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Transversality of Homoclinic Orbits and Exponential Dichotomies for Parabolic Equations
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
216, 466]480 Ž1997.
AY975662
Transversality of Homoclinic Orbits and Exponential Dichotomies for Parabolic Equations* Zeng Weiyao† Research Institute of Sciences, Changsha Railway Uni¨ ersity, Changsha, Hunan, 410075, People’s Republic of China Submitted by William F. Ames Received December 10, 1996
In this paper we discuss the existence of exponential dichotomies on R of linear parabolic equations depending on small parameters and provide a tool of proving the transversality of the homoclinic orbits of parabolic equations, and by making use of the results on exponential dichotomies of this paper, we investigate the transversality of homoclinic orbits for parabolic equations. Q 1997 Academic Press
1. INTRODUCTION Let F be a diffeomorphism on a two dimensional manifold. It is well known that the existence of one transversal homoclinic point for F implies the existence of the Smale horseshoe Žthat is, the chaotic dynamics. from the Smale]Birkhoff homoclinic theorem. Recently, Palmer w1x and Meyer and Sell w2, 3x have shown that the same results in R n can be obtained by using elementary ideas from the theory of exponential dichotomies and the shadowing lemma. In the infinite case the problem was considered by Hale and Lin w4x for functional differential equations and by Blazquez w7x for parabolic equations. It is well known that one of the best tools for detecting the existence of transversal homoclinic orbits of differential equations is the Melnikov method. The Melnikov method has been extended since the paper of Melnikov w8x, therefore we have a vast literature. We refer to Wiggins w9x, Hale and Chow w10x, Palmer w1, 11x, Zeng w6x, Meyer and Sell w2x, Hale and Lin w4x, and Blazquez w7x. * This work is partially supported by NSF of China. † E-mail address: [email protected]. 466 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
EXPONENTIAL DICHOTOMIES
467
Palmer w1x used the Liapunov]Schmidt method and the theory of exponential dichotomies to generalize the Melnikov method to ndimensional non-Hamiltonian systems and gave the relation between the transversality of the homoclinic orbits and the existence of exponential dichotomies, on R, of the variational equations along the homoclinic orbits. Palmer w11x used the Melnikov integral to provide a sufficient condition assuring the existence of exponential dichotomies, on R, of the linear systems depending on a small parameter « , and then used the result to prove the transversality of the homoclinic orbits. The result in Palmer w11x on the theory of exponential dichotomies becomes a standard method of proving the transversality of homoclinic orbits of ordinary differential equations Žrefer to Zeng w6x and Battelli w15x.. Zeng w6x generalized the results of Palmer w1, 11x. Later Blazquez w7x extended the results in Palmer w1x and Hale and Lin w4x to the parabolic equations. But Blazquez w7x did not prove the result on the theory of exponential dichotomies for parabolic equations analogous to that in Palmer w11x on the existence of exponential dichotomies on R of the linear variational equations depending on small parameters, so the transversality of the homoclinic orbits was not proved rigorously in Blazquez w7x. The main purpose of this paper is to generalize the result in Palmer w11x on the existence of the exponential dichotomies depending on small parameters on R to the linear parabolic equations by using the Liapunov]Schmidt method and the Melnikov integral. Also we use the results we obtained on the exponential dichotomies to show rigorously the transversality of the homoclinic orbits for parabolic equations. We want to use the following notations. We let LŽ X b, X . s L: X b ª X, is a linear operator of X b to X 4 ; LŽ X . s L: X ª X, is a linear operator of X to X 4 ; Cb0 Ž R, X . s u: R ª X, is continuous and bounded in X for t g R4 ; Cb1 Ž R, X . s u: R ª X, is continuously differentiable and u, u ˙ are bounded in X for t g R4. Cbk Ž R, X . has the same meaning. For any operator L we denote by RŽ L. the range of L. In this paper, the definitions of the solution of parabolic equations in X, and the derivative of functions in X are the same as in Henry w13x. If A is a sectorial operator in a Banach space X then there is a real number a such that A1 s A q aI implies Re s Ž A1 . ) 0. Let 0 F b F 1. One can define a fractional power A1b of A1. Let X b s DŽ A1b . with the graph norm 5 x 5 b s 5 A1b x 5 for x g X b Žrefer to Henry w13x.. The space X b is a Banach space with the norm 5 5 b for 0 F b F 1 and X b is a dense subspace of X r with continuous inclusion for 0 F r F b F 1 Žrefer to w13, Theorem 1.4.8x.. Suppose A is a sectorial operator on X. If t ª B Ž t .: w t 0 , t 1 x ª LŽ X b, X . is Holder continuous, then, for any u 0 g X b ¨ and t 0 F t F t 1 , there exists a unique solution uŽ t . s uŽ t, t , u 0 . of the
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ZENG WEIYAO
equation u ˙ q Au s B Ž t . u on t F t F t 1 , uŽt . s u 0 and u 0 ª uŽ t, t , u 0 . is linear and bounded in X. We can define an evolution operator T Ž t, t . N t 0 F t F t F t 14 by uŽ t, t , u 0 . s T Ž t, t . u 0 , t G t . Suppose the evolution operator T Ž t, s . g LŽ X . for the equation u ˙ q AŽ t . u s 0
Ž 1.1.
is defined on a real interval J. Equation Ž1.1. is said to have an exponential dichotomy on J with the exponent b ) 0 and the bound M if there exist projections P Ž t ., QŽ t . s I y P Ž t ., t g J, such that Ža. Žb. Ž Ž R Q t ... Žc. Žd.
T Ž t, s . P Ž s . s P Ž t .T Ž t, s ., t G s in J. The restriction T Ž t, s .< RŽQŽ s.. , t G s is an isomorphism onto We define the inverse map from RŽ QŽ t .. to RŽ QŽ s .. by T Ž s, t .. 5 T Ž t, s . P Ž s .5 F Meyb Ž tys. for t G s in J. 5 T Ž t, s . QŽ s .5 F Meyb Ž syt . for s G t in J.
The theory of exponential dichotomies plays a very important role in studying the global bifurcations; we refer to Palmer w1x, Meyer and Sell w2x, Zeng w6x, and Sandstede w18x. On the theory of exponential dichotomies for finite dimensional systems, we refer to Palmer w1x, Meyer and Sell w2x, Coppel w5x, and Sacker and Sell w19x; for infinite dimensional systems, we refer to Sackel and Sell w16x, Henry w13x, and Chow and Leiva w17x. On the other related results on dynamics of parabolic equations, we refer to Shen and Yi w21, 22x and Fiedler and Rocha w23x. If A is a sectorial operator, AŽ t . y A: R ª LŽ X b, X . is locally Holder ¨ continuous, and T Ž t, s . g LŽ X b, X . is the evolution operator, by Henry’s Theorem 7.3.1 in w13x, we can define the adjoint operator T *Ž s, t . s T Ž t, s .* by
² y, T Ž t , s . x: s² T * Ž s, t . y, x: ,
x g X , y g X *,
Ž 1.2.
and T *Ž s, t . g LŽ X *., locally continuous in s - t Žbut weak*-continuous at s s t ., such that y Ž s . s T *Ž s, t 0 . y 0 satisfies the equation adjoint to Eq. Ž1.1. dy ds
Ž s . s A* Ž s . y Ž s . .
Ž 1.3.
If Eq. Ž1.1. has an exponential dichotomy on Rq and Ry with projections Pq Ž t . and Py Ž t ., respectively, then Hale and Lin w4x and Blazquez w7x proved that the adjoint equation Ž1.3. also has an exponential dichotomy
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469
on Rq and Ry with the same constants and with projections I y Ž Pq Ž t ..* and I y Ž Py Ž t ..*, respectively. Let O be an open subset of X b and let f : R = O ª X be a continuous T-periodic function in t with continuous partial derivatives f uŽ t, u.. We denote by F Ž t, j . the unique solution of the equation u ˙ q Au s f Ž t , u .
