Homoclinic Orbits and Chaos in Discretized Perturbed NLS...

J. Nonlinear Sci. Vol. 7: pp. 315–370 (1997)

© 1997 Springer-Verlag New York Inc.

Homoclinic Orbits and Chaos in Discretized PerturbedNLS Systems: Part II. Symbolic Dynamics

Y. Li 1 and S. Wiggins21 Department of Mathematics, University of California at Los Angeles,

Los Angeles, CA 90024, USAPresent Address: Department of Mathematics, 2-336,Massachusetts Institute of Technology, Cambridge, MA 02139, USA

2 Applied Mechanics, and Control and Dynamical Systems 116-81,California Institute of Technology, Pasadena, CA 91125, USA

Received September 1, 1995; revised manuscript accepted for publication September 1, 1996Communicated by Jerrold Marsden

Summary. In Part I ([9], this journal), Li and McLaughlin proved the existence ofhomoclinic orbits in certain discrete NLS systems. In this paper, we will construct Smalehorseshoes based on the existence of homoclinic orbits in these systems.

First, we will construct Smale horseshoes for a general high dimensional dynamicalsystem. As a result, a certain compact, invariant Cantor set3 is constructed. The Poincar´emap on3 induced by the flow is shown to be topologically conjugate to the shiftautomorphism on two symbols, 0 and 1. This gives rise to deterministicchaos. We applythe general theory to the discrete NLS systems as concrete examples.

Of particular interest is the fact that the discrete NLS systems possess a symmetric pairof homoclinic orbits. The Smale horseshoes and chaos created by the pair of homoclinicorbits are also studied using the general theory. As a consequence we can interpret certainnumerical experiments on the discrete NLS systems as “chaotic center-wing jumping.”

Key words. discretized nonlinear Schroedinger equations, Smale horseshoes, chaos

MSC numbers. 58F07, 58F13, 70K50

1. Introduction

In Part I of this study ([9], this journal), Li and McLaughlin established the existenceof homoclinic orbits for the followingN-particle (any 2< N < ∞) finite-differencedynamical system:

i q̇n = 1µ2

[qn+1− 2qn + qn−1] + |qn|2(qn+1+ qn−1)− 2ω2qn

+i ²[−αqn + β

µ2(qn+1− 2qn + qn−1)+ 0

], (1.1)

316 Y. Li and S. Wiggins

wherei = √−1, qn’s are complex variables, andqn+N = qn (periodic condition),qN−n = qn (even condition);

therefore, this is a 2(M + 1) dimensional system, where{M = N/2 (N even),M = (N − 1)/2 (N odd),

with

µ = 1N ,N tan πN < ω < N tan

2πN , for N > 3,

3 tanπ3 < ω 0), β (> 0), 0 (> 0) are constants.

This system is a finite-difference discretization of the following perturbed NLS system:

iqt = qxx + 2[|q|2− ω2]q + i ² [−αq + βqxx + 0] , (1.2)

whereq(x + 1) = q(x), q(−x) = q(x); π < ω < 2π , ² ∈ [0, ²1), α (> 0), β (> 0),and0 (> 0) are constants.

The motivation for this study comes from the numerical experiments showing theexistence of chaos in near-integrable systems as summarized in [11]. These numericalexperiments show that the perturbed nonlinear Schroedinger equations (1.2) possess so-lutions which consist of beautiful regular spatial patterns that evolve irregularly (chaoti-cally) in time. (See Figure 1.) These numerical studies also correlate this chaotic behaviorin the perturbed system with the presence of a hyperbolic structure, and homoclinic man-ifolds connecting this hyperbolic structure, in the unperturbed(² = 0) integrable NLSequation (see Li [8], Li and McLaughin [10]).

In Part I ([9], this journal), Li and McLaughin proved the existence of homoclinicorbits for (1.1), and we now state the main result from that paper. We denote the externalparameter space by6N (N ≥ 7) where

6N ={(ω, α, β, 0) | ω ∈

(N tan

π

N, N tan

2π

N

),

0 ∈ (0, 1), α ∈ (0, α0), β ∈ (0, β0),whereα0 andβ0 are any positive numbers(

Symbolic Dynamics 317

Fig. 1. Solution of the nonlinear Schroedinger equa-tion in the chaotic regime. The waveform “ran-domly” switches from a form localized at the centerof the interval to a form localized at the ends (or“wings”) of the interval with an intermediate pas-sage through a spatially uniform or “flat” state.

Remark 1.1. In the cases (3≤ N ≤ 6), κ is always negative as shown in Figure 2.Since we require both dissipation parametersα andβ to be positive, the relationβ = καshows that the existence of homoclinic orbits violates this positivity. WhenN is evenand> 7, there is in fact a pair of homoclinic orbits asymptotic to a fixed pointq² ; sinceif qn = f (n, t) solves (1.1), thenqn = f (n+ N/2, t) also solves (1.1).

In this paper, we show how Smale horseshoes can be constructed near these homoclinicorbits and, most importantly, how the geometry associated with the horseshoes gives amechanism for chaotic “center-wing jumping” as described in [11].

