Smale Horseshoes and Symbolic Dynamics in Perturbed...

J. Nonlinear Sci. Vol. 9: pp. 363–415 (1999)

© 1999 Springer-Verlag New York Inc.

Smale Horseshoes and Symbolic Dynamicsin Perturbed Nonlinear Schrodinger Equations

Y. Li ∗

Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90024,USA

Received September 16, 1996; revised September 23, 1997; final version received May 12, 1998Communicated by Stephen Wiggins

Summary. In [12], we gave an intensive study on the level sets of the integrable cubicnonlinear Schrödinger (NLS) equation. Based upon that study, the existence of a sym-metric pair of homoclinic orbits in certain perturbed NLS systems was established in[11]. [Stated in Theorem 1.3 below.] In this paper, the existence of Smale horseshoesand symbolic dynamics is established in the neighborhood of the symmetric pair of ho-moclinic orbits, under certain conditions (A1)–(A3), which are “except one point”–typeconditions. [Stated in Theorem 8.1.] More specifically, a list of compact Cantor sets isconstructed through a study of the Conley-Moser conditions, each of which consists ofpoints, and is invariant under the Poincaré map induced by the flow. More importantly,the Poincaré map restricted to each of the Cantor sets is topologically conjugate to theshift automorphism on four symbols. This gives rise to deterministicchaos, which of-fers an interpretation of the numerical observation on the perturbed NLS system: chaoticcenter-wing jumping, of course under the “except one point”–type conditions (A1)–(A3).This study is a generalization of the finite-dimensional study [14] to infinite-dimensionalperturbed NLS systems.

1. Introduction

In recent years, chaos in near-integrable PDEs has drawn a huge amount of numericaland analytical studies. For a beautiful survey, see [16]. The importance of such studies isreflected through the following facts: First, studying chaos in PDEs is very important inconnection with understanding intrinsic stochastic phenomena in physical systems. Sec-ond, integrable PDEs themselves are canonical ideal models of physical phenomena (forexample, nonlinear wave motions). They are the so-called soliton equations. Third, the

∗ Present Address: Department of Mathematics, 2-336, Massachusetts Institute for Technology, Cambridge,MA 02139, USA. Also, School of Mathematics, Institute of Advanced Study, Princeton, NJ 08540, USA.

364 Y. Li

complete mathematical theories on integrable PDEs are available. This enables rigorousmathematical studies on chaos in near-integrable PDEs. Fourth, studying chaos in near-integrable PDEs is natural in view of the fact that integrable PDEs are infinite-dimensionalHamiltonian systems, and there have been many studies on chaos in perturbed finite-dimensional Hamiltonian systems. Fifth, chaos in near-integrable PDEs is an exciting andpromising area that involves PDE theory, dynamical system theory, numerical analysis,statistical analysis, and computational science. Numerical studies on chaotic behaviorin the solutions to near-integrable PDEs [2] [16] [15] have inspired further analyticalstudies. In [12], through a combinational study on the Morse-Bott-Fomenko theory andthe Backlund-Darboux transformation theory for the Zakharov-Shabat linear system as-sociated with the integrable nonlinear Schrödinger equation, certain normally hyperbolicinvariant manifolds are identified for the nonlinear Schrödinger equation, representationsfor Melnikov vectors are given, and explicit expressions for heteroclinic orbits are ob-tained. In [11], the persistence of invariant manifolds and their fibrations are establishedfor certain perturbed nonlinear Schrödinger equations. Melnikov analysis is built basedupon the persistence-of-invariant-manifolds theorem, fiber theorem, and study [12]. Ho-moclinic orbits are proved to exist through Melnikov analysis, geometric arguments, andnormal form transforms. In the current paper, starting from the homoclinic orbit theoremproved in [11], we will construct Smale horseshoes, which are hallmarks ofdeterministicchaos.

There have been a lot of works on constructing Smale horseshoes in low-dimensionaldynamical systems [18] [23]. For relatively high (four or six) dimensional dynamicalsystems, there are fewer works on horseshoes; see for example, [9] [23] [15]. For arbi-trarily finite-dimensional systems, works on horseshoes are even more scarce; see forexample, [21] [6] [14]. For infinite-dimensional systems, works on horseshoes are rare.The author only knows the beautiful work of Holmes and Marsden [8], in which a class oftime-periodically perturbed evolution equations in a Banach space is studied, includingthe example of sinusoidally forced vibrations of a buckled beam. The current work isanother piece of work on horseshoes in infinite dimensions.

In this paper, we construct Smale horseshoes and study the corresponding symbolicdynamics for the following perturbed(1+ 1)-dimensional cubic nonlinear Schrödingersystem:

iqt = qζ ζ + 2[|q|2− ω2

]q

+ i ε[−αq + D2q + 0

], (1.1)

under even periodic boundary conditions

q(ζ + 1) = q(ζ ), q(−ζ ) = q(ζ ).

Hereε (> 0) is the perturbation parameter, (ω, α, 0) are real constants and satisfy theconstraint conditions

π < ω < 2π, α > 0.

The operatorD2 is a “regularized” Laplacian, specifically given by

D2q ≡ −∞∑

j=0

βj k2j qj coskj ζ,

Smale Horseshoes and Symbolic Dynamics in Perturbed Nonlinear Schrödinger Equations 365

whereqj is the Fourier transform ofq andkj = 2π j . The regularizing coefficientβj isdefined by

βj ={β for j ≤ N,α∗k−2

j for j > N,

in whichβ andα∗ are positive constants

α∗ > 16π2β,

andN is a large fixed positive integer.Denote byH1

e,p the Sobolev space of even periodic functions that are square integrablewith square integrable first derivative over the period interval [0,1]. The pde (1.1) is well-posed inH1

e,p, as the following two theorems state [3] [4]:

Theorem 1.1(Cauchy Problem).For any q0 ∈ H1e,p, there exists a unique solution

q ∈ C0((−∞,∞), H1e,p) to the pde (1.1) such that q|t=0= q0.

Let Ft denote the evolution operator for the pde (1.1), i.e.,q = Ft (q0) (−∞ < t <∞)is a solution as stated in Theorem 1.1. Then we have

Theorem 1.2(Smooth Dependence on Data).For any t ∈ (−∞,∞), Ft is a C2 dif-feomorphism in H1e,p, which is also C2 in the parameters (ω, ε, α, 0, β, α∗).

The construction of Smale horseshoes in this paper is built upon the following homo-clinic orbit theorem proved in [11]:

Theorem 1.3(Homoclinic Orbit Theorem).There exists a positive numberε0 such thatfor any ε ∈ (0, ε0), there exists a codimension-one hypersurface Eε in the externalparameter space{ω, α, 0, β, α∗}. For any external parameters(ω, α, 0, β, α∗) ∈ Eε ,there exists a symmetric pair of homoclinic orbits hk = hk(t, ζ ) (k = 1,2) in H1

e,p forthe pde (1.1), which are asymptotic to a fixed point qε . The symmetry between h1 andh2 is reflected by the relation that h2 is a half-period translate of h1, i.e., h2(t, ζ ) =h1(t, ζ + 1/2). The hypersurface Eε is a perturbation of a known surfaceβ = κ(ω)α,whereκ(ω) is shown in Figure 1.1.

Starting from this homoclinic orbit theorem 1.3, we will adapt the construction ofSmale horseshoes in [14] to the pde (1.1). The main difficulty for this adaptation is thefact that for the pde (1.1), the evolution operatorFt is onlyC0 in t (Theorem 1.1). Thisfact causes the approximation, as in [14], of the Poincaré mapP0

1 by its linearization to beinvalid, since in that approximation we differentiatedFt with respect tot . In this paper,we can avoid this differentiation int through properly defining the Poincaré section61

and controlling the consequence thereof. Using our horseshoe construction here, we areable to interpret the numerical observation [16] in Figure 1.3 using symbolic dynamicson four symbols 1, 2,−1,−2 under certain “except one point”–type conditions. We showthat the chaotic dynamics can be interpreted geometrically as the “chaotic center-wingjumping” that is seen in the numerical experiments.

366 Y. Li

Fig. 1.1.Theκ = κ(ω) curve.

For finite-dimensional dynamical systems, our construction in [14] is established in aneighborhood of the so-called Silnikov homoclinic orbits [21] [5] [6]. Our constructionis completely different from Silnikov’s. Silnikov used a so-called exponential expansiontechnique and a contraction map argument on an infinite product of spaces. We used asmooth linearization technique, a fixed point theorem, and a n-dimensional version ofthe Conley-Moser conditions. As explained in detail by Deng ([5] p. 159, line 8–19,and [6] p. 307, line 31–43), Silnikov’s exponential expansion had a fatal flaw. Thereexist counterexamples which show that Silnikov’s exponential expansion is incorrect.This is probably why few people had utilized the exponential expansion technique afterSilnikov’s work [21]. In fact, in [20], Ovsyannikov and Silnikov used a different and morerestrictive method to treat a similar problem. Until the work of Deng [5] [6], Deng claimedthat he had rigorized Silnikov’s exponential expansion under certain restrictions. Whetheror not the method of Silnikov and Deng can be generalized to infinite-dimensionaldynamical systems is an open question. Our method offers some advantages over that ofSilnikov and Deng.

1. The Conley-Moser conditions give rise to a geometric description of chaotic dynamicsin phase space that is particular to the problem being studied. Using our horseshoeconstruction, we are able to interpret the numerical observation: the “chaotic center-wing jumping” using Bernoulli shift dynamics on four symbols (−2,−1;1,2) undercertain “except one point”–type assumptions.

2. The Conley-Moser conditions allow one to conclude easily that the invariant Cantorset is structurally stable.


3. Hyperbolicity of the invariant Cantor set is an easy consequence. This allows one tobring in a variety of statistical techniques from dynamical system theory.

4. The topological argument of Conley and Moser has some advantages in dealing withnonhyperbolic fixed points.

5. The Conley-Moser topological approach can also be used for dealing with transientchaos [10].

6. The conditions of Conley and Moser are general conditions that do not rely on beingnear a homoclinic orbit for their application. They are sufficient conditions for adynamical system to possess an invariant set on which the dynamics is topologicallyconjugate to a Bernoulli shift.

The work of Conley and Moser [18] was originally for low-dimensional maps. In this low-dimensional case, one can rather easily handle the image and preimage of selected regionsin the domain of a map to construct a horseshoe intersection and subsequently to buildan invariant Cantor set. In higher dimensions, this becomes very difficult. Consequently,there are relatively few applications of this technique in higher dimensions, despite theobvious advantages. The main contribution of our construction is in overcoming thisdifficulty. We overcome this difficulty by first proving a fixed points theorem for thePoincare map and then verifying the Conley-Moser conditions in the neighborhoods ofthese fixed points.

The structural stability of the Conley-Moser construction can eliminate the restrictionposed by smooth linearization. Smooth linearization is valid for almost every point onthe codimension-one surface of external parameters on which homoclinic orbits aresupported. Since the horseshoes constructed using the Conley-Moser construction arestructurally stable, they also persist for the subset of measure zero on the codimension-one surface where smooth linearization breaks down. Thus we can establish horseshoeson the entire codimension-one surface,even its neighborhood, of external parameters onwhich homoclinic orbits are supported. Smooth linearization in fact poses no restrictionat all.