Ž 1.4.
with F Ž0, j . s j and by F: X b ª X b FŽ j . s FŽT , j . the period map for Eq. Ž1.4.. Blazquez w7x proved the following result: y g O is a transversal homoclinic point with respect to a hyperbolic fixed point j 0 if and only if FŽ t, y. y FŽ t, j0 . ª 0
as < t < ª `
Ž 1.5.
and the variational equation u ˙ q Au s f u Ž t , F Ž t , y . . u
Ž 1.6.
has an exponential dichotomy on R. So in studying the transversality of homoclinic orbits which bifurcate from the nontransversal homoclinic orbits of parabolic equations, we want to investigate the existence of exponential dichotomies on R of the linear parabolic equation depending on a small parameter u ˙ q AŽ t , « . u s 0,
Ž 1.7.
where AŽ t, « . y A: R = Žyl 0 , l 0 . ª LŽ X b, X . is locally Holder and A is a ¨ sectorial operator and l 0 ) 0 a sufficiently small constant. We want to consider the following problem: Suppose when « s 0 the linear parabolic equation u ˙ q AŽ t , 0. u s 0
Ž 1.8.
admits an exponential dichotomy on both Rq and Ry but does not have exponential dichotomies on R. Under what conditions can we guarantee that the linear parabolic equation Ž1.7. has an exponential dichotomy on R for « / 0 sufficiently small? In this paper, we want to solve the above problem and to use the result to prove the transversality of the homoclinic orbits of parabolic equations. The result in Palmer w11x is generalized to the parabolic equations. This paper is organized as follows: Section 1 is an introduction. In Section 2 we give a sufficient condition which assures the
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ZENG WEIYAO
existence of exponential dichotomies on R for Eq. Ž1.7. for « / 0 sufficiently small. In Section 3 we apply the result in Section 2 to investigate the transversality of the homoclinic orbits.
2. THE EXISTENCE OF EXPONENTIAL DICHOTOMIES In this section, we mainly study the existence of exponential dichotomies on R of the linear parabolic equation depending on a small parameter u ˙ q AŽ t , « . u s 0,
Ž 2.1.
where x g X b, AŽ t, « . y A: R = Žyl 0 , l 0 . ª LŽ X b, X . is locally Holder ¨ continuous, A is a sectorial operator, « is a small parameter, and 0 F b F 1. We need a lemma. LEMMA 1. If A is a sectorial operator, AŽ t . y A: R ª LŽ X b, X . is locally Holder continuous, such that the equation ¨ u ˙ q AŽ t . u s 0
Ž 2.2.
has an exponential dichotomy on Rq and Ry with projections Pq Ž t ., Py Ž t ., respecti¨ ely, then for the linear operator L: Cb1 Ž R, X b . ª Cb0 Ž R, X . defined by
Ž Lu . Ž t . s u˙Ž t . q A Ž t . u Ž t . we ha¨ e Ž a. L is a Fredholm operator. Ž b . N Ž L. s g g Cb1 Ž R, X .: ˙ g Ž t . q AŽ t . g Ž t . s 0, g Ž0. g RŽ Pq Ž0.. l RŽ I y Py Ž0..4 ; Ž c . RŽ L. s f g Cb0 Ž R, X .: Hq` ² Ž . Ž .: dt s 0 for all c g y` c t , f t 1Ž b . Cb R, X * satisfying the adjoint equation u ˙ y A*Ž t . u s 04 and indŽ L. s dim RQy Ž0. y dim RQq Ž0.. Lemma 1 is due to Blazquez w7x, also refer to Palmer w1x, Hale and Lin w4x, Weinian w14x, Sacker and Sell w16x, and Chow and Leiva w17x. For Eq. Ž2.1., we assume ŽH1. The derivatives AŽ t, « . and A« Ž t, « . exist continuous and bounded on R = Žyl 0 , l 0 ., where l 0 ) 0 a sufficient small constant. We let I s Žyl 0 , l 0 ..
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471
ŽH2. The equation u ˙ q AŽ t , 0. u s 0
Ž 2.3.
admits an exponential dichotomy on Rq and Ry with projections Pq Ž t ., Py Ž t ., and dim RŽ Qy Ž0.. s dim RŽ Qq Ž0.., respectively, and dim RŽ Qq Ž t .. and dim RŽ Qy Ž t .. are finite, and has a unique, up to a scalar multiple, nontrivial bounded solution w Ž t . on R in X. For the linear equation Ž2.1. we define an operator L« by L« : Cb1 Ž R, X b . ª Cb0 Ž R , X .
Ž Lu . Ž t . s u˙Ž t . q A Ž t , « . u Ž t . . We say the operator L« is the operator associated with the linear parabolic equation Ž2.1.. Since Eq. Ž2.3. admits an exponential dichotomy on Rq and Ry with projections Pq Ž t ., Py Ž t . with dim RŽ Qy Ž0.. s dim RŽ Qq Ž0.., we have that L0 is a Fredholm operator and indŽ L0 . s 0 from Lemma 1. It follows from the roughness of exponential dichotomies Žrefer to Hale and Lin w4x, Chow and Leiva w17x, and Sacker and Sell w16x., that for « / 0 sufficiently small, Eq. Ž2.1. also has an exponential dichotomy on Rq and «Ž . «Ž . Ry with projections Pq t and Py t , respectively. So it follows from Lemma 1 that the operator L« is Fredholm and indŽ L« . s 0 because the index of the Fredholm operator persists under small perturbations. Since ind Ž L0 . s 0 s dim N Ž L0 . y codim R Ž L0 . s dim R Ž Pq Ž 0 . . l R Ž I y Py Ž 0 . . y dim R Ž Py Ž 0 . * . l R Ž I y Pq Ž 0 . * . , from condition ŽH2., we have dim R Ž Py Ž 0 . * . l R Ž I y Pq Ž 0 . * . s dim R Ž Pq Ž 0 . . l R Ž I y Py Ž 0 . . s 1. Since RŽ Py Ž0.*. l RŽ I y Pq Ž0.*. and RŽ Pq Ž0.. l RŽ I y Py Ž0.. are the spaces of the initial values of bounded solutions on R in X of Eq. Ž2.3. and the adjoint equation of Eq. Ž2.3., respectively, the adjoint equation of Eq. Ž2.3. u ˙Ž t . y A* Ž t , 0 . u Ž t . s 0
Ž 2.4.
also has and only has a unique, up to a scalar multiple, nontrivial bounded solution, denoted by c Ž t . g X for t g R.
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ZENG WEIYAO
Now we consider Eq. Ž2.1.. We let T« Ž t, s . g LŽ X . be the evolution operator for Eq. Ž2.1.. The main result of this paper is as follows: THEOREM 1. Suppose A is a sectorial operator, AŽ t, « . y A: R = Žyl 0 , l 0 . ª LŽ X b, X . is locally Holder ¨ continuous, and conditions Ž H 1. and Ž H 2. are satisfied. If q`
Hy` ² c Ž t . , A Ž t , 0. w Ž t .: dt / 0 «
then for « / 0 sufficiently small Eq. Ž2.1. u ˙ q AŽ t , « . u s 0 admits an exponential dichotomy on R. Remark. Theorem 1 generalizes the result of Palmer w11x to the infinite dimensional systems. Proof of Theorem 1. For any f g Cb0 Ž R, X ., we consider the equation u ˙ q AŽ t , « . u s « 2 f Ž t . .
Ž 2.5.
We assume, without loss of generality, q`
Hy`
c 2 Ž t . dt s 1.
We make a change of variable y Ž t . s uŽ t . y w Ž t .g for Eq. Ž2.5. where g g R is a new parameter. Then Eq. Ž2.5. becomes
˙y q A Ž t , 0 . y s AŽ t , 0 . y AŽ t , « .
y q w Ž t . g q « 2 f Ž t . . Ž 2.6.
Since the linear parabolic equation ˙ y q AŽ t, 0. y s 0 admits an exponential dichotomy on both Rq and Ry, using the Liapunov]Schmidt method Žrefer to Lemma 1., we see that Eq. Ž2.6. is equivalent to the two equations
˙y q AŽ t , 0 . y s AŽ t , 0 . y AŽ t , « . q c Ž t.
q`
y q wŽ t.g q « 2 f Ž t.
Hy` ² c Ž t . ,
AŽ t , 0. y AŽ t , « .
y q wŽ t.g
q« 2 f Ž t .: dt Ž 2.7. q`
Hy` ² c Ž t . ,
AŽ t , 0. y AŽ t , « .
y q w Ž t . g q « 2 f Ž t .: dt s 0. Ž 2.8.
We can prove that for « sufficiently small Eq. Ž2.7. has a unique solution y s y Ž t, g , « . g Cb1 Ž R, X b . for t g R, and satisfies y Ž t, g , 0. s 0. More-
473
EXPONENTIAL DICHOTOMIES
over, y Ž t, g , « . is second order continuously differentiable in Žg , « .. Substituting y s y Ž t, g , « . into Eq. Ž2.8., we obtain the bifurcative equation BŽg , « . s
q`
Hy` ² c Ž t . ,
AŽ t , 0. y AŽ t , « .
yŽ t, g , « . q w Ž t.g q« 2 f Ž t .: dt. Ž 2.9.