We will first construct Smale horseshoes for a general system, then cast the discreteNLS system (1.1) into the special form of the general system. Horseshoes, and theassociated symbolic dynamics, have been constructed for the same types of systemsby Silnikov [18] and Deng [4]. However, they do not treat the case of two homoclinicorbits which arise as a result of a symmetry. Moreover, our construction uses differentmethods. We usen-dimensional versions of the Conley-Moser conditions (see Moser [13]and Wiggins [19]). This method of proof offers some advantages over that of Silnikovand Deng.

1. The Conley-Moser conditions give rise to a geometrical description of chaotic dynam-ics in phase space that is particular to the problem being studied. Using our horseshoeconstruction we are able to interpret the numerical observation in Figure 1 using


(a) (b)

(c)

Fig. 2. The slopeκ, as a function ofω, of the hyperplane in the parameter space near whichhomoclinic orbits occur. (a) The slope for different values ofN showing convergence to the PDEcurve corresponding toN = ∞. (b) Slopes forN ≥ Nc = 7. (c) Slopes forN < Nc = 7.


symbolic dynamics on four symbols (−2,−1; 1, 2). We show that geometrically thechaotic dynamics can be interpreted as the “chaotic center-wing jumping” that is seenin the numerical experiments.

2. The Conley-Moser conditions allow one to easily conclude structural stability of theinvariant Cantor set.

3. Hyperbolicity of the invariant Cantor set is an easy consequence. This allows one tobring in a variety of statistical techniques from dynamical systems 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 (see, e.g., [7]).

There is another important point to be made. The conditions of Conley and Moser aregeneral conditions that do not rely on being near a homoclinic orbit for their application.Rather, they are sufficient conditions for a dynamical system to possess an invariant seton which the dynamics is topologically conjugate to a subshift of finite type. The workof Conley and Moser was originally for two-dimensional maps. In this low-dimensionalcase one can rather easily get a handle on the geometry of the image and preimageof selected regions of the domain of a map. In higher dimensions this becomes moreproblematic. Consequently, there are relatively few applications of this technique inhigher dimensions, despite the obvious advantages. This is one of the contributions ofthis paper. We show how generalizations of the Conley-Moser conditions can be appliedfor this general class of vector fields having a homoclinic orbit. We do this by using thelocal geometry of the fixed points of the Poincaré map. This construction plays a keyrole in our equivariant construction of the symbolic dynamics which is crucial for theinterpretation of the center-wing jumping.

The structural stability of the Conley-Moser construction is useful in another area.In order to describe the dynamics near the hyperbolic fixed point we use a smooth lin-earization argument. For ann-dimensional system this requires a countable number ofnonresonance conditions on the eigenvalues of the matrix associated with the vectorfield linearized at the fixed point. In [9] it is shown that a homoclinic orbit exists ona codimension one surface in the parameter space. Each nonresonance condition alsodefines a codimension one surface in the parameter space, and it intersects the surfaceon which the homoclinic orbits exist in a set of measure zero. There are a countablenumber of such resonance conditions, so the set of parameter values on this surfacewhere nonresonance fails is still a set of measure zero. Hence, on a codimension onesurface in parameter space there is a set of full measure where the nonresonance condi-tions hold. Now since the horseshoes constructed using the Conley-Moser constructionare structurally stable, they also persist for this set of measure zero where the nonreso-nance conditions break down, since the set of full measure where nonresonance holds isdense.

This paper is organized as follows. Sections 2.1–2.5 develop the general setting al-lowing us to construct the Poincaré map in a neighborhood of the homoclinic orbit for thegeneral class of systems under consideration. In Section 2.6 we compute fixed points ofthe Poincaré map. In Section 2.7 we develop then-dimensional versions of the Conley-Moser conditions. These are slight generalizations of those given in [19], and possibly


more easy to apply in high dimensional problems. We then use them to construct Smalehorseshoes near the homoclinic orbit. In Section 3 we apply these results to homoclinicorbits in the discrete NLS system. In the case where there are two homoclinic orbits weshow how the horseshoe chaos can be interpreted as chaotic “center-wing” jumping.

2. General Theory

2.1. General Setting

We study the following (2m+ n)-dimensional system:ẋj = ²jαj xj − βj yj + Xj (x, y, z),ẏj = βj xj + ²jαj yj + Yj (x, y, z),żk = δkγkzk + Zk(x, y, z), (2.1)( j = 1, . . . ,m; k = 1, . . . ,n),

where

²j ={−1, 1≤ j ≤ m1,

1, m1 < j ≤ m.

δk ={

1, 1≤ k ≤ n1,−1, n1 < k ≤ n,

x ≡ (x1, . . . , xm)T ,y ≡ (y1, . . . , ym)T ,z ≡ (z1, . . . , zn)T ,

Xj (0, 0, 0) = Yj (0, 0, 0) = 0,gradXj (0, 0, 0) = gradYj (0, 0, 0) = 0,

( j = 1, . . . ,m),Zk(0, 0, 0) = gradZk(0, 0, 0) = 0,

(k = 1, . . . ,n).