The main theorem proved in this paper is theHorseshoe Theoremstated in The-orem 8.1. In Section 2, we discuss the linear stability of the saddleqε , which is theasymptotic point of the symmetric pair of homoclinic orbits (Theorem 1.3). Two eigen-values have positive real parts (they are actually real); the rest of the eigenvalues havenegative real parts. The eigenvalues with the greatest negative real part are a complex-conjugate pair. Therefore, except for initial points on a measure zero set, ast →−∞, thehomoclinic orbits are tangent to a straight line; ast →+∞, they are ‘spirals’ and tangentto a plane. See Figure 1.2 for a geometric illustration. In Section 3, we study equivari-ant smooth linearization of the perturbed NLS system (1.1) in the neighborhood ofqε .Through such linearization, the construction of horseshoes in later sections is greatlysimplified. First, our evolution operator for the original perturbed NLS system (1.1) is aC2 diffeomorphism for any fixed time, and we need this property for the construction ofhorseshoes; thus, we need smooth linearization. Second, the perturbed NLS system (1.1)is invariant under a symmetry group, which is the reason for the existence of a symmetricpair of homoclinic orbits, and we want to construct horseshoes reflecting the symme-try of the pair of homoclinic orbits; thus, we need equivariant smooth linearization, sothat the conjugated system still possesses the symmetry group. Nevertheless, since the

368 Y. Li

horseshoe constructed here using Conley-Moser conditions is structurally stable, thisequivariant smooth linearization in fact poses no restriction in the external parameterspace. In Section 4, we give the basic definitions for the construction of horseshoes,which include Poincaré sections and Poincaré maps. We define two Poincaré sections60 and61 in the neighborhood ofqε , and two Poincaré maps—P1

0 from60 to61, andP0

1 from61 to60. Then we define the Poincaré mapP from60 to60 as the compositeof P1

0 and P01 . The horseshoes are going to be constructed on60 under the Poincaré

mapP. In Section 5, we study the representations of the Poincaré mapsP10 andP0

1 . P10

is given by linear dynamics, andP01 is given by the evolution operatorFT at a fixed

time T . SinceFT is a C2 diffeomorphism, we can approximate it by its linearizationat some point, which leads to a half-explicit representation forP0

1 . Therefore, we havea half-explicit representation forP. The construction of horseshoes goes as follows:First, we establish the existence of an infinite sequence of fixed points for the PoincarémapP through its half-explicit representation. Then, we study topological intersectionsbetweenSl andP(Sl ) in the neighborhood of each fixed point ofP, whereSl is a blocksubset of60, which contains two of the fixed points ofP. Through such a study, Smalehorseshoes are constructed. Finally, we establish symbolic dynamics starting from horse-shoes. In Section 6, we prove the existence of an infinite sequence of fixed points forthe Poincaré mapP. Starting from its half-explicit representation, we can write downthe equations satisfied by its fixed points in rescaled combinational coordinates. First,we solve the equations to the first order, which result in a simple equation for timet∗,11 cosbt∗+12 sinbt∗ = 0, whereb,11, and12 are constants. This simple equation hasan infinite sequence of solutions. Then, we solve the full equations through an implicitfunction argument, and establish the existence of an infinite sequence of solutions. InSection 7, first we define an infinite sequence of ‘slabs’Sl , each of which contains twofixed points ofP, and their symmetry imagesSl ,σ (σ denotes the symmetry). Then, westudy the topological intersection ofSl andSl ,σ with their imagesP(Sl ) andP(Sl ,σ ) inthe neighborhood of each fixed point. Through ‘size’ estimates along different directions,horseshoe-type intersections are established under “except one point”–type conditions.In Section 8, starting from the horseshoe intersection, we study the Conley-Moser con-ditions, which lead to the establishment of a symbolic dynamics on four symbols, whichoffers a hallmark of deterministic chaos. Finally, we interpret the numerics on chaoticbehavior of the solutions to the perturbed NLS equation (1.1) in Figure 1.3 through thesymbolic dynamics results (of course, under the “except one point”–type conditions).

2. Preliminaries

In this section, we give a detailed description on the fixed pointqε and its stability. Sinceqε is the asymptotic point of the symmetric pair of homoclinic orbits (Theorem 1.3), itslinear stability is crucial for the construction of horeshoes.

The “plane of constants,”

5 ≡{

q ∈ H1e,p

∣∣∣∣ dq

dζ≡ 0

}, (2.1)


Fig. 1.2. A geometric illustration of the symmet-ric pair of homoclinic orbits.

Fig. 1.3.Chaotic output of the perturbed NLS system.

is invariant under the flow (1.1). The dynamics on this invariant plane is governed by

iqt = 2

[|q|2− ω2

]q + i ε

[− αq + 0

].

370 Y. Li

Fig. 2.1.The phase plane diagram on5.

Its phase plane diagram is shown in Figure 2.1. There are three fixed points on the plane5: a stable focusoε near the origin, another stable focuspε , and a saddleqε . The locationof qε is

|qε | = ω − ε

4ω2

√02− α2ω2+ O(ε2),

arg{qε} = tan−1

{0

αω

√1−

(αω0

)2}+ O(ε).

(2.2)

In the whole phase space,qε is also a saddle. Letq = qε + q, then the pde (1.1) istransformed into

qt = Lq +N (q), (2.3)

where

Lq = −i qζ ζ − 2i

[(2|qε |2− ω2)q + q2

ε¯q]

+ ε[− αq + D2q

],

N (q) = −2i

[2qε |q|2+ qε q

2+ |q|2q].

The eigenvalues of the linear operatorL are

λ±j = −ε[α + βj k2j ] ± 2

√|qε |4−

[1

2k2

j − (2|qε |2− ω2)

]2

, (2.4)

wherekj = 2π j , j = 0,1,2, . . . . These eigenvalues are all simple. Forj = 0,1; λ±jare real. Forj ≥ 2, λ±j are complex,λ−j = λ+j . By the formula (2.2),


1. For j = 0,

λ±0 = ±cε1/2+ O(ε),

wherec = 2ω1/2(02− α2ω2)1/4.2. For j = 1,

λ±1 = ±4π√ω2− π2+ O(ε),

whereπ < ω < 2π .3. For 2≤ j ≤ N,

λ±j = −ε[α + βk2j ] ± 2i

√[1

2k2

j − (2|qε |2− ω2)

]2

− |qε |4.

4. For j ≥ N + 1,

λ±j = −ε[α + α∗] ± 2i

√[1

2k2

j − (2|qε |2− ω2)

]2

− |qε |4,

whereα∗ > 16π2β.

Thus,

Re{λ−2 } = max

{Re{λ−j } ( j = 0,1,2, . . .)

},

Re{λ+0 } = min

{λ+0 , λ

+1

},

and

− Re{λ−2 } < Re{λ+0 }. (2.5)

Denote byD±j the constants

D±j =i

2qε2 [aj − λ±j − ibj ], j = 0,1,2, . . . ,

whereaj = −ε(α + βj k2j ), bj = k2

j − 2(2|qε |2− ω2). The fundamental solutions to thelinearized equation

qt = Lq

are given by

1. For j = 0,1,

q = ξ±j exp{λ±j t + i θ±j } coskj ζ, (2.6)

whereξ±j are real parameters;θ±j are constant phases given byθ±j = 12 arg{D±j }.

372 Y. Li

2. For j ≥ 2,

q =[aj exp{λ+j t} + D+j aj exp{λ+j t}

]coskj ζ, (2.7)

whereaj are complex parameters; hereD+j satisfy|D+j | < 1.

We close this section with a discussion of the symmetry between the two homoclinicorbitshk = hk(t, ζ ) (Theorem 1.3),h2(t, ζ ) = h1(t, ζ + 1/2). If

h1(t, ζ ) =∞∑

j=0

Hj (t) coskj ζ,

then

h2(t, ζ ) =∞∑

j=0

(−1) j Hj (t) coskj ζ.

From now on, we denote this symmetry byσ , i.e., h2 = σ ◦ h1. Notice thatσ 2 = I(identity); then

G = {I , σ } (2.8)

is the symmetry group. The system (2.3) is invariant under the symmetry groupG.

3. Equivariant Smooth Linearization

The reference for this section is [19]. In this section, we will linearize the system (2.3)in the neighborhood ofq = 0. Moreover, we need the conjugation to be smooth (atleastC2), and the conjugated system is still invariant under the groupG. Thus, we needto study equivariant smooth linearization. Through such linearization, the dynamics inthe neighborhood ofq = 0 is greatly simplified (given by linear equations). In thisvery neighborhood, we will define Poincar´e sections, and the horseshoes are going to beconstructed on one of the Poincar´e sections.

First, we study the nonresonance condition for smooth linearization.

Lemma 3.1. For any ε ∈ (0, ε0), Eε denotes the codimension-one hypersurface inTheorem 1.3, on which symmetric pairs of homoclinic orbits are supported. For almostevery(α, β, α∗, 0, ω) ∈ Eε , the eigenvaluesλ±j (2.4) satisfy the condition of “Siegeltype”: There exists a natural number s such that for any integer n≥ 2,∣∣∣∣3n −

r∑j=1

3l j

∣∣∣∣ ≥ 1

r s, (3.1)

for all r = 2,3, . . . ,n and all l1, l2, . . ., lr ∈ Z, where3n = λ+n for n ≥ 0,3n = λ−−n−1for n < 0.


Proof. It is enough to check the condition (3.1) for the real parts ofλ±j (2.4).

Re{λ±j } =λ±j , for j = 0,1;−ε[α + βk2

j ], for 2≤ j ≤ N;−ε[α + α∗], for j ≥ N + 1.

Rewriteλ±j ( j = 0,1) as follows:

λ±j = −ε[α + βk2j ] ± 2ψj ( j = 0,1),

where

ψj =√|qε |4−

[1

2k2

j − (2|qε |2− ω2)

]2

( j = 0,1). (3.2)

First we want to summarize the basic ideas of the proof. We will change the coordinates(α, β, α∗, 0, ω) into the coordinates (α, β, α∗, ψ0, ψ1). In this new coordinate system,we can study the nonresonance condition more conveniently. In this new coordinatesystem, we show that for almost every point, the nonresonance condition holds. In orderto “shadow” this fact upon the codimension-one surfaceEε (Theorem 1.3), we definea “ray-region” generated from the image ofEε in the new coordinate system. The veryconnection between the “ray-region” and the image ofEε enables the “shadowing” uponEε for nonresonance points.

Next, we present the proof. Consider the following transformation:

F : (α, β, α∗, 0, ω) −→ (α, β, α∗, ψ0, ψ1), (3.3)

whereψj = ψj (0, ω, α), ( j = 0,1) are given in (3.2). The partial derivatives ofψj havethe expressions,

∂ψ0

∂0= −2ε0|qε |[ψ0D]−1

[2ω2− 3|qε |2

],

∂ψ0

∂ω= 2ω[ψ0D]−1

[4|qε |2

√02− α2|qε |2 (2ω2− 3|qε |2)

−D(ω2− 2|qε |2)],

∂ψ0

∂α= 2εα|qε |3[ψ0D]−1

[2ω2− 3|qε |2

],

∂ψ1

∂0= −4ε0|qε |[ψ1D]−1

[2π2+ ω2− 3

2|qε |2

], (3.4)

∂ψ1

∂ω= 2ω[ψ1D]−1

[8|qε |2

√02− α2|qε |2

(2π2+ ω2− 3

2|qε |2

)−D(2π2+ ω2− 2|qε |2)

],

∂ψ1

∂α= 4εα|qε |3[ψ1D]−1

[2π2+ ω2− 3

2|qε |2

],

374 Y. Li

whereD is given as

D = 2(3|qε |2− ω2)√02− α2|qε |2 − εα2|qε |.