Since y Ž t, g , 0. s 0 and y Ž t, g , « . is second order continuously differentiable in g , « , we have that B Žg , « . is also second order continuously differentiable in g , « and B Ž g , 0 . s 0. We define a function H Žg , « . by
¡B Ž g , « . , ¢B Ž g«, 0. ,
H Ž g , « . s~
«
«/0 « s 0.
Since B Žg , 0. s 0 and B Žg , « . is second order continuously differentiable in g , « , the function H Žg , « . is continuously differentiable in g , « . Then H Ž g , 0 . s B« Ž g , 0 . q`
s
Hy` ² c Ž t . , A Ž t , 0.
s
Hy` ² c Ž t . , A Ž t , 0.
«
q`
«
y Ž t , g , 0. q w Ž t . g
wŽ t.
: dt
: dt ? g
Hence H Ž 0, 0 . s 0 and Hg Ž 0, 0 . s
q`
Hy` ² c Ž t . , A Ž t , 0. w Ž t .: dt. «
Since q`
Hy` ² c Ž t . , A Ž t , 0. w Ž t .: dt / 0, «
it follows from the implicit theorem that there exists a continuous function g s g Ž « ., satisfying g Ž0. s 0, such that HŽg Ž « . , « . s 0
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ZENG WEIYAO
holds for « sufficiently small. Hence for « / 0 sufficiently small the equation u ˙ q AŽ t , « . u s « 2 f Ž t .
Ž 2.10.
has a bounded solution uŽ t, « . s y Ž t, g Ž « ., « . q w Ž t .g Ž « . g Cb1 Ž R, X b .. So for any f g Cb0 Ž R, X . and « / 0 sufficiently small the equation u ˙ q AŽ t , « . u s f Ž t .
Ž 2.11.
has a bounded solution uŽ t, « .r« 2 s w y Ž t, g Ž « ., « . q w Ž t .g Ž « .xr« 2 g Cb1 Ž R, X b .. Thus we have for, « / 0 sufficiently small, RŽ L« . s Cb0 Ž R, X ., that is, codim RŽ L« . s 0. Since indŽ L« . s 0 for « sufficiently small, we have 0 s dim N Ž L« . y codim R Ž L« . s dim N Ž L« . and hence « N Ž L« . s R Ž Pq Ž 0 . . l R Ž Qy« Ž 0 . . s 0 4 .
From condition ŽH2. we see « dim Ž R Ž Pq Ž 0 . . q R Ž Qy« Ž 0 . . . « s dim R Ž Pq Ž 0 . . q dim R Ž Qy« Ž 0 . . y dim Ž R Ž Pq« Ž 0 . . l R Ž Qy« Ž 0 . . . « s dim R Ž Pq Ž 0 . . q dim R Ž Qy« Ž 0 . . « s dim R Ž Pq Ž 0 . . q dim R Ž Qq« Ž 0 . .
s n. Hence « R Ž Pq Ž 0 . . q R Ž Qy« Ž 0 . . s X .
Ž 2.12.
We define two closed subsets, X 1Ž t . and X 2 Ž t ., of X for « / 0 sufficiently small by X1Ž t . s
X2 Ž t . s
½ ½
« R Ž Pq Ž t..,
w N T« Ž 0, t . w g R Ž
tG0 « Pq
Ž 0. . 4 ,
« R Ž Qy Ž t..,
T« Ž t , 0. w N w g R Ž
tF0 tF0
« Qy
Ž 0. . 4 ,
t G 0.
Then from Ž2.12. we have X1Ž 0. [ X 2 Ž 0. s X .
Ž 2.13.
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475
We claim that X1Ž t . [ X 2 Ž t . s X .
Ž 2.14.
Now we want to prove Ž2.14. by the following two steps. Step 1. We show X 1Ž t . l X 2 Ž t . s 04 for t g R. When t G 0, if w g X 1Ž t . l X 2 Ž t ., then there exists w 0 g X 2 Ž0. such « Ž .. « Ž .. that w s T« Ž t, 0. w 0 g RŽ Pq t . Hence w 0 g RŽ Pq 0 . So w 0 g X 1Ž0. l X 2 Ž0.. From Ž2.13., we have w 0 s 0 and so w s 0. When t F 0, if w g X 1Ž t . l X 2 Ž t ., then T« Ž0, t . w g X 1Ž0. l X 2 Ž0.. Thus, T« Ž0, t . w s 0. Since T« Ž0, t . is an isomorphism form X 2 Ž t . to X 2 Ž0. we have w s 0. Hence for t g R we have X1Ž t . l X 2 Ž t . s 04 .
Ž 2.15.
Step 2. We show X 1Ž t . q X 2 Ž t . s X for t g R. When t F 0, since Ž2.13. holds, for any w g X, we have T Ž 0, t . w s w 1 q w 2 ,
where w 1 g X 1 Ž 0 . and w 2 g X 2 Ž 0 . .
Hence there exists a w 1 g X 1Ž t . such that w 1 s T« Ž0, t . w 1 , and so w 2 s « Ž .. « Ž .. T« Ž0, t .Ž w y w 1 . g X 2 Ž0. s RŽ Qy 0 . Hence w y w 1 g RŽ Qy t , and so « there exists a w 2 g RŽ Qy Ž t .. satisfying w s w 1 q w 2 . Hence X1Ž t . q X 2 Ž t . s X
Ž t F 0. .
When t G 0, « X 1 Ž t . s R Ž Pq Ž t.. « X 2 Ž t . s T« Ž t , 0 . w N w g R Ž Qy Ž 0. . 4 . «Ž . «Ž . Since T« Ž t, 0. is an isomorphism of RŽ Qy 0 onto RQy t , we have dim X 2 Ž t . s dim X 2 Ž0. \ m Ž t G 0.. Since
X1Ž 0. [ X 2 Ž 0. s X , we have m s codim X 1Ž0. s codim R Ž Pq« Ž0.. s codim RPq« Ž t . s dim X 1Ž t . Ž t G 0.. Hence we have X1Ž t . q X 2 Ž t . s X
Ž t G 0. .
So we have proved that X1Ž t . q X 2 Ž t . s X
Ž t g R. .
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From Ž2.15. we obtain X1Ž t . [ X 2 Ž t . s X
Ž t g R. .
Hence for « / 0 sufficiently small Eq. Ž2.1. admits an exponential dichotomy on R because Eq. Ž2.1. has an exponential dichotomy on Rq and «Ž . «Ž . Ry with projections Pq t and Py t , respectively. This completes the proof of Theorem 1.
3. THE TRANSVERSALITY OF HOMOCLINIC ORBITS In this section we consider the perturbed parabolic equation u ˙ q Au s g Ž u . q « h Ž t , u, « . ,
Ž 3.1.
where g g C k Ž X b, X ., k G 1, and A is a sectorial operator in X, « is a small parameter, and hŽ t, u, « . g C k Ž R = X b = R, X . is bounded for t g R. We have the following result. THEOREM 2. Let g g C k Ž X b, X . be defined in an open subset X of X b, such that the unperturbed equation u ˙ q Au s g Ž u . has a bounded solution r Ž t . with the bound of the orbit contained in X on R. Suppose the ¨ ariational equation along the orbit r Ž t . u ˙ q Au s g u Ž r Ž t . . u
Ž 3.2.
has an exponential dichotomy on Rq and Ry with projections Pq Ž t ., Py Ž t . satisfying dim RŽ Qq Ž0.. s dim RŽ Qy Ž0.. and ˙ r Ž t . is the unique, up to a scalar multiple, nontri¨ ial bounded solution, in X, of Eq. Ž3.2. on R then its adjoint equation u ˙ s A*x y g Uu Ž r Ž t . . u also has a unique, up to a scalar multiple, nontri¨ ial bounded solution c Ž t . g X on R. If the Melnikov function M Ž a . defined by MŽ a . s
q`
Hy` ² c Ž t . , h Ž t q a , r Ž t . , 0.: dt
has a simple zero a 0 , then there exists a continuously differentiable function a s a Ž « . such that for « / 0 sufficiently small Eq. Ž3.1. has a unique bounded solution uŽ t, « . on R satisfying uŽ t q a Ž « . , « . y r Ž t .
b
s O Ž < « <. .