Moreover,Xj (x, y, z), Yj (x, y, z), andZk(x, y, z) areC∞ functions in a neighborhoodof (0, 0, 0), andαj ,βj , andγk are positive constants. Therefore, (0, 0, 0) is a saddle point.We assume that this system (2.1) has the following two properties:

1. There is a homoclinic orbith asymptotic to (0, 0, 0).2. (a) α1 < αj , for any 2≤ j ≤ m1; α1 < γk, for anyn1 < k ≤ n.


(b) γ1 < γk, for any 1≤ k ≤ n1; γ1 < αj , for anym1 < j ≤ m.(c) α1 < γ1.

Property 2 says thatα1 is the smallest attracting rate, andγ1 is the smallest repellingrate; moreover,α1 is smaller thanγ1. We also assume that the solution operatorFt0,

Ft0: (x(0), y(0), z(0)) 7→ (x(t), y(t), z(t)),

is C1 in t, and aC2 diffeomorphism for fixed finitet .The motivation for this study comes from the study of the discrete NLS system [9],

in which the discrete NLS system can be normalized into a special form of the abovesystem (2.1) (see Section 3).

We are going to define two Poincar´e sections60 and61 in a small neighborhood of(0, 0, 0). The Poincar´e map

P: U ⊂ 60 7→ 60induced by the flow can be expressed as the composition of two maps

P = P01 ◦ P10 ,

where

P10 : U0 ⊂ 60 7→ 61,P01 : U1 ⊂ 61 7→ 60,

are also induced by the flow. In particular,P10 is induced by the flow in a neighborhoodof (0, 0, 0), andP01 is induced by the global flow outside a neighborhood of (0, 0, 0). Wedefine60 and61 in such a way that they sit in a tubular neighborhood of the homoclinicorbit h. Moreover, the “flight time” for orbits starting from points on61 to reach60 isbounded. These facts enable us to approximateP01 by a linear transformation. The “flighttime” for orbits starting from points on60 to reach61 is unbounded. Nevertheless,P10is induced by the local flow. We will construct Smale horseshoes on60 under the mapP. As a result, restricted to some compact Cantor subset of60, the dynamics ofP istopologically conjugate to the shift automorphism on symbols.

It is much more difficult to construct horseshoes in high-dimensional systems than inlow-dimensional systems. Below we list the main difficult points:

1. Since there are many different attracting and repelling rates in (2.1) at the saddle point(0, 0, 0), linearized dynamics cannot approximate the full nonlinear dynamics in theneighborhood of (0, 0, 0). A simple illustration of this fact is given in the Appendix.

2. The affine transformation, as an approximation toP01 , has the representation as anonsingular large-matrix transformation. It is very hard to track objects on61 afterthis large-matrix transformation.

3. In high dimensional systems, verification of the so-called Conley-Moser conditionsbecomes much harder.

Point 1 will be overcome through smooth normal form reduction [6]. Point 2 will beovercome through identifying the fixed points ofP and using the local structure of the


fixed points. Similarly, point 3 will be overcome through certain generic considerationsand the local geometric structure near fixed points ofP. In this way this paper sets up ageneral program for constructing Smale horseshoes in a large class of high-dimensionaldynamical systems.

2.2. Smooth Normal Form Reduction

The reference for this section is [6]. Consider the linear part of the system (2.1), denoteby

aj ( j = 1, . . . ,2m+ n)the eigenvalues,

aj =

δj γj , j = 1, . . . ,n,²kαk + iβk, j = 2k− 1+ n, k = 1, . . . ,m,²kαk − iβk, j = 2k+ n, k = 1, . . . ,m,

wherei = √−1 and

²k ={−1, 1≤ k ≤ m1,

1, m1 < k ≤ m,

δj ={

1, 1≤ j ≤ n1,−1, n1 < j ≤ n.

We assume the following nonresonance condition:

(G1) aj 6=2m+n∑k=1

lkak, mod[i 2π ], (2.2)

for all j = 1, . . . ,2m+ n; and all sets of nonnegative integersl1, . . . , l2m+n such that

2≤2m+n∑k=1

lk


More importantly, in a small neighborhoodÄ of (x′, y′, z′) = (0, 0, 0), system (2.1) isreduced to the linear system [6],

ẋ′j = ²jαj x′j − βj y′j ,ẏ′j = βj x′j + ²jαj y′j ,ż′k = δkγkz′k,

(2.3)

( j = 1, . . . ,m; k = 1, . . . ,n).Denote the solution operator of system (2.3) byLt . The solution operator of system (2.1)is Ft0. Let

Ft ≡ RFt0 R−1,then,Ft is the solution operator of the transformed system of (2.1) underR. Moreover,in Ä, Ft = Lt . From now on, we are mainly concerned with the transformed system,

ẋ′j = ²jαj x′j − βj y′j + X′j (x, y, z),ẏ′j = βj x′j + ²jαj y′j + Y′j (x, y, z),ż′k = δkγkz′k + Z′k(x, y, z),

(2.4)

( j = 1, . . . ,m; k = 1, . . . ,n),whereX′j , Y

′j , andZ

′k vanish identically insideÄ. SinceF

t0 is aC

2 diffeomorphism forfixed t , so isFt . Moreover, sinceR is independent oft , Ft0 is C

1 in t , and so isFt . Fromnow on, we shall drop the “primes” in systems (2.3;2.4).

