness of {p + (t ; a + ) p (s ; a ): s, t R } in X

(as a continuous image of a two-dimensional torus), since to each point v of this set there corre-sponds a ball BX (v, 2v ) in which the zero number is constant and we choose a nite subcoverfrom v BX (v, v ) .

The above lemmas allow us to combine the inequalities (5.27) and (6.11). As a particularresult, we deduce that there is no homoclinic connection for a hyperbolic periodic orbit.

Theorem 7.3. If u0 (W u ( ) W sloc ( + )) \ ( + ) and 2N = z(p t (0; a

)) , then

N N + and i( ) i( + ) + 1, (7.2)

which excludes a homoclinic connection for a hyperbolic periodic orbit. Moreover, if i ( + ) = 2N + , then we even have

N N + + 1. (7.3)

Proof. Let 0 > 0 be the minimum of 1 , . . . , n from Lemmas 7.1, 7.2. Since (7.1) implies forlarge t > 0

u(t ; u 0) p + (t ; a + ) X < 0 and u( t ; u 0) p ( t ; a ) X < 0 ,

we have

z(u 0 a ) z u(t ; u 0) p + (t ; a + ) + p + (t ; a + ) p (t ; a )

= z p + (t ; a + ) p (t ; a ) = z p ( t ; a ) p + ( t ; a + )

= z u( t ; u 0) p ( t ; a ) + p ( t ; a ) p + ( t ; a + ) z(u 0 a + ).

Here we have assumed that a = a + , but if a = a + we have z(u 0 a ) = z(u 0 a + ) imme-diately. By Theorem 5.2 we have

z(u 0 a + )i( + ) + 1 = 2N + if i ( + ) = 2N + 1,i( + ) + 2 = 2N + + 2 if i ( + ) = 2N + ,

and by Theorem 6.2 we have

z(u 0 a )i( ) 1 = 2N 2 if i ( ) = 2N 1,i( ) = 2N if i ( ) = 2N .

This leads straightforward to (7.2) and (7.3).

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8. Transversal intersection of stable and unstable manifolds of hyperbolic periodic orbits

We are going to show the transversal intersection of stable and unstable manifolds of hyper-

bolic periodic orbits following the approach used for equilibria by M. Chen, X.-Y. Chen andJ.K. Hale in [6] (see also [25]). Assume that

u 0 S( )W uloc ( ) W sloc (

+ )

with some 0. Our purpose is to show that

T u0 S( )W uloc (

) + T u 0 W sloc (

+ ) = X .

Without loss of generality we assume that u0 / + . We consider the linearization of (1.1)around u( ; u

0)

vt = vxx + b(t,x)v + d(t,x)v x , t > s, x S 1 ,v(s) =

(8.1)

and the corresponding evolution operators T ( t , s ) L (X , X ) , t s. Therefore, we haveT ( t , s ) = v(t ; s, ) . Note that each operator T ( t , s ) is injective and its adjoint is also injec-tive. Let p ( ; a ) be periodic solutions with a such that 2 N = z(p t (0; a

)) and

u(t ; u 0) p (t ; a ) 0 as t .

First we concentrate on the description of the tangent space to the local stable manifold. To thisend, let us consider the linearization of (1.1) around p + ( ; a + )

w +t = w+xx + b

+ (t,x)w + + d + (t,x)w +x , t > s, x S 1 ,

w + (s) =

and the corresponding evolution operators T + (t,s) L (X , X ) , t s . We denote by T ++ =T + (+ , 0) the period map and as in Section 4 we put its multipliers in a sequence { j }j 0 suchthat they appear according to their algebraic multiplicity and are ordered by | j | | j + 1 | .

Since the coefcients of the equations converge to each other, following the part of the proof of Lemma 5.1 we get U n = T ((n + 1)+ , n + ) T ++ 0 in L (X

, X ) . We also have theformula

v (n + 1)+ = T (n + 1)+ , n + v(n + ) = T ++ v(n + ) + U n v(n + )

and the assumptions (B.1)(B.2) of [6] are satised.For any X and any m N we write

(m, ) = limsupn

v(n + ; m+ , )1nX = limsup

n

T (n + , m + )1nX .