EXPONENTIAL DICHOTOMIES
477
Moreover, for « / 0 sufficiently small the variational equation of Eq. Ž3.1. along the bounded solution uŽ t, « .
j˙q A j s g u Ž u Ž t , « . . q « h u Ž t , u Ž t , « . , « . 4 j
Ž 3.3.
admits an exponential dichotomy on R. Remark. Theorem 2 of this paper generalizes Theorem 3.3 of Blazquez w7x. Now we give an important application of Theorem 2. COROLLARY 3. Let the conditions of Theorem 2 for A, g, and f be satisfied, h is periodic in t with period T. In addition, we assume Eq. Ž3.2. has a hyperbolic equilibrium u 0 g X and a homoclinic orbit r Ž t . g X connecting u 0 Ž that is, 5 r Ž t . y u 0 5 b ª 0 as t ª"`.. Then for « sufficiently small Eq. Ž3.1. has a unique hyperbolic T-periodic solution u 0 Ž t, « . g X satisfying u 0 Ž t, 0. s u 0 and for « / 0 sufficiently small there exists a bounded solution u*Ž t, « . g X such that lim
< t <ª`
u* Ž t , « . y u 0 Ž t , « .
b
s0
and the ¨ ariational equation along u*Ž t, « .
j˙q A j s g u Ž u* Ž t , « . . q « h u Ž t , u* Ž t , « . , « . 4 j
Ž 3.4.
admits an exponential dichotomy on R. The proof of Corollary 3 follows from Theorem 2. Remark. The existence of the exponential dichotomy on R of Ž3.4. implies the bounded solution u*Ž t, « . is a transversal orbit homoclinic to the hyperbolic T-periodic solution u 0 Ž t, « . Žrefer to Blazquez w7x.. The transversality of the homoclinic orbit was not proved rigorously in Blazquez w7x. Moreover, if the conditions of Corollary 3 are satisfied Blazquez w7x proved that Eq. Ž3.1. admits a Bernoulli bundle under the condition of transversality. We can also show the existence of a Bernoulli bundle Žchaotic dynamics. by using the method of Meyer and Sell w2x. Proof of Theorem 2. The proof of the existence of the bounded solution uŽ t, « . is almost the same as the one in Blazquez w7x; we only give the outline of the proof because we will need the form of uŽ t, « . in proving that Eq. Ž3.4. has an exponential dichotomy on R. Let y Ž t . s uŽ t q a . y r Ž t ., where a g R is a new parameter. Then Eq. Ž3.1. reads
˙y q Ay s g u Ž r Ž t . . y q « h Ž t q a , y q r Ž t . , « . y g Ž r Ž t . . q g Ž y q r Ž t . . y g u Ž r Ž t . . y.
Ž 3.5.
478
ZENG WEIYAO
For simplicity, we let G Ž t , y, a , « . s « h Ž t q a , y q r Ž t . , « . y g Ž r Ž t . . q g Ž y q r Ž t . . y g u Ž r Ž t . . y.
Ž 3.6.
Then Eq. Ž3.5. can be written as
˙y q Ay s g u Ž r Ž t . . y q G Ž t , y, a , « . .
Ž 3.7.
Using the Liapunov]Schmidt method Žrefer to Blazquez w7x., we can prove that if the Melnikov function MŽ a . s
q`
Hy` ² c Ž t . , h Ž t q a , r Ž t . , 0.: dt
has a simple zero a s a 0 Žthat is, M Ž a 0 . s 0 / M9Ž a 0 .. then for « / 0 sufficiently small Eq. Ž3.5. has a unique bounded solution y Ž t, « . g X satisfying y Ž t, 0. s 0. Hence for « / 0 sufficiently small Eq. Ž3.1. has a bounded solution uŽ t, « . s y Ž t y a Ž « .. q r Ž t y a Ž « .. g X for t g R. Now we want to prove, by making use of Theorem 1, that for « / 0 sufficiently small Eq. Ž3.4. admits an exponential dichotomy on R. Equivalently, we want to prove that for « / 0 sufficiently small the equation
j˙q A j s g u Ž y Ž t , « . . q « h u Ž t q a Ž « . , y Ž t , « . q r Ž t . , « . j Ž 3.8. admits an exponential dichotomy on R. We let A Ž t , « . s yA q g u Ž y Ž t , « . q r Ž t . . q « h u Ž t q a Ž « . , y Ž t , « . q r Ž t . , « . . Then we have A Ž t , 0 . s yA q g u Ž r Ž t . . ,
Ž 3.9.
A« Ž t , 0 . s g uu Ž r Ž t . . y« Ž t , 0 . q h u Ž t , r Ž t . , 0 . .
Ž 3.10.
Since
˙y Ž t , « . q Ay Ž t , « . s g Ž y Ž t , « . q r Ž t . . q « h Ž t q a , y Ž t , « . qr Ž t . , « . y g Ž r Ž t . . , differentiating both sides of the above equation with respect to « and setting « s 0, we have
¨y« Ž t , 0 . q Ay« Ž t , 0 . s g u Ž r Ž t . . y« Ž t , 0 . q h Ž t q a 0 , r Ž t . , 0 . . Ž 3.11.
479
EXPONENTIAL DICHOTOMIES
Differentiating both sides of the above equation with respect to t, we obtain
¨y« Ž t , 0 . q Ay˙« Ž t , 0 . s g u Ž r Ž t . . ˙y« Ž t , 0 . q h t Ž t q a 0 , r Ž t . , 0 . q g uu Ž r Ž t . . y« Ž t , 0 . q h u Ž t q a 0 , r Ž t . , 0 . ˙ rŽ t. ,
Ž 3.12. hence ˙ y« Ž t, 0. is the bounded solution in X on R of the equation
˙y q Ay s g u Ž r Ž t . . y q h t Ž t q a 0 , r Ž t . , 0 . q g u u Ž r Ž t . . y« Ž t , 0 . q h u Ž t q a 0 , r Ž t . , 0 . ˙ rŽ t. . It follows from Lemma 1 that q`
Hy` ² c Ž t . , h Ž t q a t
0,
r Ž t . , 0.
q g u u Ž r Ž t . . y« Ž t , 0 . q h u Ž t q a 0 , r Ž t . , 0 . ˙ r Ž t .: dt s 0. Hence q`
Hy` ² c Ž t . , A Ž t , 0. ˙r Ž t .: dt «
s
q`
Hy` ² c Ž t . ,
g u u Ž r Ž t . . y« Ž t , 0 . q h u Ž t q a 0 , r Ž t . , 0 . ˙ r Ž t .: dt
q`
Hy` ² c Ž t . , h Ž t q a
sy
t
0,
r Ž t . , 0 .: dt s yM9 Ž a 0 . / 0,
so we follow from Theorem 1 that for « / 0 sufficiently small the variational equation Ž3.4. admits an exponential dichotomy on R. The proof of Theorem 2 is complete.
ACKNOWLEDGMENTS We thank Professor Sell, Professor Palmer, Professor Y. Yi, Professor Z. Xia, and the referees for their help.
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REFERENCES 1. K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 Ž1984., 225]256. 2. K. R. Meyer and G. R. Sell, Melnikov transforms, Bernoulli bundles and almost periodic perturbations, Trans. Amer. Math. Soc. 314 Ž1989., 63]105. 3. K. R. Meyer and G. R. Sell, An analytic proof of the shadowing lemma, Funkcial. Ek¨ ac. 30 Ž1987., 127]133. 4. J. K. Hale and X. B. Lin, Exponential dichotomies and homoclinic orbits in functional differential equations, J. Differential Equations 63 Ž1986., 227]254. 5. W. A. Coppel, ‘‘Dichotomies in Stability Theory,’’ Springer-Verlag, New York, 1978. 6. Z. Weiyao, Exponential dichotomies and transversal homoclinic orbits in degenerate cases, J. Dynamics Differential Equations 7 Ž1995.. 7. C. Miguel Blazquez, Transverse homoclinic orbits in periodically perturbed parabolic equations, Nonlinear Anal. 10 Ž1986., 1277]1291. 8. V. K. Melnikov, On the stability of the center for time-periodic perturbations, Trans. Moscow Math. Soc. 12 Ž1964., 1]57. 9. S. Wiggins, ‘‘Global Bifurcations and Chaos,’’ Appl. Math. Sci., Vol. 73, Springer-Verlag, New York, 1988. 10. J. K. Hale and S. N. Chow, ‘‘Methods of Bifurcations,’’ Springer-Verlag, New YorkrBerlin, 1982. 11. K. J. Palmer, Transversal heteroclinic orbits and Cherry’s example of a non-integrable Hamiltonian systems, J. Differential Equations 65 Ž1986., 321]360. 12. S. N. Chow, J. K. Hale, and J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations 37 Ž1980., 351]373. 13. D. Henry, ‘‘Geometric Theory of Semilinear Parabolic Equations,’’ Lectures Notes in Mathematics, Vol. 840, Springer-Verlag, New YorkrBerlin, 1981. 14. Z. Weinian, The Fredholm alternative and exponential dichotomies for parabolic equations, J. Math. Anal. Appl. 191 Ž1995., 180]201. 15. F. Battelli, Heteroclinic orbits in singular systems: A unifying approach, J. Dynamics Differential Equations 6 Ž1994., 147]173. 16. R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations 113 Ž1994., 17]67. 17. S. N. Chow and H. Leiva, Existence and roughness of the exponential dichotomies for skew-product semiflow in Banach spaces, J. Differential Equations 120 Ž1995., 429]477. 18. B. Sandstede, ‘‘Verzweigungstheorie homokliner Verdopplungen,’’ Doctoral thesis, University of Stuttgart, 1993. 19. R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear systems, I, J. Differential Equations 15 Ž1974., 429]458. 20. W. Shen and Y. Yi, Asymptotic almost periodicity of scalar parabolic equations with almost periodic time dependence, J. Differential Equations 122 Ž1979., 373]379. 21. W. Shen and Y. Yi, On minimal sets of scalar parabolic equations with skew-product structures, Trans. Amer. Math. Soc. 347 Ž1995., 4413]4431. 22. J. K. Hale, L. T. Magalhaes, and W. M. Oliva, ‘‘An Introduction to Infinite Dimensional Dynamical Systems}Geometric Theory,’’ Springer-Verlag, New YorkrBerlin, 1984. 23. B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Differential Equations 125 Ž1996., 239]281.