Remark 2.1. The requirement thatXj , Yj , and Zk in (2.1) are of classC∞ can bereplaced by requiring that they are of classCN , where [6]

N = N(2m+ n; {aj }, j = 1, . . . ,2m+ n).

2.3. Some Definitions

In this section we will define two Poincar´e sections60 and61 and two Poincar´e maps,

P10 : U0 ⊂ 60 7→ 61,P01 : U1 ⊂ 61 7→ 60.

We will denote the stable and unstable manifolds of (0, 0, 0) by Ws andWu, respec-tively. We know that

h ⊂ Ws ∩Wu.We denote the components of

Ä ∩Ws and Ä ∩Wu

containing the origin by

Wsloc and Wuloc,


respectively. We know thatWsloc andWuloc coincide with the corresponding stable and

unstable subspaces, respectively. In particular, they have the following representations:

Wsloc ≡{(x, y, z)

∣∣ xj = 0, yj = 0,m1 < j ≤ m; zk = 0, 1≤ k ≤ n1} ,Wuloc ≡

{(x, y, z)

∣∣ xj = 0, yj = 0, 1≤ j ≤ m1; zk = 0, n1 < k ≤ n} .We refer toh ∩Wsloc ≡ h+ as the theforward time segmentandh ∩Wuloc ≡ h− as thebackward time segment, respectively. By assumption 2 on system (2.1),α1 is the smallestattracting rate,γ1 is the smallest repelling rate. Therefore, generically,

• (G2) h+ is tangent to the (x1, y1)-plane at (0, 0, 0), h− is tangent to the positivez1-axis at (0, 0, 0).

Specifically,h+ andh− can be parametrized as follows:x+j (t) = e−αj t (x+j (0) cosβj t − y+j (0) sinβj t),y+j (t) = e−αj t (x+j (0) sinβj t + y+j (0) cosβj t); (1≤ j ≤ m1),z+k (t) = e−γkt z+k (0) (n1 < k ≤ n),

(2.5)

0≤ t


Definition 2. The Poincaré section61 is defined by the constraints,

z1 = η,|zk| < η, k = 2, . . . ,n,x2j + y2j < η2, j = 1, . . . ,m.

Definition 3. The Poincaré mapP10 from60 to61 is defined as

P10 : U0 ⊂ 60 7→ 61,∀q ∈ U0, P10 (q) = Ft∗(q) ∈ 61,

wheret∗ = t∗(q) > 0 is the smallest time such thatFt∗(q) ∈ 61.

Definition 4. The Poincaré mapP01 from61 to 6̄0 (≡ 60 ∪ ∂60) is defined as

P01 : U1 ⊂ 61 7→ 6̄0,∀q ∈ U1, P01 (q) = FT(q)(q) ∈ 6̄0,

whereT(q) > 0 is the smallest time such thatFT(q)(q) ∈ 6̄0.

If η is sufficiently small,

60 ⊂ Ä, 61 ⊂ Ä.By the representation (2.5) ofh+, we can chooseη, such thath+ intersects the (z1 = 0)boundary of60 at

q+ ≡ h+ ∩ ∂60,whereq+ has the coordinates

x1 = x+1 , y1 = 0,xj = x+j , yj = y+j , (2≤ j ≤ m1),zk = z+k , (n1 < k ≤ n),xj = 0, yj = 0, (m1 < j ≤ m),zk = 0, (1≤ k ≤ n1),

with η1 < x+1 < η, andx

+j , y

+j , andz

+k satisfy the corresponding inequalities in the

definition of60. Similarly, h− intersects61 at

q− ≡ h− ∩61,


whereq− has the coordinates

z1 = η,xj = x−j , yj = y−j (m1 < j ≤ m),zk = z−k (2≤ k ≤ n1),xj = 0, yj = 0, (1≤ j ≤ m1),zk = 0, (n1 < k ≤ n),

andx−j , y−j , andz

−k satisfy the corresponding inequalities in the definition of61.

Finally, we have the crucial fact,

P01 (q−) = q+. (2.9)

Lemma 2.1. If η is sufficiently small, then the Poincaré sections60 and61 are transver-sal to the vector field.