The Jacobian of the transformationF is

JF = det

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

∂ψ0

∂α0 0 ∂ψ0

∂0

∂ψ0

∂ω

∂ψ1

∂α0 0 ∂ψ1

∂0

∂ψ1

∂ω

= 8ε0ωπ2|qε |3[ψ0ψ1D]−1. (3.5)

In a certain neighborhoodUE of the codimension-one hypersurfaceEε (which is aperturbation of the cylindrical surfaceβ = κ(ω)α) given by Theorem 1.3, all the partialderivatives ofψj ( j = 0,1) given in (3.4) are bounded and continuous, and the Jacobiangiven in (3.5),JF 6= 0,∞. Thus, in this neighborhoodUE, the transformation (3.3)is a diffeomorphism. Under the diffeomorphismF , the codimension-one hypersurfaceEε is transformed into a codimension-one hypersurfaceEε that is a perturbation ofthe hypersurfaceβ = κ(ω(α,ψ0, ψ1))α in the new parameter space{α, β, α∗, ψ0, ψ1},where

ω(α,ψ0, ψ1) =√(

ψ1

2π

)2

+ π2 + O(ε). (3.6)

Then, the hypersurfaceβ = κ(ω(α,ψ0, ψ1))α is a perturbation of the cylindrical surfaceβ = κ(ψ1)α, where

κ(ψ1) = κ([ψ21(2π)

−2+ π2]1/2), (3.7)

which isκ(ω) evaluated at the first-order term in the expression (3.6) forω. Thus, thecodimension-one hypersurfaceEε is a perturbation of the cylindrical surfaceβ = κ(ψ1)α

in the new parameter space{α, β, α∗, ψ0, ψ1}.

Definition 1. We say that a vectorEÄ ∈ Rn is of type (C, s) if

|〈 EÄ, El 〉| =∣∣∣∣∣ n∑

j=1

Äj l j

∣∣∣∣∣ ≥ C|El |−s, for all El ∈ Zn/{0},

where|El | =∑nj=1 |l j |.


Fig. 3.1.A geometric illustration of the “ray-region”Uδ.

Fact. Almost everyEÄ ∈ Rn is of type (C, s) for someC ands.

For a reference on this fact, see for example [22]. The linear combination in (3.1) is aspecial case of the following linear combination:

l1α + l2β + l3α∗ + l4ψ0+ l5ψ1, (3.8)

whereα = εα, β = εβπ2, α∗ = εα∗, El ∈ Z5/{0}. By the aboveFact, for almost every(α, β, α∗, ψ0, ψ1),

|l1α + l2β + l3α∗ + l4ψ0+ l5ψ1| ≥ C|El |−s, for all El ∈ Z5/{0}, (3.9)

for someC ands. Next we define a “ray-region”Uδ generated from the codimension-onesurfaceEε ,

Uδ ={(α, β, α∗, ψ0, ψ1)

∣∣∣∣ (α, β, α∗, ψ0, ψ1) = ρ(α(0), β(0), α(0)∗ , ψ(0)0 , ψ

(0)1 ),

(α(0), β(0), α(0)∗ , ψ(0)0 , ψ

(0)1 ) ∈ Eε,

ρ ∈ (1− δ,1+ δ), for some smallδ > 0

}.

See Figure 3.1 for a geometric illustration of this “ray-region”Uδ.Notice thatEε is a perturbation of a cylindrical surfaceβ = κ(ψ1)α, whereκ is given

in (3.7), in whichκ is given in Figure 1.1; thus,Uδ is a neighborhood ofEε . Then foralmost every(α, β, α∗, ψ0, ψ1) ∈ Uδ, the inequality (3.9) is true for someC ands. Eachsuch point

(α, β, α∗, ψ0, ψ1) ∈ Uδ (3.10)

376 Y. Li

corresponds to a point (α(0), β(0), α(0)∗ , ψ(0)0 , ψ

(0)1 ) on the surfaceEε ,

(α, β, α∗, ψ0, ψ1) = ρ(α(0), β(0), α(0)∗ , ψ(0)0 , ψ

(0)1 );

thus,

|l1α(0) + l2β(0) + l3α

(0)∗ + l4ψ

(0)0 + l5ψ

(0)1 | ≥ Cρ−1|El |−s, (3.11)

for all El ∈ Z5/{0},

for theC ands in (3.9) corresponding to the point (3.10). Therefore, for almost every(α(0), β(0), α

(0)∗ , ψ

(0)0 , ψ

(0)1 ) ∈ Eε , the inequality (3.11) is true for someC, s, andρ ∈

(1 − δ,1 + δ). Notice that the ‘r ’ in (3.1) starts from the integer 2; we can chooses = s− s1 ands1 large enough such that 2s1Cρ−1 ≥ 1; then inequality (3.1) followsfrom (3.11). Thus, applying the inverse of the diffeomorphismF , we have that for almostevery(α, β, α∗, 0, ω) ∈ Eε , the inequality (3.1) is true for some natural numbers. Theproof of the lemma is completed.

For the rest of this paper, we always choose the values of the external parameters(α, β, α∗, 0, ω) on Eε so that the Siegel condition (3.1) is satisfied.

Remark 3.1. As shown in subsequent sections, the horseshoes constructed using Conley-Moser conditions are structurally stable; thus, the horseshoes alsopersistfor externalparameters on the measure zero subset ofEε where the Siegel condition (3.1) breaksdown. Therefore, the nonresonance Siegel condition (3.1) in fact poses no restriction onexternal parameters. Furthermore, the horseshoes even persist for external parameters inaneighborhoodof the codimension-one cylindrical surfaceEε , where homoclinic orbitshave broken.

The condition (3.1) was posed in Theorem 4.1 of [19] for analytic linearization. Sincethe nonlinear termN (q) in (2.3) is a polynomial in (q, ¯q), by Theorem 4.1 of [19] (page83), the PDE (2.3) is analytically equivalent to the linear pde,

qt = Lq, (3.12)

in a neighborhood ofq = 0. There exists an analytic map

H: q→ q

that transforms the pde (2.3) to the linear pde (3.12) in a neighborhood ofq = 0. LetHG be the average ofH over the symmetry groupG (2.8):

HG = 1

2[H+ σ−1Hσ ];

thenq = HG(q) satisfies the pde:

qt = Lq + N (q), (3.13)


whereL is given in (2.3), andN vanishes identically in a neighborhood ofq = 0. Moreimportantly, system (3.13) is invariant under the symmetry groupG (2.8). Denote by

hk = HG ◦ hk (3.14)

the transformed homoclinic orbits; then

h2 = σ ◦ h1.

Denote byF t and F t the evolution operators of (3.13) and (2.3), respectively; then

F t = HG ◦ F t ◦H−1G .

Thus F t (q) is still C2 in q andC0 in t .

4. Basic Definitions

In this section, we will define Poincar´e sections and Poincar´e maps, which are the basicobjects for the horseshoe construction.

First we will rewrite (3.13) in a more convenient form for later horseshoe construction.Using the eigenvector basis (2.6; 2.7),

q = z1 exp{i θ+0 } + z2 exp{i θ+1 } cosk1ζ

+[(x + iy)+ D+2 (x − iy)

]cosk2ζ + Q, (4.1)

where

Q =1∑

j=0

ξ−j exp{i θ−j } coskj ζ +∞∑

j=3

[aj + D+j aj ] coskj ζ.

In terms of the new coordinates,

EQ ≡ (z1, z2, x, y, Q), (4.2)

(3.13) can be rewritten as

x = −ax− by+ Nx( EQ),y = bx− ay+ Ny( EQ),

z1 = γ1z1+ Nz1(EQ), (4.3)

z2 = γ2z2+ Nz2(EQ),

Q = L Q+ NQ( EQ),

wherea = −Re{λ+2 }, b = Im{λ+2 }, γ1 = λ+0 , γ2 = λ+1 ; N ’s vanish identically in aneighborhoodÄ of EQ = 0, andL is given in (2.3).

378 Y. Li

Definition 2. In terms of the new coordinatesEQ, ‖ EQ‖ is defined as

‖ EQ‖2 = x2+ y2+ z21 + z2

2 + ‖Q‖2,

where‖Q‖ denotes theH1 norm of Q:

‖Q‖2 =∫ 1

0

[|Q(ζ )|2+

∣∣∣∣d Q

dζ

∣∣∣∣2]

dζ.

Now the problem is reduced to constructing Smale horseshoes for the system (4.3)given the following properties:

• There exists a symmetric pair of homoclinic orbitshk (k = 1,2), h2 = σ ◦ h1. Thesystem (4.3) is invariant under the symmetry groupG (2.8).• The rate condition (2.5) is valid, i.e.,a < γ1. Herea is the smallest decay rate andγ1

is the smallest growth rate for the linear part of the system (4.3).

See Figure 1.2 for a geometric illustration.

Remark 4.1. The action ofσ on the system (4.3) has the representation,

σ ◦ {x, y, z1, z2, Q} = {x, y, z1,−z2, σ ◦ Q}.

Recall thatÄ is the neighborhood ofEQ = 0 in which the nonlinear terms in (4.3) vanishidentically. Denote byWu and Ws the unstable and stable manifolds ofEQ = 0, anddenote byWu

loc andWsloc the local unstable and stable manifoldsWu

loc = Ä ∩ Wu andWs

loc = Ä∩Ws. Denote byh+k = hk∩Wsloc andh−k = hk∩Wu

loc the forward time segmentsand backward time segments, respectively. Without loss of generality, we assume thath−1 lies in the first quadrant andh−2 lies in the fourth quadrant of the{z1, z2}-plane, asshown in Figure 4.1.

Lemma 4.1. As t→−∞, h1 is tangent to the positive z1-axis at EQ = 0.

Proof. This claim is equivalent to the claim thath1 intersects an unstable fiber withbase point different fromEQ = 0 (i.e.,qε (2.2)). From the Melnikov analysis in [11], theargumentγu of such an unstable-fiber base-point (which is a point on the invariant plane5 (2.1)) is a function ofω. By varyingω, we can have values ofγu that are differentfrom the phase ofqε .

Sincea is the smallest attracting rate, we assume that

• (A1) As t →+∞, h1 is tangent to{x, y}-plane at EQ = 0.

Remark 4.2. The claim: Ast → +∞, h1 is tangent to{x, y}-plane at EQ = 0, isequivalent to that for any pointq ∈ h1 ∩ Ä when t is sufficiently large, the Fouriercoefficientq2 6= 0, whereq = ∑∞

j=0 qj coskj ζ , sincea is the smallest attracting ratein (4.3). Thus, this is an “except one point forq2” assumption. It turns out that such an


Fig. 4.1.The location ofh−1 andh−2 .

assumption is almost impossible to verify analytically or numerically. It involves detailedlong-time global structures of the evolution operator for the PDE. The geometric singularperturbation nature of the homoclinic orbits poses great difficulties for a numericalverification.

Next we give several basic definitions for the construction of horseshoes.

Definition 3. The Poincar´e section60 is defined by the constraints

y = 0, η exp{−2πa/b} < x < η,

0< z1 < η, −η < z2 < η, ‖Q‖ < η,

whereη is a small parameter.

The horseshoes are going to be constructed on this Poincar´e section.

Definition 4. The auxiliary section6+0 is defined by the constraints

y = 0, η exp{−2πa/b} < x < η,

−η < zk < η, k = 1,2; ‖Q‖ < η.

By choosingη appropriately,hk (k = 1,2) intersect the (z1 = 0) boundary of60 at EQ+k(k = 1,2):

EQ+k = hk ∩ ∂60, (4.4)

where EQ+k have the coordinates

x = x(+,k), y = 0, z1 = z2 = 0, Q = Q(+,k). (4.5)

380 Y. Li

Fig. 4.2. A geometric illustration of the definitions of the Poincar´esections60 and61.