216, 466]480 Ž1997.
AY975662
Transversality of Homoclinic Orbits and Exponential Dichotomies for Parabolic Equations* Zeng Weiyao† Research Institute of Sciences, Changsha Railway Uni¨ ersity, Changsha, Hunan, 410075, People’s Republic of China Submitted by William F. Ames Received December 10, 1996
In this paper we discuss the existence of exponential dichotomies on R of linear parabolic equations depending on small parameters and provide a tool of proving the transversality of the homoclinic orbits of parabolic equations, and by making use of the results on exponential dichotomies of this paper, we investigate the transversality of homoclinic orbits for parabolic equations. Q 1997 Academic Press
1. INTRODUCTION Let F be a diffeomorphism on a two dimensional manifold. It is well known that the existence of one transversal homoclinic point for F implies the existence of the Smale horseshoe Žthat is, the chaotic dynamics. from the Smale]Birkhoff homoclinic theorem. Recently, Palmer w1x and Meyer and Sell w2, 3x have shown that the same results in R n can be obtained by using elementary ideas from the theory of exponential dichotomies and the shadowing lemma. In the infinite case the problem was considered by Hale and Lin w4x for functional differential equations and by Blazquez w7x for parabolic equations. It is well known that one of the best tools for detecting the existence of transversal homoclinic orbits of differential equations is the Melnikov method. The Melnikov method has been extended since the paper of Melnikov w8x, therefore we have a vast literature. We refer to Wiggins w9x, Hale and Chow w10x, Palmer w1, 11x, Zeng w6x, Meyer and Sell w2x, Hale and Lin w4x, and Blazquez w7x. * This work is partially supported by NSF of China. † E-mail address: [email protected]. 466 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
EXPONENTIAL DICHOTOMIES
467
Palmer w1x used the Liapunov]Schmidt method and the theory of exponential dichotomies to generalize the Melnikov method to ndimensional non-Hamiltonian systems and gave the relation between the transversality of the homoclinic orbits and the existence of exponential dichotomies, on R, of the variational equations along the homoclinic orbits. Palmer w11x used the Melnikov integral to provide a sufficient condition assuring the existence of exponential dichotomies, on R, of the linear systems depending on a small parameter « , and then used the result to prove the transversality of the homoclinic orbits. The result in Palmer w11x on the theory of exponential dichotomies becomes a standard method of proving the transversality of homoclinic orbits of ordinary differential equations Žrefer to Zeng w6x and Battelli w15x.. Zeng w6x generalized the results of Palmer w1, 11x. Later Blazquez w7x extended the results in Palmer w1x and Hale and Lin w4x to the parabolic equations. But Blazquez w7x did not prove the result on the theory of exponential dichotomies for parabolic equations analogous to that in Palmer w11x on the existence of exponential dichotomies on R of the linear variational equations depending on small parameters, so the transversality of the homoclinic orbits was not proved rigorously in Blazquez w7x. The main purpose of this paper is to generalize the result in Palmer w11x on the existence of the exponential dichotomies depending on small parameters on R to the linear parabolic equations by using the Liapunov]Schmidt method and the Melnikov integral. Also we use the results we obtained on the exponential dichotomies to show rigorously the transversality of the homoclinic orbits for parabolic equations. We want to use the following notations. We let LŽ X b, X . s L: X b ª X, is a linear operator of X b to X 4 ; LŽ X . s L: X ª X, is a linear operator of X to X 4 ; Cb0 Ž R, X . s u: R ª X, is continuous and bounded in X for t g R4 ; Cb1 Ž R, X . s u: R ª X, is continuously differentiable and u, u ˙ are bounded in X for t g R4. Cbk Ž R, X . has the same meaning. For any operator L we denote by RŽ L. the range of L. In this paper, the definitions of the solution of parabolic equations in X, and the derivative of functions in X are the same as in Henry w13x. If A is a sectorial operator in a Banach space X then there is a real number a such that A1 s A q aI implies Re s Ž A1 . ) 0. Let 0 F b F 1. One can define a fractional power A1b of A1. Let X b s DŽ A1b . with the graph norm 5 x 5 b s 5 A1b x 5 for x g X b Žrefer to Henry w13x.. The space X b is a Banach space with the norm 5 5 b for 0 F b F 1 and X b is a dense subspace of X r with continuous inclusion for 0 F r F b F 1 Žrefer to w13, Theorem 1.4.8x.. Suppose A is a sectorial operator on X. If t ª B Ž t .: w t 0 , t 1 x ª LŽ X b, X . is Holder continuous, then, for any u 0 g X b ¨ and t 0 F t F t 1 , there exists a unique solution uŽ t . s uŽ t, t , u 0 . of the
468
ZENG WEIYAO
equation u ˙ q Au s B Ž t . u on t F t F t 1 , uŽt . s u 0 and u 0 ª uŽ t, t , u 0 . is linear and bounded in X. We can define an evolution operator T Ž t, t . N t 0 F t F t F t 14 by uŽ t, t , u 0 . s T Ž t, t . u 0 , t G t . Suppose the evolution operator T Ž t, s . g LŽ X . for the equation u ˙ q AŽ t . u s 0
Ž 1.1.
is defined on a real interval J. Equation Ž1.1. is said to have an exponential dichotomy on J with the exponent b ) 0 and the bound M if there exist projections P Ž t ., QŽ t . s I y P Ž t ., t g J, such that Ža. Žb. Ž Ž R Q t ... Žc. Žd.
T Ž t, s . P Ž s . s P Ž t .T Ž t, s ., t G s in J. The restriction T Ž t, s .< RŽQŽ s.. , t G s is an isomorphism onto We define the inverse map from RŽ QŽ t .. to RŽ QŽ s .. by T Ž s, t .. 5 T Ž t, s . P Ž s .5 F Meyb Ž tys. for t G s in J. 5 T Ž t, s . QŽ s .5 F Meyb Ž syt . for s G t in J.
The theory of exponential dichotomies plays a very important role in studying the global bifurcations; we refer to Palmer w1x, Meyer and Sell w2x, Zeng w6x, and Sandstede w18x. On the theory of exponential dichotomies for finite dimensional systems, we refer to Palmer w1x, Meyer and Sell w2x, Coppel w5x, and Sacker and Sell w19x; for infinite dimensional systems, we refer to Sackel and Sell w16x, Henry w13x, and Chow and Leiva w17x. On the other related results on dynamics of parabolic equations, we refer to Shen and Yi w21, 22x and Fiedler and Rocha w23x. If A is a sectorial operator, AŽ t . y A: R ª LŽ X b, X . is locally Holder ¨ continuous, and T Ž t, s . g LŽ X b, X . is the evolution operator, by Henry’s Theorem 7.3.1 in w13x, we can define the adjoint operator T *Ž s, t . s T Ž t, s .* by
² y, T Ž t , s . x: s² T * Ž s, t . y, x: ,
x g X , y g X *,
Ž 1.2.
and T *Ž s, t . g LŽ X *., locally continuous in s - t Žbut weak*-continuous at s s t ., such that y Ž s . s T *Ž s, t 0 . y 0 satisfies the equation adjoint to Eq. Ž1.1. dy ds
Ž s . s A* Ž s . y Ž s . .