Proof. First we will show that6̄0 is transversal to the vector field at the pointq+. Letn0 be the unit normal vector tō60; thenn0 has the coordinate representation asy1 = 1,and all other coordinates are zeros. The vector fieldv+ at q+ can be obtained throughdifferentiating (2.5) with respect tot . Notice that for the calculation of the inner product

〈n0, v+〉,we only need to know they1-coordinate ofv+, which is

x+1 (0) cosβ1t+ − y+1 (0) sinβ1t+,wheret+ satisfies the equation

x+1 (0) sinβ1t+ + y+1 (0) cosβ1t+ = 0. (2.10)Assume that

〈n0, v+〉 = 0;then,

x+1 (0) cosβ1t+ − y+1 (0) sinβ1t+ = 0. (2.11)Eqs. (2.10), (2.11) imply that

x+1 (0) = y+1 (0) = 0.This contradicts the generic condition (2.7). Thus,

〈n0, v+〉 6= 0.This, together with the smoothness of the vector field, implies that for sufficiently smallη,60 is transversal to the vector field at any point of60. Similarly for61. This completesthe proof of the lemma.


Lemma 2.2. If η is sufficiently small, then the Poincaré map P10 is 1-1; moreover, if inaddition U1 is inside a sufficiently small neighborhood of q−, then P01 is 1-1.

Proof. The claim thatP01 is 1-1 follows immediately from the fact thatFt is C1 in t and

a C2 diffeomorphism for fixedt . For details, see Section 2.5. Next, we show thatP10 is1-1. Let p1, p2 be two different points inU0 ⊂ 60, and assume that

P10 (p1) = P10 (p2). (2.12)

We will only need to study the (x1, y1, z1) coordinates of pointsp1, p2: P10 (p1) andP10 (p2). Let t1 andt2 be the time for orbits starting fromp1 andp2 to reachP

10 (p1) and

P10 (p2), respectively. Then,

t1 = 1γ1

ln η/z1|p1, t2 =1

γ1ln η/z1|p2.

We introduce the notationz1|p1 to denote thez1 coordinate ofp1. Sincez1|p1 > 0,z1|p2 > 0, botht1 andt2 are finite. The relations

x1|P10 (p1) = x1|P10 (p2), y1|P10 (p1) = y1|P10 (p2),

lead to

x1|p1e−α1t1 cosβ1t1 = x1|p2e−α1t2 cosβ1t2, (2.13)x1|p1e−α1t1 sinβ1t1 = x1|p2e−α1t2 sinβ1t2, (2.14)

since botht1 andt2 are finite; moreover, bothx1|p1 andx1|p2 satisfy the constraint,

η1 < x1 < η, (2.15)

whereη1 > η exp{−2πα1/β1}. Then from (2.13;2.14), we have

tanβ1t1 = tanβ1t2,

and thus,

β1t1 = β1t2+ jπ, for some j ∈ Z.If j is odd, then neither (2.13) nor (2.14) will hold. Therefore,j is even (= 2 j0). Finally,from (2.13;2.14), we have

x1|p1x1|p2

= exp{2α1 j0π /β1}.

This relation contradicts the constraint (2.15), except for the casej0 = 0. But, if j0 =0, thent1 = t2. In this case, the assumption (2.12) contradicts the fact thatFt1 is adiffeomorphism. Thus, in any case, the assumption (2.12) is not valid, which shows thatP10 is 1-1. This completes the proof of the lemma.


2.4. The Poincaré Map P10

Denote the coordinates on60 by(x01; {x0j , y0j }, j = 2, . . . ,m; z0k, k = 1, . . . ,n

).

Denote the coordinates on61 by({x1j , y1j }, j = 1, . . . ,m; z1k, k = 2, . . . ,n) .In Ä, the solution operatorFt = Lt has the representation

xj (t) = e²j αj t(xj (0) cosβj t − yj (0) sinβj t

),

yj (t) = e²j αj t(xj (0) sinβj t + yj (0) cosβj t

),

zk(t) = eδkγkt zk(0)( j = 1, . . . ,m; k = 1, . . . ,n).

Let t∗ be the “flight” time for an orbit starting from a point on60 to reach61. Then,

t∗ = 1γ1

ln η/z01. (2.16)

Using this expression for the solution operator and the “flight” time,P10 is given by{x11 = e−α1t∗x01 cosβ1t∗,y11 = e−α1t∗x01 sinβ1t∗,{x1j = e−αj t∗(x0j cosβj t∗ − y0j sinβj t∗),y1j = e−αj t∗(x0j sinβj t∗ + y0j cosβj t∗) (2≤ j ≤ m1),

z1k = e−γkt∗z0k (n1 < k ≤ n), (2.17){x1j = eαj t∗(x0j cosβj t∗ − y0j sinβj t∗),y1j = eαj t∗(x0j sinβj t∗ + y0j cosβj t∗) (m1 < j ≤ m),

z1k = eγkt∗z0k (1≤ k ≤ n1). (2.18)

2.5. The Poincaré Map P01

Let 6̂0 be the enlargement of60 specified by the condition

−η < z1 < η.

The Poincar´e mapP01 can be naturally extended tôP01 , and we denote the extension by

P̂01 : Û1 ⊂ 61 7→ 6̂0.


We know from (2.9) that

P̂01 (q−) = q+.

Denote byT(q) the flight time for the orbit starting from the pointq on61 to reach6̂0.Then,

FT(q−)(q−) = q+.