Here EQ+2 = σ ◦ EQ+1 , i.e.,

x(+,2) = x(+,1), Q(+,2) = σ ◦ Q(+,1).

Denote byF t the evolution operator for the system (4.3). There existsT > 0 such thatthe points

EQ−k = F−T ( EQ+k ) (4.6)

on hk have thez1 coordinates equal toη. Denote the coordinates ofEQ−k by

z1 = η, z2 = z(−,k)2 , x = y = Q = 0. (4.7)

Here EQ−2 = σ ◦ EQ−1 , i.e.,z(−,2)2 = −z(−,1)2 . See Figure 4.2 for a geometric illustration.Recall thatÄ is the neighborhood ofEQ = 0 in which the nonlinear terms in (4.3) vanishidentically. Whenη is sufficiently small,60 ⊂ 6+0 ⊂ Ä. Then we define the Poincar´esection61 as follows:


Definition 5. The Poincaré section61 is defined as

61 =(

F−T ◦6+0)∩Ä,

whereT is given in (4.6).

For a geometric illustration, see Figure 4.2.

Definition 6. The Poincaré mapP10 from60 to61 is defined as

P10 : U0 ⊂ 60 7→ 61,

∀ EQ ∈ U0, P10 (EQ) = F t∗( EQ) ∈ 61,

wheret∗ = t∗( EQ) > 0 is the smallest timet such thatF t ( EQ) ∈ 61.

Definition 7. The Poincaré mapP01 from61 to 60 (= 60 ∪ ∂60) is defined as

P01 : U1 ⊂ 61 7→ 60,

∀ EQ ∈ 61, P01 (EQ) = FT ( EQ) ∈ 60,

whereT is specified in (4.6).

Remark 4.3. We are going to approximate the Poincaré mapP01 by its linearization

at the point EQ−1 or EQ−2 . Since the evolution operatorFt is only C0 in t , we define thePoincare section61 by “pull-back” of6+0 for a fixed time, to avoid differentiation intwhen linearizingP0

1 . Such a set-up is very different from that in [14], in which61 isdefined by fixingz1 coordinate, and linearizingP0

1 does involve differentiation ofFt

in t .

Definition 8. The Poincaré mapP under which horseshoes are constructed is definedas

P : U ⊂ 60 7→ 60,

P = P01 ◦ P1

0 .

In the next section, a half-explicit representation forP will be established.

5. Representations

In this section, we study the representations of the Poincaré mapsP10 and P0

1 . First,we study the Poincaré section61 in more detail. The evolution operatorFT has the

382 Y. Li

representation,

∀ EQ = (x, y, z1, z2, Q),

FT ( EQ) =(

fx( EQ), fy( EQ), fz1(EQ), fz2(

EQ), fQ( EQ)).

Then by Definition 5, the Poincar´e section61 is defined by

∀ EQ1 = (x1, y1, z11, z

12, Q1) ∈ 61,

fy(x1, y1, z1

1, z12, Q1) = 0. (5.1)

We study (5.1) in the neighborhood ofEQ−1 (the study forEQ−2 follows from the symmetry):

fy( EQ−1 ) = 0. (5.2)

Lemma 5.1. ∂ fy

∂z1( EQ−1 ) and ∂ fy

∂z2( EQ−1 ) cannot be zero simultaneously.

Proof. Assume that

∂ fy

∂z1( EQ−1 ) =

∂ fy

∂z2( EQ−1 ) = 0. (5.3)

Sinceh−1 lies in (z1, z2)-plane, (5.3) implies that the directional derivative

τ · ∇ fy( EQ−1 ) = 0, (5.4)

whereτ is the tangent vector ofh−1 at EQ−1 , “∇” denotes gradient. Then (5.4) togetherwith (4.6) implies thath+1 is tangent to6+0 at EQ+1 . On the other hand,h+1 is transversalto6+0 at EQ+1 . This contradiction proves the lemma.

Corollary 1. In a neighborhood ofEQ−1 , the Poincare section61 can be representedeither by the C2 function

z11 = z1

1(x1, y1, z1

2, Q1), (5.5)

or by the C2 function

z12 = z1

2(x1, y1, z1

1, Q1), (5.6)

which in turn can be approximated by linear functions

z11(x

1, y1, z12, Q1) = η

+B1(x1, y1, z1

2 − z(−,1)2 , Q1)+41, (5.7)

z12(x

1, y1, z11, Q1) = z(−,1)2

+B2(x1, y1, z1

1 − η, Q1)+42, (5.8)


Fig. 5.1.Tangent spaces of61 at EQ−1 .

whereB1 andB2 are linear operators,B1 = ∇z11(EQ−1 ), andB2 = ∇z1

2(EQ−1 ),41 and42

are higher order terms,

41, 42 ∼ O(‖ EQ1− EQ−1 ‖2),EQ1 = (x1, y1, z1

1, z12, Q1),

EQ−1 = (0,0, η, z(−,1)2 ,0).

Proof. The representations (5.5; 5.6) follow immediately from (5.2) and Lemma 5.1when we apply the implicit function theorem to (5.1). Representations (5.7; 5.8) areTaylor expansions of the functions atEQ−1 to the first order.

Remark 5.1. By the representations (5.7; 5.8), the hyperplanes

z11 = η + B1(x

1, y1, z12 − z(−,1)2 , Q1),

or

z12 = z(−,1)2 + B2(x

1, y1, z11 − η, Q1),

are tangent spaces of61 at EQ−1 . See Figure 5.1.

Next, we study the representation ofP10 . Denote the coordinates on60 by

(x0, z01, z

02, Q0).

384 Y. Li

Denote the coordinates on61 by

(x1, y1, z11, z

12, Q1),

where in a neighborhood ofEQ−1 , either (5.7)

z11 = η + B1(x

1, y1, z12 − z(−,1)2 , Q1)+41

holds, or (5.8)

z12 = z(−,1)2 + B2(x

1, y1, z11 − η, Q1)+42

holds. In the neighborhoodÄ of EQ = 0, the solution of (4.3) is given by

x(t) = e−at

[x(0) cosbt − y(0) sinbt

],

y(t) = e−at

[x(0) sinbt + y(0) cosbt

],

z1(t) = z1(0)eγ1t ,

z2(t) = z2(0)eγ2t ,

Q(t) =1∑

j=0

ξ−j exp{λ−j t + i θ−j } coskj x

+∞∑

j=3

[aj exp{λ+j t} + D+j aj exp{λ+j t}

]coskj x.

Let t∗ be the “flight” time for an orbit starting from a point inU0 ⊂ 60 to reach61; thenP1

0 is given by

x1 = e−at∗x0 cosbt∗,y1 = e−at∗x0 sinbt∗,z1

1 = z01eγ1t∗ ,

z12 = z0

2eγ2t∗ , (5.9)

Q1 =1∑

j=0

ξ−j exp{λ−j t∗ + i θ−j } coskj x

+∞∑

j=3

[aj exp{λ+j t∗} + D+j aj exp{λ+j t∗}

]coskj x ≡ F(Q0),

whereQ0 is given by

Q0 =1∑

j=0

ξ−j exp{i θ−j } coskj x

+∞∑

j=3

[aj + D+j aj

]coskj x.


Remark 5.2. The “flight” time t∗ in (5.9) can be obtained by substituting (5.9) intoeither (5.7) or (5.8):

z01eγ1t∗ = η + f1(t∗), (5.10)

or

z02eγ2t∗ = z(−,1)2 + f2(t∗); (5.11)

where f1(t∗) and f2(t∗) are of orderO(‖ EQ1− EQ−1 ‖) when EQ1→ EQ−1 .

By the representation (5.9), we can prove the lemma.

Lemma 5.2. The Poincare map P is 1-1.

Proof. By Definition 7, it is obvious that the Poincar´e mapP01 is 1-1. Next we show

that the Poincar´e mapP10 is 1-1. Let EQ1 and EQ2 be two different points inU0 ⊂ 60, and

assume that

P10 (EQ1) = P1

0 (EQ2). (5.12)

By the representation (5.9) ofP10 , the coordinate equations

x1 |P10 (EQ1)= x1 |P1

0 (EQ2), y1 |P1

0 (EQ1)= y1 |P1

0 (EQ2)

lead to

e−at(1)∗ x0(1) cosbt(1)∗ = e−at(2)∗ x0

(2) cosbt(2)∗ , (5.13)

e−at(1)∗ x0(1) sinbt(1)∗ = e−at(2)∗ x0

(2) sinbt(2)∗ , (5.14)

where the index (k) corresponds toEQk (k = 1,2). From (5.13; 5.14), we have

tanbt(1)∗ = tanbt(2)∗ ;thus

bt(1)∗ = bt(2)∗ + jπ, for some j ∈ Z.

If j is odd, then neither (5.13) nor (5.14) can hold. Therefore,j is even (= 2 j1). Finallyfrom (5.13; 5.14), we have

x0(1)

/x0(2) = exp{2aj1π /b}.

This relation contradicts the constraint

η exp{−2πa/b} < x0(1), x0

(2) < η,

except for the casej1 = 0. But if j1 = 0, thent (1)∗ = t (2)∗ . In this case, the assumption(5.12) contradicts the fact that for fixedt , F t is a diffeomorphism. Thus in any case, theassumption (5.12) is not valid, which shows thatP1

0 is 1-1. ThusP = P01 ◦ P1

0 is 1-1.

386 Y. Li

Next, we study the representation ofP01 . The Poincar´e map P0

1 can be extendednaturally to

P01 : U1 ⊂ 61 7→ 6+0 .

By (4.6) and Definition 5,

P01 (EQ−k ) = EQ+k (k = 1,2).

We take EQ−1 as the example for the discussion below. The discussion forEQ−2 followssimilarly from the symmetry. For anyEQ1 in a sufficiently small neighborhood ofEQ−1 in61,

P01 (EQ1) ≡ FT ( EQ1) = FT ( EQ−1 )+∇ FT ( EQ−1 ) ◦ ( EQ1− EQ−1 )+43, (5.15)

where∇ FT ( EQ−1 ) is the gradient of the evolution operator with respect toEQ and evaluatedat EQ−1 ,

43 ∼ O(‖ EQ1− EQ−1 ‖2), as‖ EQ1− EQ−1 ‖ → 0.