Ž 1.3.
If Eq. Ž1.1. has an exponential dichotomy on Rq and Ry with projections Pq Ž t . and Py Ž t ., respectively, then Hale and Lin w4x and Blazquez w7x proved that the adjoint equation Ž1.3. also has an exponential dichotomy
EXPONENTIAL DICHOTOMIES
469
on Rq and Ry with the same constants and with projections I y Ž Pq Ž t ..* and I y Ž Py Ž t ..*, respectively. Let O be an open subset of X b and let f : R = O ª X be a continuous T-periodic function in t with continuous partial derivatives f uŽ t, u.. We denote by F Ž t, j . the unique solution of the equation u ˙ q Au s f Ž t , u .
Ž 1.4.
with F Ž0, j . s j and by F: X b ª X b FŽ j . s FŽT , j . the period map for Eq. Ž1.4.. Blazquez w7x proved the following result: y g O is a transversal homoclinic point with respect to a hyperbolic fixed point j 0 if and only if FŽ t, y. y FŽ t, j0 . ª 0
as < t < ª `
Ž 1.5.
and the variational equation u ˙ q Au s f u Ž t , F Ž t , y . . u
Ž 1.6.
has an exponential dichotomy on R. So in studying the transversality of homoclinic orbits which bifurcate from the nontransversal homoclinic orbits of parabolic equations, we want to investigate the existence of exponential dichotomies on R of the linear parabolic equation depending on a small parameter u ˙ q AŽ t , « . u s 0,
Ž 1.7.
where AŽ t, « . y A: R = Žyl 0 , l 0 . ª LŽ X b, X . is locally Holder and A is a ¨ sectorial operator and l 0 ) 0 a sufficiently small constant. We want to consider the following problem: Suppose when « s 0 the linear parabolic equation u ˙ q AŽ t , 0. u s 0
Ž 1.8.
admits an exponential dichotomy on both Rq and Ry but does not have exponential dichotomies on R. Under what conditions can we guarantee that the linear parabolic equation Ž1.7. has an exponential dichotomy on R for « / 0 sufficiently small? In this paper, we want to solve the above problem and to use the result to prove the transversality of the homoclinic orbits of parabolic equations. The result in Palmer w11x is generalized to the parabolic equations. This paper is organized as follows: Section 1 is an introduction. In Section 2 we give a sufficient condition which assures the
470
ZENG WEIYAO
existence of exponential dichotomies on R for Eq. Ž1.7. for « / 0 sufficiently small. In Section 3 we apply the result in Section 2 to investigate the transversality of the homoclinic orbits.
2. THE EXISTENCE OF EXPONENTIAL DICHOTOMIES In this section, we mainly study the existence of exponential dichotomies on R of the linear parabolic equation depending on a small parameter u ˙ q AŽ t , « . u s 0,
Ž 2.1.
where x g X b, AŽ t, « . y A: R = Žyl 0 , l 0 . ª LŽ X b, X . is locally Holder ¨ continuous, A is a sectorial operator, « is a small parameter, and 0 F b F 1. We need a lemma. LEMMA 1. If A is a sectorial operator, AŽ t . y A: R ª LŽ X b, X . is locally Holder continuous, such that the equation ¨ u ˙ q AŽ t . u s 0
Ž 2.2.
has an exponential dichotomy on Rq and Ry with projections Pq Ž t ., Py Ž t ., respecti¨ ely, then for the linear operator L: Cb1 Ž R, X b . ª Cb0 Ž R, X . defined by
Ž Lu . Ž t . s u˙Ž t . q A Ž t . u Ž t . we ha¨ e Ž a. L is a Fredholm operator. Ž b . N Ž L. s g g Cb1 Ž R, X .: ˙ g Ž t . q AŽ t . g Ž t . s 0, g Ž0. g RŽ Pq Ž0.. l RŽ I y Py Ž0..4 ; Ž c . RŽ L. s f g Cb0 Ž R, X .: Hq` ² Ž . Ž .: dt s 0 for all c g y` c t , f t 1Ž b . Cb R, X * satisfying the adjoint equation u ˙ y A*Ž t . u s 04 and indŽ L. s dim RQy Ž0. y dim RQq Ž0.. Lemma 1 is due to Blazquez w7x, also refer to Palmer w1x, Hale and Lin w4x, Weinian w14x, Sacker and Sell w16x, and Chow and Leiva w17x. For Eq. Ž2.1., we assume ŽH1. The derivatives AŽ t, « . and A« Ž t, « . exist continuous and bounded on R = Žyl 0 , l 0 ., where l 0 ) 0 a sufficient small constant. We let I s Žyl 0 , l 0 ..
EXPONENTIAL DICHOTOMIES
471
ŽH2. The equation u ˙ q AŽ t , 0. u s 0
Ž 2.3.
admits an exponential dichotomy on Rq and Ry with projections Pq Ž t ., Py Ž t ., and dim RŽ Qy Ž0.. s dim RŽ Qq Ž0.., respectively, and dim RŽ Qq Ž t .. and dim RŽ Qy Ž t .. are finite, and has a unique, up to a scalar multiple, nontrivial bounded solution w Ž t . on R in X. For the linear equation Ž2.1. we define an operator L« by L« : Cb1 Ž R, X b . ª Cb0 Ž R , X .
Ž Lu . Ž t . s u˙Ž t . q A Ž t , « . u Ž t . . We say the operator L« is the operator associated with the linear parabolic equation Ž2.1.. Since Eq. Ž2.3. admits an exponential dichotomy on Rq and Ry with projections Pq Ž t ., Py Ž t . with dim RŽ Qy Ž0.. s dim RŽ Qq Ž0.., we have that L0 is a Fredholm operator and indŽ L0 . s 0 from Lemma 1. It follows from the roughness of exponential dichotomies Žrefer to Hale and Lin w4x, Chow and Leiva w17x, and Sacker and Sell w16x., that for « / 0 sufficiently small, Eq. Ž2.1. also has an exponential dichotomy on Rq and «Ž . «Ž . Ry with projections Pq t and Py t , respectively. So it follows from Lemma 1 that the operator L« is Fredholm and indŽ L« . s 0 because the index of the Fredholm operator persists under small perturbations. Since ind Ž L0 . s 0 s dim N Ž L0 . y codim R Ž L0 . s dim R Ž Pq Ž 0 . . l R Ž I y Py Ž 0 . . y dim R Ž Py Ž 0 . * . l R Ž I y Pq Ž 0 . * . , from condition ŽH2., we have dim R Ž Py Ž 0 . * . l R Ž I y Pq Ž 0 . * . s dim R Ž Pq Ž 0 . . l R Ž I y Py Ž 0 . . s 1. Since RŽ Py Ž0.*. l RŽ I y Pq Ž0.*. and RŽ Pq Ž0.. l RŽ I y Py Ž0.. are the spaces of the initial values of bounded solutions on R in X of Eq. Ž2.3. and the adjoint equation of Eq. Ž2.3., respectively, the adjoint equation of Eq. Ž2.3. u ˙Ž t . y A* Ž t , 0 . u Ž t . s 0
Ž 2.4.
also has and only has a unique, up to a scalar multiple, nontrivial bounded solution, denoted by c Ž t . g X for t g R.
472
ZENG WEIYAO
Now we consider Eq. Ž2.1.. We let T« Ž t, s . g LŽ X . be the evolution operator for Eq. Ž2.1.. The main result of this paper is as follows: THEOREM 1. Suppose A is a sectorial operator, AŽ t, « . y A: R = Žyl 0 , l 0 . ª LŽ X b, X . is locally Holder ¨ continuous, and conditions Ž H 1. and Ž H 2. are satisfied. If q`
Hy` ² c Ž t . , A Ž t , 0. w Ž t .: dt / 0 «
then for « / 0 sufficiently small Eq. Ž2.1. u ˙ q AŽ t , « . u s 0 admits an exponential dichotomy on R. Remark. Theorem 1 generalizes the result of Palmer w11x to the infinite dimensional systems. Proof of Theorem 1. For any f g Cb0 Ž R, X ., we consider the equation u ˙ q AŽ t , « . u s « 2 f Ž t . .
Ž 2.5.
We assume, without loss of generality, q`
Hy`
c 2 Ž t . dt s 1.