SinceFt is C1 in t and aC2 diffeomorphism for fixedt , the implicit function theoremimplies that there is a small neighborhoodU−1 of q

− in61 in whichT(q) is aC1 functionof q. Restricted toU−1 , P̂

01 has the representation,

P̂01 (q− +1q) = q+ + 〈gradP̂01 (q−),1q〉 + N(q−,1q), (2.19)

where

‖N(q−,1q)‖ ∼ o(‖1q‖), as‖1q‖ → 0;〈, 〉 and‖ ‖ are the usual Cartesian inner product and norm; moreover,

gradP̂01 (q−) = ∂

∂tFT(q

−)(q−) · ∂∂q

T(q−)+ ∂∂q

FT(q−)(q−).

Next, we introduce new coordinates on60 and61 with q+ andq− as origins, respectively.On60,

x01 = x+1 + x̃01,{x0j = x+j + x̃0j ,y0j = y+j + ỹ0j (2≤ j ≤ m1),

z0k = z+k + z̃0k (n1 < k ≤ n),{x0j = x̃0j ,y0j = ỹ0j (m1 < j ≤ m),

z0k = z̃0k (1≤ k ≤ n1).

On61,

z1k = z−k + z̃1k (2≤ k ≤ n1),{x1j = x−j + x̃1j ,y1j = y−j + ỹ1j (m1 < j ≤ m),

z1k = z̃1k (n1 < k ≤ n),{x1j = x̃1j ,y1j = ỹ1j (1≤ j ≤ m1).


In terms of the new coordinates, Eq. (2.19) can be written as x̃0ỹ0z̃0

= A x̃1ỹ1

z̃1

+ B, (2.20)where

x̃0 ≡ (x̃01, . . . , x̃0m)T ,ỹ0 ≡ (ỹ02, . . . , ỹ0m)T ,z̃0 ≡ (z̃01, . . . , z̃0n)T ,x̃1 ≡ (x̃11, . . . , x̃1m)T ,ỹ1 ≡ (ỹ11, . . . , ỹ1m)T ,z̃1 ≡ (z̃12, . . . , z̃1n)T .

A is a(2m+n−1)× (2m+n−1) constant matrix,B is a(2m+n−1) column vectorfunction ofq− and1q; moreover,

‖B‖ ∼ o(‖1q‖), as‖1q‖ → 0,

1q = x̃1ỹ1

z̃1

.A convenient notation forA is the following block form:

A(x,x)jk A(x,y)jk A

(x,z)jk

A(y,x)jk A(y,y)jk A

(y,z)jk

A(z,x)jk A(z,y)jk A

(z,z)jk

, (2.21)

where the indexj in a block runs through the dimension of the first superscript and theindexk runs through the dimension of the second superscript. A similar notation is usedfor the entries ofB:

B(x)j , B(y)k , etc.

2.6. Fixed Points of the Poincar´e Map P ≡ P01 ◦ P10The Poincar´e mapP is defined as

P: U ⊂ 60 7→ 60,P = P01 ◦ P10 . (2.22)


A fixed point of P corresponds to a periodic orbit of system (2.4).Supposeq ∈ U ⊂ 60is a fixed point ofP, with coordinates(

x̃01; {x̃0j , ỹ0j }, j = 2, . . . ,m; z̃0k, k = 1, . . . ,n). (2.23)

Then,

P(q) = q (2.24)gives (2m+ n − 1) equations with (2m+ n − 1) variables (2.23). The task is to findsolutions to the system (2.24). However, the set of variables (2.23) is not the best forsolving system (2.24). We will next give a set of variables that is more suited to thispurpose.

2.6.1. The Silnikov Variables. We consider a set of variables coordinatizing a regionS, and constructed from the variables on60 and61,given by(

x̃01; {x̃0j , ỹ0j }, 2≤ j ≤ m1; {x̃1j , ỹ1j },m1 < j ≤ m;

t∗; z̃1k, 2≤ k ≤ n1; z̃0k, n1 < k ≤ n),

which are defined by the transformation

T : S → 60,

x̃01 → x̃01,{x̃0j , ỹ0j } → {x̃0j , ỹ0j }, 2≤ j ≤ m1,

z̃0k → z̃0k, n1 < k ≤ n,

x̃1j → x̃0j = e−αj t∗[(x−j + x̃1j ) cosβj t∗ + (y−j + ỹ1j ) sinβj t∗

], m1 < j ≤ m,

ỹ1j → ỹ0j = e−αj t∗[−(x−j + x̃1j ) sinβj t∗ + (y−j + ỹ1j ) cosβj t∗

], m1 < j ≤ m,

z̃1k → z̃0k = e−γkt∗(z−k + z̃1k), 2≤ k ≤ n1,t∗ → z̃01 = e−γ1t∗η, (2.25)

with

T−1 : 60→ S,

x̃01 → x̃01,{x̃0j , ỹ0j } → {x̃0j , ỹ0j }, 2≤ j ≤ m1,

z̃0k → z̃0k, n1 < k ≤ n,


x̃0j → x̃1j = −x−j + eαj t∗[x̃0j cosβj t∗ − ỹ0j sinβj t∗

], m1 < j ≤ m,

ỹ0j → ỹ1j = −y−j + eαj t∗[x̃0j sinβj t∗ + ỹ0j cosβj t∗

], m1 < j ≤ m,

z̃0k → z̃1k = −z−k + eγkt∗ z̃0k, 2≤ k ≤ n1,

z̃01 → t∗ =1

γ1log

η

z̃01. (2.26)