Now we introduce new coordinates on60 and61 by translating the origin toEQ+1 andEQ−1 , respectively. On60,

x0 = x0− x(+,1), z0k = z0

k (k = 1,2),

Q0 = Q0− Q(+,1). (5.16)

On61,

x1 = x1, y1 = y1, Q1 = Q1,

z11 = z1

1 − η, z12 = z1

2 − z(−,1)2 , (5.17)

where in a neighborhood ofEQ−1 , either (5.7)

(Case 1)z11 = B1(x

1, y1, z12, Q1)+41 (5.18)

holds, or (5.8)

(Case 2)z12 = B2(x

1, y1, z11, Q1)+42 (5.19)

holds. Here,

B1(x1, y1, z1

2, Q1) = b11x1+ b12y1+ b13z

12 + B1(Q

1), (5.20)

B2(x1, y1, z1

1, Q1) = b21x1+ b22y1+ b23z

11 + B2(Q

1), (5.21)

wherebjk ( j = 1,2; k = 1,2,3) are real constants.Bk (k = 1,2) are linear operators.In terms of the new coordinates (5.16; 5.17), the representation (5.15) can be rewrittenas

x0 = a11x1+ a12y1+ a13z

11 + a14z

12 + A15(Q

1)+431,

z01 = a21x

1+ a22y1+ a23z11 + a24z

12 + A25(Q

1)+432,

z02 = a31x

1+ a32y1+ a33z11 + a34z

12 + A35(Q

1)+433,

Q0 = A41(x1)+ A42(y

1)+ A43(z11)+ A44(z

12)+ A45(Q

1)+434,


whereajk ( j = 1,2,3; k = 1,2,3,4) are real constants,A4 j ( j = 1,2,3,4) andAj 5

( j = 1,2,3,4) are linear operators; eitherz11 or z1

2 are respectively given by (5.18; 5.20)or (5.19; 5.21). Thus the Poincar´e mapP0

1 can be represented either by

(Case 1)

x0

z01

z02

Q0

= C(1)

x1

y1

z12

Q1

+4(1) (5.22)

or by

(case 2)

x0

z01

z02

Q0

= C(2)

x1

y1

z11

Q1

+4(2), (5.23)

which correspond respectively to case 1 (5.18; 5.20) or case 2 (5.19; 5.21), where

4(1) ∼ O

((x1)2+ (y1)2+ (z1

2)2+ ‖Q1‖2

),

4(2) ∼ O

((x1)2+ (y1)2+ (z1

1)2+ ‖Q1‖2

),

C(k) =

c(k)11 c(k)12 c(k)13 C(k)14

c(k)21 c(k)22 c(k)23 C(k)24

c(k)31 c(k)32 c(k)33 C(k)34

C(k)41 C(k)

42 C(k)43 C(k)

44

, k = 1,2,

in whichc(k)j l (k = 1,2; j, l = 1,2,3) are real constants,C(k)j 4 (k = 1,2; j = 1,2,3,4),

andC(k)4l (k = 1,2; l = 1,2,3) are linear operators.

Lemma 5.3. The linear operatorC(1) or C(2) (5.22; 5.23) is invertible.

Proof. SinceFT is a diffeomorphism, the restriction∇ F−T ( EQ+1 ) |6+0∇ F−T ( EQ+1 ) |6+0 : T EQ+1 6

+0 7→ T EQ−1 61

is invertible. (HereT EQ denotes tangent space atEQ.) The tangent spaceT EQ−1 61 can berepresented (Remark 5.1) either by [cf. (5.18; 5.20)]

z11 = B1(x

1, y1, z12, Q1),

or by [cf. (5.19; 5.21)]

z12 = B2(x

1, y1, z11, Q1).

388 Y. Li

The map

G(1)r : (x1, y1, z1

2, Q1)

7→(

x1, y1, z11 = B1(x

1, y1, z12, Q1), z1

2, Q1

)or the map

G(2)r : (x1, y1, z1

1, Q1)

7→(

x1, y1, z11, z

12 = B2(x

1, y1, z11, Q1), Q1

)is invertible. Then the composition

G(1)r ◦

(∇ F−T ( EQ+1 ) |6+0

)or G(2)

r ◦(∇ F−T ( EQ+1 ) |6+0

)is invertible, which implies that eitherC(1) or C(2) is invertible.

6. Fixed Points of the Poincare Map P

In this section, we study the fixed points of the Poincar´e mapP. We will study the fixedpoints in the neighborhood ofEQ+1 as the example; the study in the neighborhood ofEQ+2follows similarly from symmetry.

In Case 1 (5.22), first we write the equations for the fixed points ofP in the coordinates,

{t∗, z12, x0, Q0}. (6.1)

By the representations (5.9; 5.22) ofP10 and P0

1 , we have the equations for the fixedpoints ofP,

x0

(η + z11)e−γ1t∗

(z(−,1)2 + z12)e−γ2t∗

Q0

= C(1)(x(+,1) + x0)e−at∗ cosbt∗(x(+,1) + x0)e−at∗ sinbt∗

z12

F(Q(+,1) + Q0)

+4(1), (6.2)

wherez11 is given in (5.18) andF is defined in (5.9). By rescaling the coordinates (6.1)

as follows, {t∗, z1

2 = eat∗ z12, x0 = eat∗ x0, Q0 = eat∗ Q0

},

we can rewrite (6.2) asx0

00

Q0

= C(1)

x(+,1) cosbt∗x(+,1) sinbt∗

z120

+ 4(1), (6.3)


where4(1) ∼ O(e−ν1t∗) ast∗ → +∞ for someν1 > 0. In Case 2 (5.23), first we writethe equations for the fixed points ofP in the coordinates{

t∗, z11, x0, Q0

}. (6.4)

By the representations (5.9; 5.23) ofP10 and P0

1 , we have the equations for the fixedpoints ofP,

x0

(η + z11)e−γ1t∗

(z(−,1)2 + z12)e−γ2t∗

Q0

= C(2)(x(+,1) + x0)e−at∗ cosbt∗(x(+,1) + x0)e−at∗ sinbt∗

z11

F(Q(+,1) + Q0)

+4(2), (6.5)

wherez12 is given in (5.19) andF is defined in (5.9). By rescaling the coordinates (6.4)

as follows: {t∗, z1

1 = eat∗ z11, x0 = eat∗ x0, Q0 = eat∗ Q0

},

we can rewrite (6.5) asx0

00

Q0

= C(2)


z110

+ 4(2), (6.6)

where4(2) ∼ O(e−ν2t∗) ast∗ → +∞ for someν2 > 0.

6.1. 4(k) = 0 (k = 1,2) Solutions

Setting4(1) = 0 in (6.3), we havex0

00

Q0

= C(1)


z120

. (6.7)

Explicitly, the second and third equations in (6.7) are

x(+,1)[c(1)21 cosbt∗ + c(1)22 sinbt∗

]+ c(1)23 z1

2 = 0,

(6.8)

x(+,1)[c(1)31 cosbt∗ + c(1)32 sinbt∗

]+ c(1)33 z1

2 = 0.

Lemma 6.1. c(1)23 and c(1)33 in (6.8) cannot be zero simultaneously.

390 Y. Li

Proof. Recall thatWu and Ws denote the global unstable and stable manifolds ofEQ = 0 (Remark 4.1),5 denotes the “plane of constants” (2.1).Wu is two-dimensional,

and can be parametrized by a phase variableθ in5 and an initial time variable. LetWcs

denote the persistent codimension-one center-stable manifold studied in [11]. Then theMelnikov measurement in [11] is to measure the signed distance betweenWu andWcs

on a codimension-one section6: For fixed (ω, α, 0, β, α∗),

d = signed distance{Wu,Wcs} |6= d(θ);there existsθ0 such that

d(θ0) = 0, d′(θ0) 6= 0. (6.9)

Relation (6.9) is also true for (ω, α, 0, β, α∗) on Eε (Theorem 1.3). Thus at anyEQ ∈ h1,

dim

{T EQWu ∩ T EQWcs

}= 1, (6.10)

whereT EQ denotes tangent space atEQ. Notice thatWs ⊂ Wcs; then at EQ(+,1) ∈ h1,

dim

{T EQ(+,1)Wu ∩ T EQ(+,1)Ws

}= 1. (6.11)

Recall the representation (5.22) ofP01 , sinceC(1) is an invertible linear operator, for any

nonzeroz12,

v0 = C(1)

00z1

20

6= 0. (6.12)

Moreover,v0 ∈ T EQ(+,1)Wu. Assume thatc(1)23 and c(1)33 are zero simultaneously; thenv0 ∈ T EQ(+,1)Ws. Thus,

v0 ∈ T EQ(+,1)Wu ∩ T EQ(+,1)Ws. (6.13)

We also know that

T EQ(+,1) h1 ∈ T EQ(+,1)Wu ∩ T EQ(+,1)Ws. (6.14)

SinceT EQ(+,1) h1 is transversal to6+0 , while v0 lies in6+0 and is nonzero (6.12),T EQ(+,1) h1

andv0 are linearly independent; thus (6.13; 6.14),

dim

{T EQ(+,1)Wu ∩ T EQ(+,1)Ws

}= 2,

which contradicts (6.11). Thusc(1)23 andc(1)33 cannot be zero simultaneously.

From (6.8) and Lemma 6.1, we have

11 cosbt∗ +12 sinbt∗ = 0, (6.15)


where

11 = c(1)21 c(1)33 − c(1)31 c(1)23 ,

12 = c(1)22 c(1)33 − c(1)32 c(1)23 .

If we assume that the condition that

(A2) 11 and12 do not vanish simultaneously,

is true, then (6.15) has infinitely many solutions:

t (l )∗ =1

b[lπ − ϕ1], l ∈ Z; (6.16)

where

ϕ1 = arctan{11/12}.

Remark 6.1. (A2) is also an “except one point”–type condition. If at some parametervalue,11 and12 vanish simultaneously, a perturbation of the parameter may remove11 and12 away from vanishing simultaneously. It turns out that such an assumptionis almost impossible to verify analytically or numerically. It involves detailed long-time global structures of the evolution operator for the PDE. The geometric singularperturbation nature of the homoclinic orbits poses great difficulties for a numericalverification.

By virtue of Lemma 6.1, without loss of generality, we assumec(1)23 6= 0. Then, solving(6.8), we have

z12 = z(l )2 ≡ −x(+,1)[c(1)23 ]−1{c(1)21 cosbt(l )∗ + c(1)22 sinbt(l )∗ }. (6.17)

Solving (6.7), we have

x0 = x(l )1 ≡ x(+,1)[c(1)11 cosbt(l )∗ + c(1)12 sinbt(l )∗

]+ c(1)13 z(l )2 , (6.18)

Q0 = Q(l )1 ≡ x(+,1)

[C(1)

41 cosbt(l )∗ + C(1)42 sinbt(l )∗

]+ C(1)

43 z(l )2 . (6.19)

Similarly, setting4(2) = 0 in (6.6), we havex0

00

Q0

= C(2)


z110

. (6.20)

Explicitly, the second and third equations in (6.20) are

x(+,1)[c(2)21 cosbt∗ + c(2)22 sinbt∗

]+ c(2)23 z1

1 = 0,

(6.21)

x(+,1)[c(2)31 cosbt∗ + c(2)32 sinbt∗

]+ c(2)33 z1

1 = 0.

The same proof as of Lemma 6.1 shows that

392 Y. Li

Lemma 6.2. c(2)23 and c(2)33 in (6.21) cannot be zero simultaneously.

From (6.21) and Lemma 6.2, we have

13 cosbt∗ +14 sinbt∗ = 0, (6.22)

where

13 = c(2)21 c(2)33 − c(2)31 c(2)23 ,

14 = c(2)22 c(2)33 − c(2)32 c(2)23 .

If we assume that the condition that

(A2) 13 and14 do not vanish simultaneously,

is true, then (6.22) has infinitely many solutions:

t (l )∗ =1

b[lπ − ϕ2], l ∈ Z; (6.23)

where

ϕ2 = arctan{13/14}.By virtue of Lemma 6.2, without loss of generality, we assumec(2)23 6= 0. Then, solving(6.21), we have

z11 = z(l )1 ≡ −x(+,1)[c(2)23 ]−1{c(2)21 cosbt(l )∗ + c(2)22 sinbt(l )∗ }. (6.24)

Solving (6.20), we have

x0 = x(l )2 ≡ x(+,1)[c(2)11 cosbt(l )∗ + c(2)12 sinbt(l )∗

]+ c(2)13 z(l )1 , (6.25)

Q0 = Q(l )2 ≡ x(+,1)

[C(2)

41 cosbt(l )∗ + C(2)42 sinbt(l )∗

]+ C(2)

43 z(l )1 . (6.26)

6.2. Fixed Points

Starting from solutions obtained in the last subsection, we want to solve systems (6.3)and (6.6) by the implicit function theorem. Since

4(1) ∼ O(e−γ1t∗), 4(2) ∼ O(e−γ2t∗), as t∗ → +∞,

by the implicit function theorem, we have the following theorem.