Let be the set of all nonnegative numbers r such that

T ++ z C : |z | = r = .

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We also set r j = | j | , j 0. For any integer j 0 and m N we dene the spaces

F +j (m) = X : (m, ) r j .

From [6, Corollary B.3] it follows that for every X and m N there exists r such that

limn

v(n + ; m+ , )1nX = r.

Therefore, we have

X = F +0 (m) F +1 (m) F

+2 (m) .

Since it is well known that linear equations like (8.1) do not possess nontrivial solutionsdecaying faster than any exponential (see [16]), as in [6, Theorem 3.1] we see that for all = 0and m N we have (m, ) > 0. In other words, we get

j = 0

F +j (m) = { 0},

because (m, 0) = 0, m N .From the asymptotic behavior of the solution v(n + ; m+ , ) we are able to characterize

the zero number z( ) of the initial condition . Assume that

limn

v(n + ; m+ , )1nX = r j .

We have the following two cases. If j is odd and rj > r j + 1 , then F +j (m) \ F +j + 1(m) and

by [6, Theorems B.2, B.4] there exists E j + 12

( + ) , X = 1 and a subsequence {t nk }

{n+ : n m} such that

v(t nk ; m+ , )v(t nk ; m+ , )X

.

Therefore, we have for large k N

z( ) z v(t n k ; m+ , ) = z

v(t n k ; m+ , )

v(t n k ; m+ , )X = z( ) = j + 1.

If j is even, then F +j (m) \ F +j + 1(m) and by [6, Theorems B.2, B.4] there exists

E j 2

( + ) , X = 1 and a subsequence {t nk } {n+ : n m} such that

v(t nk ; m+ , )

v(t nk ; m+ , )X .

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Therefore, we have for large k N

z( ) z v(t nk ; m+ , ) = z

v(t nk ; m+ , )

v(t nk; m+ , )X

= z( ) = j.

Fix m N , F +k (m) , = 0 and let j k be such that F +j (m) \ F

+j + 1(m) . Suppose

that k is even. If j is even, then z( ) j k , whereas if j is odd, then z( ) j + 1 k.Suppose now that k is odd. If j is even, then z( ) j k + 1 due to the difference of parity of k and j . Moreover, if j is odd, then z( ) j + 1 k + 1.

Since the local stable manifold W sloc ( + ) of the hyperbolic periodic orbit + coincides

locally with the local center-stable manifold W csloc (a+ ) of a xed point a + + for the map

S( + ) , it follows from [6, Theorem C.4] that

T u 0

W sloc

( + ) = X : limsupn

v(n + ; 0, )1n

X 1 .

If i( + ) = 2N + , then r2N + = 2N + = 1 and by [6, Theorem B.7] for sufciently largem0 N we see that T u(m 0+ ;u0 ) W

sloc (

+ ) = F +2N + (m 0) is isomorphic to cl X (span{p+t (0; a

+ )}E N + + 1( + ) ) . Therefore, z( ) 2N + for T u(m 0+ ;u0 ) W

sloc (

+ ) \ {0} andcodim T u(m 0+ ;u0 ) W

sloc (

+ ) = 2N + .If i( + ) = 2N + 1, then r2N + 1 = 2N + 1 = 1 and by [6, Theorem B.7] for suf-

ciently large m0 N we see that T u(m 0+ ;u 0) W sloc (

+ ) = F +2N + 1(m 0) is isomorphic toclX (E N + ( + ) E N + + 1( + ) ) . Therefore, z( ) 2N + for T u(m 0+ ;u0 ) W

sloc (

+ ) \{0} and codim T

u(m 0+

;u0 )W s

loc( + ) = 2N + 1.