We refer to these variables as theSilnikov variablessince they were first used by Sil-nikov [15], [16], [17], [18], and further developed by Deng [3]. These variables will beparticularly useful for finding fixed points of the Poincar´e map. In particular, we will usethem by computing the conjugate ofP01 ◦ P10 with T , i.e.,

T−1 ◦ P01 ◦ P10 ◦ T, (2.27)

and seeking fixed points of this map. Clearly, these fixed points correspond to fixedpoints ofP01 ◦ P10 under the mapT . Using (2.17), (2.20), (2.25), and (2.26), we computethe components of the conjugated map. Multiplying each equation byeα1t∗ , and usingthe ordering of the eigenvalues given in Section 2.1, we obtain the following equations:

eα1t∗ σ̃ 0′

k = (x+1 + x̃01)(

A(σ,x)k1 cosβ1t∗ + A(σ,y)k1 sinβ1t∗)

+m∑

j=m1+1

(A(σ,x)k j e

α1t∗ x̃1j + A(σ,y)k j eα1t∗ ỹ1j)

+n1∑

j=2A(σ,z)k j e

α1t∗ z̃1j + C(σ )k , (2.28)σ = x, 1< k ≤ m1,σ = y, 2< k ≤ m1,σ = z, n1 < k ≤ n.

0 = (x+1 + x̃01)(

A(z,x)k1 cosβ1t∗ + A(z,y)k1 sinβ1t∗)

+m∑

j=m1+1

(A(z,x)k j e

α1t∗ x̃1j + A(z,y)k j eα1t∗ ỹ1j)

+n1∑

j=2A(z,z)k j e

α1t∗ z̃1j + C(z)k , 1≤ k ≤ n1, (2.29)

0 = (x+1 + x̃01){(

A(x,x)k1 cosβ1t∗ + A(x,y)k1 sinβ1t∗)

cosβkt∗

−(

A(y,x)k1 cosβ1t∗ + A(y,y)k1 sinβ1t∗)

sinβkt∗}


+{

m∑j=m1+1

(A(x,x)k j e

α1t∗ x̃1j + A(x,y)k j eα1t∗ ỹ1j)

+n1∑

j=2A(x,y)k j e

α1t∗ z̃1j

}cosβkt∗

−{

m∑j=m1+1

(A(y,x)k j e

α1t∗ x̃1j + A(y,y)k j eα1t∗ ỹ1j)

+n1∑

j=2A(y,z)k j e

α1t∗ z̃1j

}sinβkt∗

+C(x)k , m1 < k ≤ m, (2.30)

0 = (x+1 + x̃01){(

A(x,x)k1 cosβ1t∗ + A(x,y)k1 sinβ1t∗)

sinβkt∗

+(

A(y,x)k1 cosβ1t∗ + A(y,y)k1 sinβ1t∗)

cosβkt∗}

+{

m∑j=m1+1

(A(x,x)k j e

α1t∗ x̃1j + A(x,y)k j eα1t∗ ỹ1j)

+n1∑

j=2A(x,y)k j e

α1t∗ z̃1j

}sinβkt∗

+{

m∑j=m1+1

(A(y,x)k j e

α1t∗ x̃1j + A(y,y)k j eα1t∗ ỹ1j)

+n1∑

j=2A(y,z)k j e

α1t∗ z̃1j

}cosβkt∗

+C(y)k , m1 < k ≤ m, (2.31)where the primes on the variables in (2.28) denote the images of the unprimed coordinatesand the functionsC(σ )k → 0, ast∗ → +∞.The common factor ofeα1t∗ makes it naturalto consider these equations as functions of the following scaled coordinates:

t∗; ẑ1k ≡ eα1t∗ z̃1k, 2≤ k ≤ n1; (2.32)(x̂1j ≡ eα1t∗ x̃1j , ŷ1j ≡ eα1t∗ ỹ1j

), m1 < j ≤ m;

x̂01 ≡ eα1t∗ x̃01; ẑ0k ≡ eα1t∗ z̃0k, n1 < k ≤ n;(x̂0j ≡ eα1t∗ x̃0j , ŷ0j ≡ eα1t∗ ỹ0j

), 2≤ j ≤ m1. (2.33)

Note that the coefficient multiplying sinβkt∗ in (2.30) is the same as the coefficientmultiplying cosβkt∗ in (2.31). Similarly, the coefficient multiplying cosβkt∗ in (2.30) is


the same as the coefficient multiplying sinβkt∗ in (2.31). Hence, equivalent equationsthat the fixed points must satisfy are given by

0 = x+1(


+m∑

j=m1+1

(A(σ,x)k j x̂

1j + A(σ,y)k j ŷ1j

)+

n1∑j=2

A(σ,z)k j ẑ1j + C(σ )k , (2.34)

{σ = x, y, m1 < k ≤ m,σ = z, 1≤ k ≤ n1.