We make a change of variable y Ž t . s uŽ t . y w Ž t .g for Eq. Ž2.5. where g g R is a new parameter. Then Eq. Ž2.5. becomes
˙y q A Ž t , 0 . y s AŽ t , 0 . y AŽ t , « .
y q w Ž t . g q « 2 f Ž t . . Ž 2.6.
Since the linear parabolic equation ˙ y q AŽ t, 0. y s 0 admits an exponential dichotomy on both Rq and Ry, using the Liapunov]Schmidt method Žrefer to Lemma 1., we see that Eq. Ž2.6. is equivalent to the two equations
˙y q AŽ t , 0 . y s AŽ t , 0 . y AŽ t , « . q c Ž t.
q`
y q wŽ t.g q « 2 f Ž t.
Hy` ² c Ž t . ,
AŽ t , 0. y AŽ t , « .
y q wŽ t.g
q« 2 f Ž t .: dt Ž 2.7. q`
Hy` ² c Ž t . ,
AŽ t , 0. y AŽ t , « .
y q w Ž t . g q « 2 f Ž t .: dt s 0. Ž 2.8.
We can prove that for « sufficiently small Eq. Ž2.7. has a unique solution y s y Ž t, g , « . g Cb1 Ž R, X b . for t g R, and satisfies y Ž t, g , 0. s 0. More-
473
EXPONENTIAL DICHOTOMIES
over, y Ž t, g , « . is second order continuously differentiable in Žg , « .. Substituting y s y Ž t, g , « . into Eq. Ž2.8., we obtain the bifurcative equation BŽg , « . s
q`
Hy` ² c Ž t . ,
AŽ t , 0. y AŽ t , « .
yŽ t, g , « . q w Ž t.g q« 2 f Ž t .: dt. Ž 2.9.
Since y Ž t, g , 0. s 0 and y Ž t, g , « . is second order continuously differentiable in g , « , we have that B Žg , « . is also second order continuously differentiable in g , « and B Ž g , 0 . s 0. We define a function H Žg , « . by
¡B Ž g , « . , ¢B Ž g«, 0. ,
H Ž g , « . s~
«
«/0 « s 0.
Since B Žg , 0. s 0 and B Žg , « . is second order continuously differentiable in g , « , the function H Žg , « . is continuously differentiable in g , « . Then H Ž g , 0 . s B« Ž g , 0 . q`
s
Hy` ² c Ž t . , A Ž t , 0.
s
Hy` ² c Ž t . , A Ž t , 0.
«
q`
«
y Ž t , g , 0. q w Ž t . g
wŽ t.
: dt
: dt ? g
Hence H Ž 0, 0 . s 0 and Hg Ž 0, 0 . s
q`
Hy` ² c Ž t . , A Ž t , 0. w Ž t .: dt. «
Since q`
Hy` ² c Ž t . , A Ž t , 0. w Ž t .: dt / 0, «
it follows from the implicit theorem that there exists a continuous function g s g Ž « ., satisfying g Ž0. s 0, such that HŽg Ž « . , « . s 0
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ZENG WEIYAO
holds for « sufficiently small. Hence for « / 0 sufficiently small the equation u ˙ q AŽ t , « . u s « 2 f Ž t .
Ž 2.10.
has a bounded solution uŽ t, « . s y Ž t, g Ž « ., « . q w Ž t .g Ž « . g Cb1 Ž R, X b .. So for any f g Cb0 Ž R, X . and « / 0 sufficiently small the equation u ˙ q AŽ t , « . u s f Ž t .
Ž 2.11.
has a bounded solution uŽ t, « .r« 2 s w y Ž t, g Ž « ., « . q w Ž t .g Ž « .xr« 2 g Cb1 Ž R, X b .. Thus we have for, « / 0 sufficiently small, RŽ L« . s Cb0 Ž R, X ., that is, codim RŽ L« . s 0. Since indŽ L« . s 0 for « sufficiently small, we have 0 s dim N Ž L« . y codim R Ž L« . s dim N Ž L« . and hence « N Ž L« . s R Ž Pq Ž 0 . . l R Ž Qy« Ž 0 . . s 0 4 .
From condition ŽH2. we see « dim Ž R Ž Pq Ž 0 . . q R Ž Qy« Ž 0 . . . « s dim R Ž Pq Ž 0 . . q dim R Ž Qy« Ž 0 . . y dim Ž R Ž Pq« Ž 0 . . l R Ž Qy« Ž 0 . . . « s dim R Ž Pq Ž 0 . . q dim R Ž Qy« Ž 0 . . « s dim R Ž Pq Ž 0 . . q dim R Ž Qq« Ž 0 . .
s n. Hence « R Ž Pq Ž 0 . . q R Ž Qy« Ž 0 . . s X .
Ž 2.12.
We define two closed subsets, X 1Ž t . and X 2 Ž t ., of X for « / 0 sufficiently small by X1Ž t . s
X2 Ž t . s
½ ½
« R Ž Pq Ž t..,
w N T« Ž 0, t . w g R Ž
tG0 « Pq
Ž 0. . 4 ,
« R Ž Qy Ž t..,
T« Ž t , 0. w N w g R Ž
tF0 tF0
« Qy
Ž 0. . 4 ,
t G 0.
Then from Ž2.12. we have X1Ž 0. [ X 2 Ž 0. s X .
Ž 2.13.
EXPONENTIAL DICHOTOMIES
475
We claim that X1Ž t . [ X 2 Ž t . s X .
Ž 2.14.
Now we want to prove Ž2.14. by the following two steps. Step 1. We show X 1Ž t . l X 2 Ž t . s 04 for t g R. When t G 0, if w g X 1Ž t . l X 2 Ž t ., then there exists w 0 g X 2 Ž0. such « Ž .. « Ž .. that w s T« Ž t, 0. w 0 g RŽ Pq t . Hence w 0 g RŽ Pq 0 . So w 0 g X 1Ž0. l X 2 Ž0.. From Ž2.13., we have w 0 s 0 and so w s 0. When t F 0, if w g X 1Ž t . l X 2 Ž t ., then T« Ž0, t . w g X 1Ž0. l X 2 Ž0.. Thus, T« Ž0, t . w s 0. Since T« Ž0, t . is an isomorphism form X 2 Ž t . to X 2 Ž0. we have w s 0. Hence for t g R we have X1Ž t . l X 2 Ž t . s 04 .
Ž 2.15.
Step 2. We show X 1Ž t . q X 2 Ž t . s X for t g R. When t F 0, since Ž2.13. holds, for any w g X, we have T Ž 0, t . w s w 1 q w 2 ,
where w 1 g X 1 Ž 0 . and w 2 g X 2 Ž 0 . .
Hence there exists a w 1 g X 1Ž t . such that w 1 s T« Ž0, t . w 1 , and so w 2 s « Ž .. « Ž .. T« Ž0, t .Ž w y w 1 . g X 2 Ž0. s RŽ Qy 0 . Hence w y w 1 g RŽ Qy t , and so « there exists a w 2 g RŽ Qy Ž t .. satisfying w s w 1 q w 2 . Hence X1Ž t . q X 2 Ž t . s X
Ž t F 0. .
When t G 0, « X 1 Ž t . s R Ž Pq Ž t.. « X 2 Ž t . s T« Ž t , 0 . w N w g R Ž Qy Ž 0. . 4 . «Ž . «Ž . Since T« Ž t, 0. is an isomorphism of RŽ Qy 0 onto RQy t , we have dim X 2 Ž t . s dim X 2 Ž0. \ m Ž t G 0.. Since
X1Ž 0. [ X 2 Ž 0. s X , we have m s codim X 1Ž0. s codim R Ž Pq« Ž0.. s codim RPq« Ž t . s dim X 1Ž t . Ž t G 0.. Hence we have X1Ž t . q X 2 Ž t . s X
Ž t G 0. .
So we have proved that X1Ž t . q X 2 Ž t . s X
Ž t g R. .
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ZENG WEIYAO
From Ž2.15. we obtain X1Ž t . [ X 2 Ž t . s X
Ž t g R. .
Hence for « / 0 sufficiently small Eq. Ž2.1. admits an exponential dichotomy on R because Eq. Ž2.1. has an exponential dichotomy on Rq and «Ž . «Ž . Ry with projections Pq t and Py t , respectively. This completes the proof of Theorem 1.