Since the adjoint operator of the evolution operator T (m 0+ , 0) is injective (thusT (m 0+ , 0) has dense range by [14, Theorem 7.3.3]) and T u0 W

sloc (

+ ) is the preimage of T u(m 0+ ;u 0) W

sloc (

+ ) under T (m 0+ , 0) , we see that if i ( + ) = 2N + , then

codim T u 0 W sloc (

+ ) = codim F +2N + (0) = 2N +

and z( ) 2N + for T u 0 W sloc (

+ ) \ {0}, whereas if i ( + ) = 2N + 1, then

codim T u 0 W sloc (

+ ) = codim F +2N + 1(0) = 2N + 1

and z( ) 2N + for T u 0 W sloc ( + ) \ {0}.

In both cases of i( + ) we now consider r2N + + 1 < 1 and the subspace F +2N + + 1(0) of

T u 0 W sloc (

+ ) , which is isomorphic to cl X (E N + + 1( + ) ) . Then we have z( ) 2N + + 2for F +2N + + 1(0) \ {0} and codim F

+2N + + 1(0) = 2N

+ + 1.From the above considerations the following result follows.

Lemma 8.1. We have

z(v) 2N + , v T u 0 W sloc (

+ ) \ {0}.

Moreover, there exists a subspace W + of T u 0 W sloc ( + ) such that

z(v) 2N + + 2, v W + \ {0},

and codim W + = 2N + + 1.

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Using Lemma 8.1 we nally prove the main result of the paper.

Theorem 8.2. The stable and unstable manifolds of two hyperbolic periodic orbits for the

problem (1.1) have a transversal intersection

W u ( ) W sloc ( + ),

i.e. if u0 S( )W uloc ( ) W sloc (

+ ) with some 0 , then

T u 0 S( )W uloc (

) + T u0 W sloc (

+ ) = X .

Proof. The proof is in the same vein as the proof of Lemma 8.1 and is based on the descriptionof the tangent space to the global unstable manifold.

First assume i( + ) = 2N + 1. Thus we have codim W sloc ( + ) = 2N + 1. From theinequality N N + (see Theorem 7.3) we see, in a similar way as in the local sta-ble manifold case, that for sufciently large m0 N there exists a subspace F m0 ,N + of T u( m0 ;u0 ) S( )W

uloc (

) such that

dim F m0 ,N + = dim E 0( ) E N + 1(

) = 2N + 1,

z(v) 2N + 2, v F m0 ,N + \ {0}.

Now note that F 0,N + = T (0, m0 )F m0 ,N + is a subspace of T u0 S( )W uloc ( ) with

dim F 0,N + = dim F m0 ,N +

and z(v) 2N + 2, v F 0,N + \ {0},

since the evolution operator T ( , ) for the linearized equation around the solution u( ; u 0) isinjective and does not increase the zero number. Thus we have

dim F 0,N + = 2N + 1 = codim W sloc (

+ )

and F 0,N + T u 0 W sloc (

+ ) = {0}, since z(v) 2N + 2 for v F 0,N + \ {0} and by Lemma 8.1we have z(v) 2N + for v T u0 W

sloc (

+ ) \ {0}. This proves that

T u 0 S( )W uloc (

) + T u0 W sloc (

+ ) = X

in this case.Let now i ( + ) = 2N + . Then by Theorem 7.3 we have N N + + 1 and we see again that

for sufciently large m0 N there exists a subspace F m0 ,N + of T u( m0 ;u 0) S( )W uloc (

) suchthat

dim F m0 ,N + = dim E 0( ) E N + (

) = 2N + + 1,

z(v) 2N + , v F m0 ,N + \ {0}.

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As before, note that F 0,N + = T (0, m0 )F m0 ,N + is a subspace of T u 0 S( )W

uloc (

) with

dim F 0,N + = dim F m0 ,N +

and z(v) 2N + , v F 0,N + \ {0}.

Thus we get

dim F 0,N + = 2N + + 1 = codim W + ,

where W + is the subspace of T u 0 W sloc (

+ ) given in Lemma 8.1. Moreover, we have

F 0,N + W + = { 0},

since z(v) 2N + for v F 0,N

+ \ {0} and z(v) 2N + + 2 for v W + \ { 0}, by Lemma 8.1.Therefore, we obtain F 0,N + W

+ = X . This shows that

T u0 S( )W uloc (

) + T u0 W sloc (

+ ) = X

also in this case and concludes the proof.