Theorem 6.1. There exists an integer l0, such that there are infinitely many solutions,labeled by l (l≥ l0), to system (6.3):

t∗ = T (1)l , x0 = x(1,l ), Q0 = Q(1,l ), z1

2 = z(1,l )2 ,


where as l→+∞,

T (1)l = 1

b[lπ − ϕ1] + o(1),

x(1,l ) = x(l )1 + o(1),

Q(1,l ) = Q(l )1 + o(1),

z(1,l )2 = z(l )2 + o(1),

in whichx(l )1 , Q(l )1 , andz(l )2 are given in (6.18), (6.19), (6.17); or to system (6.6):

t∗ = T (2)l , x0 = x(2,l ), Q0 = Q(2,l ), z1

1 = z(2,l )1 ,

where as l→+∞,

T (2)l = 1

b[lπ − ϕ2] + o(1),

x(2,l ) = x(l )2 + o(1),

Q(2,l ) = Q(l )2 + o(1),

z(2,l )1 = z(l )1 + o(1),

in whichx(l )2 , Q(l )2 , andz(l )1 are given in (6.25), (6.26), and (6.24).

Proof. The proof is a standard fixed point argument. Here we give a sketch of the proof.Let

t∗ = 1

β1[2sπ + τ ], τ ∈ [0,2π ], s ∈ Z+,

andv denote the rest of the variables in (6.3) or (6.6). Then, (6.3) or (6.6) can be writtenas

f (τ, v) ≡ g(τ, v)+ C(s; τ, v) = 0, (6.27)

where ass→+∞, C(s; τ, v)→ 0. By the study in the last subsection, there exist twosolutions to

g(τ, v) = 0,

which are denoted by

(τ1, v1) and (τ2, v2).

Moreover,

∇g(τi , vi ), i = 1,2, (6.28)

are linear diffeomorphisms. Forτ ∈ [0,2π ], v in some bounded regionD1; D ≡[0,2π ] × D1,

sup(τ,v)∈D

‖C(s; τ, v)‖ → 0, ass→∞; (6.29)

sup(τ,v)∈D

‖∇C(s; τ, v)‖ → 0, ass→∞. (6.30)

394 Y. Li

We know that

f (τi , vi ) = C(s; τi , vi ).

We want to find(τ ′i , v′i ), such that

f (τi + τ ′i , vi + v′i )− f (τi , vi ) = −C(s; τi , vi ). (6.31)

Whens is sufficiently large, (6.31) is equivalent to

(τ ′i , v′i ) = −[∇ f (τi , vi )]

−1[C(s; τi , vi )+ R(s; τi , vi ; τ ′i , v′i )], (6.32)

where

R(s; τi , vi ; τ ′i , v′i ) = f (τi+τ ′i , vi+v′i )− f (τi , vi )−∇ f (τi , vi )•(τ ′i , v′i ) = o(‖(τ ′i , v′i )‖).

Then, a fixed point argument [1] [7] for (6.32) implies the theorem. This completes theproof of the theorem.

Remark 6.2. By this theorem, there are infinitely many periodic orbits, in a neighbor-hood of the homoclinic orbith1. Moreover, by the asymptotic representations of the fixedpoints given in the theorem, this sequence of periodic orbits approaches the homoclinicorbit h1 asl →+∞.

7. Smale Horseshoes

In this section, starting from the fixed point theorem 6.1, we construct Smale horseshoesthrough a topological intersection study in the neighborhood of each fixed point. Theconstruction is a generalization of that given in [14] to infinite dimensions. Since thePoincare section61 is not flat, we will introduce its tangent space atEQ−k in order tomake estimates easier.

7.1. Definition of Slabs

We begin by defining the notion of a “slab.”

Definition 9. For sufficiently large natural numberl , we define slabsS(1)l or S(2)l in 60

as follows:1. In Case 1 (5.7; 5.10),

S(1)l ≡{EQ ∈ 60

∣∣∣∣ η exp{−γ1

(T (1)

2(l+1) −π

2b

)}≤ z0

1(EQ) ≤ η exp

{−γ1

(T (1)

2l −π

2b

)},

|x0( EQ)| ≤ η exp

{−1

2aT(1)2l

},


|z12(P

10 (EQ))| ≤ η exp

{−1

2aT(1)2l

},

‖Q1(P10 (EQ))‖ ≤ η exp

{−1

2aT(1)2l

}}.

2. In Case 2 (5.8; 5.11),

S(2)l ≡{EQ ∈ 60

∣∣∣∣ z(−,1)2 exp{−γ2

(T (2)

2(l+1) −π

2b

)}≤ z0

2(EQ) ≤ z(−,1)2 exp

{−γ2

(T (2)

2l −π

2b

)},


{−1

2aT(2)2l

},

|z11(P

10 (EQ))| ≤ η exp

{−1

2aT(2)2l

},

‖Q1(P10 (EQ))‖ ≤ η exp

{−1

2aT(2)2l

}},

where the notationsx0( EQ), z12(P

10 (EQ)), etc., denote thex0 coordinate of the pointEQ, the

z12 coordinate of the pointP1

0 (EQ), etc.

S(k)l (k = 1,2) is defined so that it includes two fixed points ofP (see Theorem 6.1).

Remark 7.1. The definitions ofS(1)l andS(2)l are respectively based upon the represen-tations (5.10) and (5.11). For all points inP1

0 (S(k)l ), k = 1,2; f1(t∗) and f2(t∗) are of

orderO(exp{− 12aT(1)2l }) andO(exp{− 1

2aT(2)2l }), respectively.

From now on, we takeS(1)l as the example for the construction, and the construction forS(2)l follows similarly; moreover, we denoteS(1)l simply by Sl . We denote the two fixedpoints inSl by p+l and p−l , wherep+l corresponds toT (1)

2l , andp−l corresponds toT (1)2l+1

in Theorem 6.1. It follows that

z01(p+l ) > z0

1(p−l ).

Now we apply the symmetryσ of the system (4.3) to the objectsSl andp+l andp−l , andwe have

Sl ,σ = σ ◦ Sl , p+l ,σ = σ ◦ p+l , p−l ,σ = σ ◦ p−l . (7.1)

If l is sufficiently large,P10 (Sl ) is included in a ball centered atEQ−1 (4.6) on61, with

radius of order

O

(exp

{−1

2aT(1)2l

}).

Therefore,

P10 (Sl ) ⊂ U1,

396 Y. Li

whereU1 is the domain of definition ofP01 (see Definition 7). Thus,

Sl ⊂ U,

whereU is the domain of definition ofP (see Definition 8). We also define a larger slabSl as follows:

Sl ≡{EQ ∈ 60

∣∣∣∣ η exp{−γ1

(T (1)

2(l+1) −π

2b

)}≤ z0

1(EQ) ≤ η exp

{−γ1

(T (1)

2l −π

2b

)},

|x0( EQ)− x(+,1)| ≤ η exp

{−1

2aT(1)2l

},

|z12(P

10 (EQ))| ≤ |z(−,1)2 | + η exp

{−1

2aT(1)2l

},

‖Q1(P10 (EQ))‖ ≤ η exp

{−1

2aT(1)2l

}}.

Then

Sl ∪ Sl ,σ ⊂ Sl .

Remark 7.2. In Case 2 (i.e., forS(2)l in Definition 9), Sl should be defined as

Sl ≡{EQ ∈ 60

∣∣∣∣ |z02(EQ)| ≤ z(−,1)2 exp

{−γ2

(T (2)

2l −π

2b

)},


{−1

2aT(2)2l

},

|z11(P

10 (EQ))| ≤ η exp

{−1

2aT(2)2l

},

‖Q1(P10 (EQ))‖ ≤ η exp

{−1

2aT(2)2l

}}.

7.2. Sl and P10(Sl)

In this subsection, we describe the geometry ofP10 (Sl ). We choose a basis on the tangent

spaceT EQ−1 61 represented in the coordinates (x1, y1, z11, z

12, Q1) as follows:

Ex1 = (1,0,B1(1,0,0,0),0,0),

Ey1 = (0,1,B1(0,1,0,0),0,0),

Ez12= (0,0,B1(0,0,1,0),1,0), (7.2)

EQ1 = (0,0,B1(0,0,0,1),0,1),

where1 represents a basis for theQ1 coordinate. Define the projectionπ as follows:

π : 61 7→ T EQ−1 61, (7.3)


Fig. 7.1.The product representation ofSl .

π

(x1, y1, z1

1 = B1(x1, y1, z1

2, Q1)+41, z12, Q1

)=(

x1, y1, z11 = B1(x

1, y1, z12, Q1), z1

2, Q1

).

Denote

P10(Sl ) = π(P1

0 (Sl )).

Denote by

{ex0,ez01,ez0

2,eQ0} (7.4)

the unit vectors along (x0, z01, z

02, Q0)-directions in60. In the coordinate frame

{ex0,ez01,ez0

2,eQ0}, Sl has the product representation as shown in Figure 7.1. In the co-

ordinate frame{Ex1, Ey1, Ez12,EQ1}, P1

0(Sl ) has the product representation as shown inFigure 7.2.

7.3. P(Sl)

By (5.15),

P01 (EQ1) = L( EQ1)+43, (7.5)

where

L( EQ1) = FT ( EQ−1 )+∇ FT ( EQ−1 ) ◦ ( EQ1− EQ−1 )

398 Y. Li

Fig. 7.2.The product representation ofP10(Sl ).

is the linear approximation ofP01 . Denote byP(Sl ) the linear approximation ofP(Sl ),

P(Sl ) = L(P10(Sl )).

We can estimate the “size” ofP(Sl ) easily. Then for sufficiently largel , P(Sl ) has thesame “size” estimate asP(Sl ). UnderL, the coordinate frame{Ex1, Ey1, Ez1

2,EQ1} is

mapped into a coordinate frame

{Ex1, Ey1, Ez12, EQ1} (7.6)

on60 with origin at EQ+1 . In this coordinate frame,P(Sl ) has the product representationas shown in Figure 7.3. Notice that the intersection ofP(Sl ) with the (Ex1, Ey1)-plane isan annulus. This annulus has the width of order

O

(exp

{−3

2aT(1)2l

}), asl →+∞,

and radius of order

O(exp{−aT(1)2l }), asl →+∞.

7.4. Topological Intersections and Horseshoes

First, we define certain boundaries ofSl .


Fig. 7.3.The product representation ofP(Sl ).

Definition 10. Let ¯Sl be the closure ofSl in 60. The connected components of thestable boundary ofSl are defined as

∂+s Sl ≡{EQ ∈ ¯Sl

∣∣∣∣ x0( EQ) = η exp

{−1

2aT(1)2l

}},

∂−s Sl ≡{EQ ∈ ¯Sl

∣∣∣∣ x0( EQ) = −η exp

{−1

2aT(1)2l

}},

∂Qs Sl ≡

{EQ ∈ ¯Sl

∣∣∣∣ ‖Q0‖ = η}.