3. THE TRANSVERSALITY OF HOMOCLINIC ORBITS In this section we consider the perturbed parabolic equation u ˙ q Au s g Ž u . q « h Ž t , u, « . ,
Ž 3.1.
where g g C k Ž X b, X ., k G 1, and A is a sectorial operator in X, « is a small parameter, and hŽ t, u, « . g C k Ž R = X b = R, X . is bounded for t g R. We have the following result. THEOREM 2. Let g g C k Ž X b, X . be defined in an open subset X of X b, such that the unperturbed equation u ˙ q Au s g Ž u . has a bounded solution r Ž t . with the bound of the orbit contained in X on R. Suppose the ¨ ariational equation along the orbit r Ž t . u ˙ q Au s g u Ž r Ž t . . u
Ž 3.2.
has an exponential dichotomy on Rq and Ry with projections Pq Ž t ., Py Ž t . satisfying dim RŽ Qq Ž0.. s dim RŽ Qy Ž0.. and ˙ r Ž t . is the unique, up to a scalar multiple, nontri¨ ial bounded solution, in X, of Eq. Ž3.2. on R then its adjoint equation u ˙ s A*x y g Uu Ž r Ž t . . u also has a unique, up to a scalar multiple, nontri¨ ial bounded solution c Ž t . g X on R. If the Melnikov function M Ž a . defined by MŽ a . s
q`
Hy` ² c Ž t . , h Ž t q a , r Ž t . , 0.: dt
has a simple zero a 0 , then there exists a continuously differentiable function a s a Ž « . such that for « / 0 sufficiently small Eq. Ž3.1. has a unique bounded solution uŽ t, « . on R satisfying uŽ t q a Ž « . , « . y r Ž t .
b
s O Ž < « <. .
EXPONENTIAL DICHOTOMIES
477
Moreover, for « / 0 sufficiently small the variational equation of Eq. Ž3.1. along the bounded solution uŽ t, « .
j˙q A j s g u Ž u Ž t , « . . q « h u Ž t , u Ž t , « . , « . 4 j
Ž 3.3.
admits an exponential dichotomy on R. Remark. Theorem 2 of this paper generalizes Theorem 3.3 of Blazquez w7x. Now we give an important application of Theorem 2. COROLLARY 3. Let the conditions of Theorem 2 for A, g, and f be satisfied, h is periodic in t with period T. In addition, we assume Eq. Ž3.2. has a hyperbolic equilibrium u 0 g X and a homoclinic orbit r Ž t . g X connecting u 0 Ž that is, 5 r Ž t . y u 0 5 b ª 0 as t ª"`.. Then for « sufficiently small Eq. Ž3.1. has a unique hyperbolic T-periodic solution u 0 Ž t, « . g X satisfying u 0 Ž t, 0. s u 0 and for « / 0 sufficiently small there exists a bounded solution u*Ž t, « . g X such that lim
< t <ª`
u* Ž t , « . y u 0 Ž t , « .
b
s0
and the ¨ ariational equation along u*Ž t, « .
j˙q A j s g u Ž u* Ž t , « . . q « h u Ž t , u* Ž t , « . , « . 4 j
Ž 3.4.
admits an exponential dichotomy on R. The proof of Corollary 3 follows from Theorem 2. Remark. The existence of the exponential dichotomy on R of Ž3.4. implies the bounded solution u*Ž t, « . is a transversal orbit homoclinic to the hyperbolic T-periodic solution u 0 Ž t, « . Žrefer to Blazquez w7x.. The transversality of the homoclinic orbit was not proved rigorously in Blazquez w7x. Moreover, if the conditions of Corollary 3 are satisfied Blazquez w7x proved that Eq. Ž3.1. admits a Bernoulli bundle under the condition of transversality. We can also show the existence of a Bernoulli bundle Žchaotic dynamics. by using the method of Meyer and Sell w2x. Proof of Theorem 2. The proof of the existence of the bounded solution uŽ t, « . is almost the same as the one in Blazquez w7x; we only give the outline of the proof because we will need the form of uŽ t, « . in proving that Eq. Ž3.4. has an exponential dichotomy on R. Let y Ž t . s uŽ t q a . y r Ž t ., where a g R is a new parameter. Then Eq. Ž3.1. reads
˙y q Ay s g u Ž r Ž t . . y q « h Ž t q a , y q r Ž t . , « . y g Ž r Ž t . . q g Ž y q r Ž t . . y g u Ž r Ž t . . y.
Ž 3.5.
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ZENG WEIYAO
For simplicity, we let G Ž t , y, a , « . s « h Ž t q a , y q r Ž t . , « . y g Ž r Ž t . . q g Ž y q r Ž t . . y g u Ž r Ž t . . y.
Ž 3.6.
Then Eq. Ž3.5. can be written as
˙y q Ay s g u Ž r Ž t . . y q G Ž t , y, a , « . .
Ž 3.7.
Using the Liapunov]Schmidt method Žrefer to Blazquez w7x., we can prove that if the Melnikov function MŽ a . s
q`
Hy` ² c Ž t . , h Ž t q a , r Ž t . , 0.: dt
has a simple zero a s a 0 Žthat is, M Ž a 0 . s 0 / M9Ž a 0 .. then for « / 0 sufficiently small Eq. Ž3.5. has a unique bounded solution y Ž t, « . g X satisfying y Ž t, 0. s 0. Hence for « / 0 sufficiently small Eq. Ž3.1. has a bounded solution uŽ t, « . s y Ž t y a Ž « .. q r Ž t y a Ž « .. g X for t g R. Now we want to prove, by making use of Theorem 1, that for « / 0 sufficiently small Eq. Ž3.4. admits an exponential dichotomy on R. Equivalently, we want to prove that for « / 0 sufficiently small the equation
j˙q A j s g u Ž y Ž t , « . . q « h u Ž t q a Ž « . , y Ž t , « . q r Ž t . , « . j Ž 3.8. admits an exponential dichotomy on R. We let A Ž t , « . s yA q g u Ž y Ž t , « . q r Ž t . . q « h u Ž t q a Ž « . , y Ž t , « . q r Ž t . , « . . Then we have A Ž t , 0 . s yA q g u Ž r Ž t . . ,
Ž 3.9.
A« Ž t , 0 . s g uu Ž r Ž t . . y« Ž t , 0 . q h u Ž t , r Ž t . , 0 . .
Ž 3.10.
Since
˙y Ž t , « . q Ay Ž t , « . s g Ž y Ž t , « . q r Ž t . . q « h Ž t q a , y Ž t , « . qr Ž t . , « . y g Ž r Ž t . . , differentiating both sides of the above equation with respect to « and setting « s 0, we have
¨y« Ž t , 0 . q Ay« Ž t , 0 . s g u Ž r Ž t . . y« Ž t , 0 . q h Ž t q a 0 , r Ž t . , 0 . . Ž 3.11.
479
EXPONENTIAL DICHOTOMIES
Differentiating both sides of the above equation with respect to t, we obtain
¨y« Ž t , 0 . q Ay˙« Ž t , 0 . s g u Ž r Ž t . . ˙y« Ž t , 0 . q h t Ž t q a 0 , r Ž t . , 0 . q g uu Ž r Ž t . . y« Ž t , 0 . q h u Ž t q a 0 , r Ž t . , 0 . ˙ rŽ t. ,
Ž 3.12. hence ˙ y« Ž t, 0. is the bounded solution in X on R of the equation
˙y q Ay s g u Ž r Ž t . . y q h t Ž t q a 0 , r Ž t . , 0 . q g u u Ž r Ž t . . y« Ž t , 0 . q h u Ž t q a 0 , r Ž t . , 0 . ˙ rŽ t. . It follows from Lemma 1 that q`
Hy` ² c Ž t . , h Ž t q a t
0,
r Ž t . , 0.
q g u u Ž r Ž t . . y« Ž t , 0 . q h u Ž t q a 0 , r Ž t . , 0 . ˙ r Ž t .: dt s 0. Hence q`
Hy` ² c Ž t . , A Ž t , 0. ˙r Ž t .: dt «
s
q`
Hy` ² c Ž t . ,
g u u Ž r Ž t . . y« Ž t , 0 . q h u Ž t q a 0 , r Ž t . , 0 . ˙ r Ž t .: dt
q`
Hy` ² c Ž t . , h Ž t q a
sy
t
0,
r Ž t . , 0 .: dt s yM9 Ž a 0 . / 0,
so we follow from Theorem 1 that for « / 0 sufficiently small the variational equation Ž3.4. admits an exponential dichotomy on R. The proof of Theorem 2 is complete.
ACKNOWLEDGMENTS We thank Professor Sell, Professor Palmer, Professor Y. Yi, Professor Z. Xia, and the referees for their help.
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ZENG WEIYAO
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