9. Concluding remarks

As pointed out in the introduction, the transversality between stable and unstable manifolds

is one of the main ingredients for the structural stability of the semiow generated by (1.1). Fur-thermore, from the point of view of applications the discussion of structural stability is essentialfor dynamical systems. As concluding remarks we overview here some of the available resultsdealing with this topic.

The discussion of structural stability involves the characterization of the semiow on theglobal attractor A f and its dependence on the nonlinearity f C 2 considered as a parameter.In this innite-dimensional setting it requires the comparison between different global attractorsusing topological equivalence (for a reference see [13]). If A f and A g denote the global attractorscorresponding to the semiows generated by (1.1) with nonlinearities f and g , respectively, A f and A g are orbit equivalent , A f = A g , if there exists a homeomorphism h : A f A g takingorbits of (1.1) f to orbits of (1.1) g preserving the time direction. Then, a global attractor A f is structurally stable if there exists a neighborhood N (f ) of f (in the space of C 2 functionswith the adequate topology) such that A g = A f for all g N (f ) . Therefore, a structurallystable global attractor A f is invariant up to a homeomorphism under small perturbations of thenonlinearity f .

Structural stability is known to hold in the class of MorseSmale semiows (see [24] fordetails). The class of semiows considered enjoys a number of properties related to the existenceof global attractors and includes many semiows generated by partial differential equations, likein our case, and delay or functional differential equations. For completeness we recall that sucha semiow has the MorseSmale property if: (i) the corresponding global attractor A

f has only

a nite number of equilibria and periodic orbits, which are all hyperbolic, (ii) all the stable andunstable manifolds of these critical elements are transversal, and (iii) the nonwandering set (f )(the set of points of A f for which any neighborhood U is visited after an arbitrarily large timeT > 0 by an orbit starting in U ) contains only the equilibria and the periodic orbits. Property

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(iii) is expected to hold for (1.1) under condition (i) due to the zero number decay propertyand the result [8, Theorem 1] mentioned in the introduction. Therefore, the verication of theMorseSmale property of A f should involve only the conrmation of property (i), which holds

generically in the above space of C2

functions, and property (ii) regarding the transversalitybetween stable and unstable manifolds of the critical elements. Here, in view of our transversalityresult, it only remains to check transversality for pairs of critical elements where at least one isan equilibrium.

In restricted classes of nonlinear functions f we are able to exhibit the MorseSmale prop-erty. In fact, this is the case when f = f (u, u x ) does not depend explicitly on x . Then, S 1-equivariance of (1.1) forces the global attractor A f to be invariant under the S 1-action andreduces the number of possibilities for the critical elements of the semiow. In the generic situa-tion, the set of critical elements is composed of a nite set of hyperbolic homogeneous equilibria,corresponding to the solutions of f(e, 0) = 0, and a nite set of hyperbolic rotating waves as

mentioned in the introduction. From the results in [9] it follows that, in the class of nonlinearitiesf = f(u,u x ) , the generic global attractor A f is structurally stable.

The situation is quite different in the class of x -dependent nonlinearities f = f(x,u,u x ) . Aspointed out before, the time periodic orbits need not be rotating waves. Furthermore, A f maycontain hyperbolic nonhomogeneous equilibria and orbits homoclinic to equilibria. The exis-tence of these homoclinic orbits follows from a result in [31] asserting that the ow of any planarvector eld can be realized locally by a ow embedding in a two-dimensional invariant subspaceof (1.1). Therefore, nontransversal intersection between stable and unstable manifolds of equi-libria actually takes place. Also, the occurrence of nonhomogeneous equilibria is a distinguishedimportant feature, and their heteroclinic connections need to be analyzed.

Finally, to understand the global geometry of A f when structural stability fails, it is importantto consider local bifurcation problems that explore the boundary of the structurally stable setof global attractors. Here, in view of the result [31, Theorem 2], the set of codimension onebifurcations for vector elds in the plane should play a role.

In conclusion, all these considerations just show that the geometry of the global attractorof (1.1), in general, is far from understood.

References

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