The union of all the connected components of the stable boundary ofSl is referred to asthe stable boundary ofSl , denoted by∂sSl . Similarly, the connected components of theunstable boundary ofSl are defined as

∂(1)u Sl ≡{EQ ∈ ¯Sl

∣∣∣∣ z01(EQ) = η exp

{−γ1

(T (1)

2l −π

2b

)}},

∂(2)u Sl ≡{EQ ∈ ¯Sl

∣∣∣∣ z01(EQ) = η exp

{−γ1

(T (1)

2(l+1) −π

2b

)}},

∂(3)u Sl ≡{EQ ∈ ¯Sl

∣∣∣∣ z12(P

10 (EQ)) = |z(−,1)2 | + η exp

{−1

2aT(1)2l

}},

∂(4)u Sl ≡{EQ ∈ ¯Sl

∣∣∣∣ z12(P

10 (EQ)) = −

(|z(−,1)2 | + η exp

{−1

2aT(1)2l

})}.

The union of all the connected components of the unstable boundary ofSl is referred toas the unstable boundary ofSl , denoted by∂uSl .

400 Y. Li

We now define “slices of slabs.”

Definition 11. A stable sliceV in Sl is a subset ofSl defined as the region swept outthrough homeomorphically moving and deforming∂sSl in such a way that the part

∂sSl ∩ ∂uSl

of ∂sSl only moves and deforms inside∂uSl . The new boundary obtained through suchmoving and deforming of∂sSl is called the stable boundary ofV , which is denoted by∂sV . The rest of the boundary ofV is called its unstable boundary, which is denoted by∂uV . An unstable slice ofSl , denoted byH , is defined similarly.

By definition,

∂uV ⊂ ∂uSl , ∂sH ⊂ ∂sSl .

From the definition ofSl , we know thatSl andSl ,σ are unstable slices ofSl . In Figure 7.1,we have marked two pieces of∂sSl and∂uSl by (1,2) and (3,4), respectively. In Figures 7.2and 7.3, we also marked the corresponding images of these boundaries by the sameletters. Note the contraction, expansion, and bending in the deformation process fromthe rectangle in the (ex0,ez0

1)-plane in Figure 7.1 to the annulus in the (Ex1, Ey1)-plane

in Figure 7.2.In the coordinates (x0, z0

1, z02, Q0) on60, let G be a hyperplane inSl , specified by the

condition, {z0

1 = constant, z02 = constant

}.

In general,G ∩ V consists of several singly connected regions

G ∩ V =K∑

k=1

Gk.

See Figure 7.4 for an illustration.

Definition 12. The diameter of the stable sliceV is defined as

d(V) ≡ supG

{sup

k

{supEQ1, EQ2∈Gk

{|x0( EQ1)− x0( EQ2)|

+‖Q0( EQ1)− Q0( EQ2)‖}}}

.

The diameter of an unstable sliceH is defined similarly.

Denote by

{x(1), y(1), z(1)2 , Q(1)} (7.7)

the coordinate on the tangent spaceT EQ−1 61 with respect to the base

{Ex1, Ey1, Ez12,EQ1}


Fig. 7.4.The intersection betweenG andV .

defined in (7.2). Introduce the polar coordinates (r , θ ) with respect to the affine coordi-nates (x(1), y(1)) on the (Ex(1) , Ey(1) )-plane as follows:

r =√(x(1))2+ (y(1))2, θ = arctan

{y(1)

/x(1)

}.

Then the mapπ ◦ P10 restricted to the (Ex1, Ey1)-plane has the following representation

(whereπ is the projection defined in (7.3)):

r = e−at∗(x0+ x(+,1)), θ = bt∗.

From Theorem 6.1, the “time of flight” of the fixed points ofP from60 to61 is givenby

t∗ = T (1)l =

1

b[lπ − ϕ1] + o(1), asl →+∞.

Consequently, for any small positiveθ0, there existsl1, such that for anyl ≥ l1, thecoordinate components (x(1), y(1)) of π ◦ P1

0 (p+l ) andπ ◦ P1

0 (p−l ) are in two sectorss+

ands− of angle 2θ0:

(s+) θ(π ◦ P10 (p

+l1))− θ0 ≤ θ (π ◦ P1

0 (p+l )) ≤ θ (π ◦ P1

0 (p+l1))+ θ0, (7.8)

(s−) θ(π ◦ P10 (p

−l1))− θ0 ≤ θ (π ◦ P1

0 (p−l )) ≤ θ (π ◦ P1

0 (p−l1))+ θ0. (7.9)

See Figure 7.5 for an illustration. Under the linearizationL of P01 defined in (7.5), the

two sectorss+ ands− are mapped to two sectors,

s+ = L(s+), (7.10)

s− = L(s−), (7.11)

402 Y. Li

Fig. 7.5.The two sectorss+ ands− on the (Ex1, Ey1)-plane.

on the (Ex1, Ey1)-plane. See Figure 7.6 for an illustration. The boundaries (1,2) includeparts of the boundaries ofs+ ands−. We introduce a system of curvilinear coordinates(ξu, ξs) on the (Ex1, Ey1)-plane such that

{ξu = 0}, {ξu = bu(constant)}, {ξs = 0}, {ξs = bs(constant)}, (7.12)

correspond to the boundaries 3, 4; 1, 2 of the annulus on the (Ex1, Ey1)-plane. Cf. Figure 7.6and Figure 7.7. From now on, we restrict the coordinates (ξu, ξs) to the two sectorss+ands−.

Definition 13. Define two subsets ofP(Sl ) as follows:

S+l ≡{EQ ∈ P(Sl )

∣∣∣∣ (ξu, ξs)( EQ) ∈ s+

},

S−l ≡{EQ ∈ P(Sl )

∣∣∣∣ (ξu, ξs)( EQ) ∈ s−

}.


Fig. 7.6.The two sectorss+ ands− on the (Ex1, Ey1)-plane.

Without loss of generality, we focus attention onp+l . LetE+u be the tangent vector to theξu coordinate atp+l ,

E+u ≡ Tp+lξu.

Then we make the assumption atp+l ,

(A3) Span

{ex0,eQ0, E+u , Ez1

2

}= 60, (7.13)

whereex0 andeQ0 are defined in (7.4), andEz12

is defined in (7.6).

Remark 7.3. (A3) is also an “except one point”–type assumption. If we denote byθ the angle between the vectorex0 and the subspace Span{eQ0, E+u , Ez1

2} of 60, then

ex0 ∈ Span{eQ0, E+u , Ez12} is equivalent toθ = 0. It turns out that such an assumption

is almost impossible to verify analytically or numerically. It involves detailed long-time global structures of the evolution operator for the PDE. The geometric singularperturbation nature of the homoclinic orbits poses great difficulties for a numericalverification.

Denote by

{x(0), y(0), z(0)2 , Q(0)} (7.14)

404 Y. Li

Fig. 7.7.Curvilinear coordinates (ξu, ξs) on the (Ex1, Ey1)-plane.

the coordinate on60 with respect to the base

{Ex1, Ey1, Ez12, EQ1}

defined in (7.6). Then we introduce the system of coordinates

{x0, Q0, ξu, z(0)2 } (7.15)

on60, where{x0, Q0} are defined in (5.16),z(0)2 is defined above in (7.14), andξu isthe curvilinear coordinate defined in (7.12); the vectors in (7.13) are tangent vectors tothis system of coordinates atp+l . Now in the system of coordinates (7.15), we give thefollowing definitions.

Definition 14. We define the diameters ofSl andP(Sl ) as follows:

du(Sl ) ≡ supC

{|ξu( EQ1)− ξu( EQ2)| + |z(0)2 ( EQ1)− z(0)2 ( EQ2)|

},


where

C ≡ { EQ1, EQ2 ∈ Sl ; x0( EQ1) = x0( EQ2), Q0( EQ1) = Q0( EQ2)},

ds(P(Sl )) ≡ supC

{|x0( EQ1)− x0( EQ2)|

+ ‖Q0( EQ1)− Q0( EQ2)‖},

where

C ≡ { EQ1, EQ2 ∈ S+l ⊂ P(Sl ); ξu( EQ1) = ξu( EQ2), z(0)2 ( EQ1) = z(0)2 ( EQ2)}.

The diameters have the following estimates:

du(Sl ) ∼ O(exp{−γ1T (1)2l }), asl →∞, (7.16)

ds(P(Sl )) ∼ o(exp{−aT(1)2l }), asl →∞. (7.17)

Definition 15. Define two sections atp+l as follows:

5s(Sl ) ≡{EQ ∈ Sl

∣∣∣∣ ξu( EQ) = ξu(p+l ), z

(0)2 ( EQ) = z(0)2 (p+l )

},

5u(P(Sl )) ≡{EQ ∈ S+l ⊂ P(Sl )

∣∣∣∣ x0( EQ) = x0(p+l ), Q0( EQ) = Q0(p+l )}.

Then there exists an order

O(exp{−1

2aT(1)2l }), asl →∞, (7.18)

neighborhood ofp+l in 5s(Sl ), and an order

O(exp{−aT(1)2l }), asl →∞, (7.19)

neighborhood ofp+l in 5u(P(Sl )).Notice that for sufficiently largel , P(Sl ) has the same “size” estimate asP(Sl ), and

we have

Proposition 7.1. Under the assumption (A3), there exists a sufficiently large l2, suchthat for all l ≥ l2, P(Sl ) and P(Sl ,σ ) respectively intersectSl into four disjoint stableslices{V1,V2} and {V−1,V−2} in Sl . Vj ’s ( j = 1,2,−1,−2) do not intersect∂sSl ;moreover,

∂sVi ⊂ P(∂sSl ), (i = 1,2), ∂sVi ⊂ P(∂sSl ,σ ), (i = −1,−2). (7.20)

406 Y. Li

Proof. We begin by showing the “disjointness.” It is obvious that{V1,V2} are disjointfrom {V−1,V−2}. Next, for example, we show thatV1 andV2 are disjoint. Assume theyare not disjoint, then there exists a curveg connectingp+l and p−l , such that

g ⊂ Sl , (7.21)

and

g ⊂ P(Sl ). (7.22)

Now we use the curvilinear coordinateξu (7.12) to define

w(g) ≡ supEQ1, EQ2∈g

{|ξu( EQ1)− ξu( EQ2)|

}.

Then by (7.21), there is a constantD1, such that

w(g) < D1 exp{−γ1T (1)2l };

by (7.22), there is a constantD2, such that

w(g) > D2 exp{−aT(1)2l }.Notice thata < γ1; for l sufficiently large, this is a contradiction. Therefore,V1 andV2

are disjoint. The rest of the claim in the proposition is obvious from the estimates (7.16;7.17; 7.18; 7.19), the assumption (A3), and the fact thatP(Sl ) is a linear approximationof P(Sl ). For a geometric illustration, see Figure 7.8.

We now define

Hj = P−1(Vj ), ( j = 1,2,−1,−2). (7.23)

Corollary 2. Hj ’s ( j = 1,2,−1,−2) are unstable slices.

Proof. This is immediate from (7.20) in Proposition 7.1.

The unstable boundaries ofHj ( j = 1,2,−1,−2) are

∂u Hj = P−1(∂uVj ), ( j = 1,2,−1,−2). (7.24)

In summary, from Proposition 7.1, (7.23), Corollary 2, and (7.24), we have Vj = P(Hj ),

∂sVj = P(∂sHj ), ( j = 1,2,−1,−2),∂uVj = P(∂u Hj ).

(7.25)

In the next section, we will establish shift dynamics on eachSl . Consequently, on eachSl , we have a Smale horseshoe. Thus, we have infinitely many Smale horseshoes labeledby l in 60.


Fig. 7.8. (a) An illustration of the intersection ofP(Sl )with Sl in the neighborhood ofp+l . (b) Anillustration of the Smale horseshoe.

8. Symbolic Dynamics

In this section, we will construct an invariant Cantor subset3 of Sl and show that thePoincare mapP, restricted to3, is topologically conjugate to the shift automorphismon four symbols, 1,2,−1,−2.

408 Y. Li

8.1. The Shift Automorphism

LetW be a set that consists of elements of the doubly infinite sequence form:

a = (. . .a−2a−1a0,a1a2 . . .),

whereak ∈ {1,2,−1,−2}; k ∈ Z. We introduce a topology inW by taking as theneighborhood the basis of

a∗ = (. . .a∗−2a∗−1a∗0,a∗1a∗2 . . .),

the set

Wj ={

a ∈W∣∣∣∣ ak = a∗k (|k| < j )

}for j = 1,2, . . . . This makesW a topological space. The shift automorphismχ isdefined onW by

χ : W 7→W,∀a ∈W, χ(a) = b, wherebk = ak+1.

The shift automorphismχ exhibitssensitive dependence on initial conditions, which isa hallmark ofchaos.

8.2. Conley-Moser Conditions

The Conley-Moser conditions are sufficient conditions for establishing the topologicalconjugacy between the Poincar´e mapP restricted to a Cantor set in60, and the shiftautomorphism on symbols [18] [23]. Specifically, the Conley-Moser conditions are:

Conley-Moser condition (i): Vj = P(Hj ),

∂sVj = P(∂sHj ), ( j = 1,2,−1,−2),∂uVj = P(∂u Hj ).

Conley-Moser condition (ii):There exists a constant 0< ν < 1, such that for anystable sliceV ⊂ Vj ( j = 1,2,−1,−2), the diameter decay relation

d(V) ≤ νd(V)

holds, where the diameterd(·) is defined in Definition 12, andV = P(V ∩ Hk), (k =1,2,−1,−2); for any unstable sliceH ⊂ Hj ( j = 1,2,−1,−2), the diameter decayrelation

d(H) ≤ νd(H)

holds, whereH = P−1(H ∩ Vk), (k = 1,2,−1,−2).


Remark 8.1. Here we modify the Conley-Moser condition (i) by dropping the Lipschitzcondition for the boundaries of the slices. (cf. [18] [23].) In our case, a stable sliceVand an unstable sliceH can possibly intersect into more than one regions. But we onlypick any one of them for the invariant set that we construct.

The above Conley-Moser condition (i) has been verified in the last section. (See relation(7.25).) Next we discuss the Conley-Moser condition (ii). By the representation (5.9) ofP1

0 and the representation (5.22) ofP01 , we have

d(V) ≤ ν1d(V),

where

ν1 ∼ O

(exp{−aT(1)2l }

), asl →+∞;

d(H) ≤ ν2d(H),

where

ν2 ∼ O

(exp{−γ1T (1)

2l }), asl →+∞.

Thus the Conley-Moser condition (ii) is also verified. Next we discuss the nested se-quences of slices.

Lemma 8.1. If H (1) ⊃ H (2) ⊃ · · · is an infinite sequence of unstable slices, andd(H (k))→ 0 as k→∞, then

∞⋂k=1

H (k) ≡ H (∞)

is a codimension-two continuous surface in60, and its boundary is included in∂sH (1):

∂H (∞) ⊂ ∂sH (1).

Similarly, if V(1) ⊃ V (2) ⊃ · · · is an infinite sequence of stable slices, and d(V (k))→ 0as k→∞, then

∞⋂k=1

V (k) ≡ V (∞)

is a two-dimensional continuous surface in60, and its boundary is included in∂uV (1):

∂H (∞) ⊂ ∂uV (1).

Proof. The dimension ofH (∞) follows from the fact thatd(H (k))→ 0 ask→∞. Since∂sH (k) ⊂ ∂sH (1) for all k, ∂H (∞) ⊂ ∂sH (1). SinceP is a homeomorphism,∂u H (k) iscontinuous for eachk. Since∂u H (k) shrinks toH (∞) ask→∞, H (∞) is also continuous.Similarly for V (∞).

410 Y. Li

8.3. Topological Conjugacy

Let

a = (. . .a−2a−1a0,a1a2 . . .),

be any element ofW. Define inductively fork ≥ 2 the stable slices

Va0a−1 = P(Ha−1) ∩ Ha0,

Va0a−1...a−k = P(Va−1...a−k) ∩ Ha0.

By the Conley-Moser condition (ii),

d(Va0a−1...a−k) ≤ ν1d(Va0a−1...a−(k−1) ) ≤ · · · ≤ νk−11 d(Va0a−1).

By Lemma 8.1,

V(a) =∞⋂

k=1

Va0a−1...a−k

defines a two-dimensional continuous surface in60; moreover,

∂V(a) ⊂ ∂uSl . (8.1)

Similarly, define inductively fork ≥ 1 the unstable slices

Ha0a1 = P−1(Ha1 ∩ Va0),

Ha0a1...ak = P−1(Ha1...ak ∩ Va0).

By the Conley-Moser condition (ii),

d(Ha0a1...ak) ≤ ν2d(Ha0a1...ak−1) ≤ · · · ≤ νk2d(Ha0).

By Lemma 8.1,

H(a) =∞⋂

k=0

Ha0a1...ak

defines a codimension-two continuous surface in60; moreover,

∂H(a) ⊂ ∂sSl . (8.2)

By (8.1; 8.2) and dimension count,

V(a) ∩ H(a) 6= ∅consists of points. Let

p ∈ V(a) ∩ H(a)

be any point in the intersection set. Now we define the mapping

φ : W 7→ Sl ,

φ(a) = p.


By the above construction,

P(p) = φ(χ(a)).That is,

P ◦ φ = φ ◦ χ.Let

3 ≡ φ(W),then3 is a compact Cantor subset ofSl , and is invariant under the Poincar´e mapP.Moreover, with the topology inherited fromSl for 3, φ is a homeomorphism fromWto3. For more detailed discussion, see [18] [23]. Thus we have the theorem.

Theorem 8.1(Horseshoe Theorem).Under the “except one point”–type assumptions(A1)–(A3) for the perturbed nonlinear Schrodinger system (1.1), there exists a compactCantor subset3 of Sl , and3 consists of points and is invariant under P. P restricted to3 is topologically conjugate to the shift automorphismχ on four symbols,1,2,−1,−2.That is, there exists a homeomorphism

φ: W 7→ 3,

such that the following diagram commutes:

W φ−→ 3

χ

y yP

W −→φ

3

(8.3)

Remark 8.2. The above theorem is proved for almost every external parameter on thecodimension-one cylindrical surfaceEε on which the symmetric pair of homoclinicorbits is supported. Since our horseshoe construction is by verifying the Conley-Moserconditions, the invariant Cantor set3 is structurally stable. The invariant Cantor set3 together with the shift dynamics,persistsfor every external parameter in an openneighborhoodof the codimension-one cylindrical surfaceEε .

Remark 8.3. Topological conjugacy ofP to shift or subshift dynamics on many sym-bols can also be established; but we omit them here. For a relevant discussion on suchtopics, see [23].

Remark 8.4. The construction in this paper can be summarized as follows: The fixedpoint theorem 6.1 leads to four fixed points ofχ ,

a( j ) = (. . .a( j )−2a( j )−1a( j )

0 ,a( j )1 a( j )

2 . . .), ( j = 1,2,−1,−2),

wherea( j )k = j , ∀k ∈ Z. Then starting from these four fixed points, all other points in

W can be constructed through the construction in the last section and in this section.

412 Y. Li

Fig. 8.1. Hyperbolic structures and the correspondence with the chaotic center-wing jump-ing.

8.4. Interpretation of Numerical Observation on the Perturbed NLS System: TheChaotic Center-Wing Jumping

In the chaotic regime, typical numerical output [16] on the perturbed NLS system (1.1)is shown in Figure 1.3. Notice that there are two typical profiles at a fixed time: One isa breather-type profile with its hump located at the center of the spatial period intervaland the other is also a breather-type profile, but with its hump located at the boundaries(wing) of the spatial period interval. More importantly, these two types of profiles arehalf-spatial-period translates of each other. If we label the profiles with their humps atthe center of the spatial period interval by “C,” and those profiles with their humps at thewing of the spatial period interval by “W,” then

“W” = σ ◦ “C, ” (8.4)

whereσ is the symmetry group element. More importantly, the time series of the out-put in Figure (1.3) is a chaotic jumping between “C” and “W,” which we refer to aschaotic center-wing jumping. By the relation (8.4) and the study in the last subsection,we interpret the chaotic center-wing jumping as the numerical realization of the shift au-tomorphismχ on four symbols, 1,2,−1,−2, of course with the tolerance of the “exceptone point”–type assumptions (A1)–(A3).


Fig. 8.2.Spatial-temporal profile realization ofLC in Figure 8.1.

We make this more precise in terms of the phase space geometry. In [12], the hyper-bolic structure for the (ε = 0) integrable case of (1.1) was identified. This hyperbolicstructure projected onto the plane of the Fourier component cosk1ζ (4.1) is illustratedin Figure 8.1 and is labeled byLC andLW. From the symmetry, we know that

LW = σ ◦ LC.

LC has the spatial-temporal profile realization as in Figure 8.2 [17].LW corresponds tothe half-spatial-period translate of the spatial-temporal profile realization as in Figure 8.2,in particular, with the hump located at the boundaries. An orbit insideLC, LCin has aspatial-temporal profile realization as in Figure 8.3 [17]. The half-period translate ofLCin, LWin is insideLW. An orbit outsideLC and LW, Lout has the spatial-temporalprofile realization as in Figure 8.4 [17]. The two slabs (unstable slices ofSl ), Sl andSl ,σ , projected onto the plane of “cosk1ζ ,” are also illustrated in Figure 8.1. Then, thechaotic center-wing jumping (Figure 1.3) as the realization of the shift automorphismon four symbols, 1,2,−1,−2, in Sl ∪ Sl ,σ becomes more apparent. Concerning the“except one point”–type assumptions (A1)–(A3), we believe that first of all, they are notverifiable analytically or numerically; secondly, the exceptional cases are numericallyunobservable, and even numerical error can remove the exceptional cases away.

Fig. 8.3.Spatial-temporal profile realization ofLCin in Figure 8.1.

414 Y. Li

Fig. 8.4.Spatial-temporal profile realization ofLout in Figure 8.1.

9. Conclusion

In [12], we gave an intensive Morse study of the level sets of the integrable NLS equation[(ε = 0) in (1.1)]. Based upon that study, the existence of homoclinic orbits in theperturbed NLS system (1.1) was established in [11] (stated in Theorem 1.3 above). In thispaper, based upon the above studies [12] [11] and the studies for perturbed discrete NLSsystems [13] [14], the existence of Smale horseshoes and symbolic dynamics (which arethe hallmark ofchaos) is established in the neighborhood of the homoclinic orbits undercertain “except one point”–type assumptions [Theorem 8.1]. Such a study leads to theinterpretation of the numerical observation:chaotic center-wing jumping(Figure 1.3) onthe perturbed NLS system (1.1) through shift dynamics with the tolerance of the “exceptone point”–type assumptions. This study is a generalization of the finite-dimensionalstudy [14] to the infinite-dimensional system (1.1).

Acknowledgments

This work is built upon my collaborated works with the brilliant mathematicians DaveMcLaughlin, Jalal Shatah, and Steve Wiggins, from whom I have greatly learnt overthe years. I have greatly benefited from conversations with Professors Peter Bates, NickErcolani, Greg Forest, Yannis Keverekidis, Dave Levermore, and Jerry Marsden; Drs. An-nalisa Calini, George Haller, Ben Luce, Connie Schober, and Mark Winograd.

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