σ̂ 0k = x+1(


+m∑

j=m1+1

(A(σ,x)k j x̂

1j + A(σ,y)k j ŷ1j

)+

n1∑j=2

A(σ,z)k j ẑ1j + C(σ )k , (2.35)

σ = x, 1≤ k ≤ m1,σ = y, 2≤ k ≤ m1,σ = z, n1 < k ≤ n.

2.6.2. C(σ )k = 0 Solutions. Next, we setC(σ )k = 0 in system (2.34;2.35), and solve theresulting system. In particular, we first want to solve system (2.34) for

t∗; ẑ1k, 2≤ k ≤ n1,(x̂1j , ŷ

1j ), m1 < j ≤ m.

Then, we substitute these values into (2.35) to get

x̂0j , 1≤ j ≤ m1,ŷ0j , 2≤ j ≤ m1,ẑ0k, n1 < k ≤ n.

Before solving system (2.34), we want to show the following fact.

Lemma 2.3. Let A1 be the[2(m−m1)+ n1] × [2(m−m1)+ n1− 1] matrix,

row(A1) =(

A(σ,x)k(m1+1) . . . A(σ,x)km A

(σ,y)k(m1+1) . . . A

(σ,y)km A

(σ,z)k2 . . . A

(σ,z)kn1

),{

σ = x, y, m1 < k ≤ m,σ = z, 1≤ k ≤ n1.

If the generic condition,

(G3) dim{Tq+Ws ∩ Tq+Wu} = 1,


is true, then

rank(A1) = 2(m−m1)+ n1− 1.

Proof. Let A0 be the [2m+ n− 1]× [2(m−m1)+ n1− 1] matrix,

row(A0) =(

A(σ,x)k(m1+1) . . . A(σ,x)km A

(σ,y)k(m1+1) . . . A

(σ,y)km A

(σ,z)k2 . . . A

(σ,z)kn1

),

σ = x, 1≤ k ≤ m,σ = y, 2≤ k ≤ m,σ = z, 1≤ k ≤ n.

Notice that {(x̂1j , ŷ

1j ), m1 < j ≤ m,

ẑ1k, 2≤ k ≤ n1;coordinatizeWuloc∩61; moreover,Wuloc∩61 is [2(m−m1)+n1−1]-dimensional. SinceP01 is a diffeomorphism, when restricted to the small neighborhoodU

−1 of q

− in 61,P01 (W

uloc∩U−1 ) is also [2(m−m1)+n1−1]-dimensional. Moreover,Tq+P01 (Wuloc∩U−1 )

has the representation

A0

x̂1ŷ1ẑ1

, (2.36)where

x̂1 = (x̂1m1+1, . . . , x̂1m)T ,ŷ1 = (ŷ1m1+1, . . . , ŷ1m)T ,ẑ1 = (ẑ12, . . . , ẑ1n1)T .

Therefore,

rank(A0) = 2(m−m1)+ n1− 1. (2.37)Assume

rank(A1) < 2(m−m1)+ n1− 1,then columns ofA1 are linearly dependent, and thus there exists a nonzero [2(m−m1)+n1− 1] column vectora, such that

A1a = 0. (2.38)

Moreover, by (2.37)

A0a 6= 0. (2.39)


By (2.38),

A0a ⊂ Wsloc ∩60 = Tq+(Wsloc ∩60

).

By (2.36),

A0a ⊂ Tq+P01 (Wuloc ∩U−1 ).Notice that

P01 (Wuloc ∩U−1 ) ⊂ Wu,

then

A0a ⊂ Tq+Ws ∩ Tq+Wu.We also know that

Tq+h ⊂ Tq+Ws ∩ Tq+Wu.Moreover,Tq+h is transversal to60, while A0a ⊂ 60. By (2.39),Tq+h and A0a arelinearly independent, and then

dim{Tq+Ws ∩ Tq+Wu} = 2.

This contradicts the assumption in the lemma, and the lemma is proved.

By this Lemma (2.3), without loss of generality, we assume the [2(m−m1)+n1−1]×[2(m−m1)+ n1− 1] matrix A2,

row(A2) =(

A(σ,x)k(m1+1) . . . A(σ,x)km A

(σ,y)k(m1+1) . . . A

(σ,y)km A

(σ,z)k2 . . . A

(σ,z)kn1

),{

σ = x, y, m1 < k ≤ m;σ = z, 2≤ k ≤ n1,

is nonsingular. This can be used to provide an equation for the variablet∗ that we caneasily solve, which is seen as follows. Writing out (2.34) in matrix form gives

A(z,x)1 j A(z,y)1 j A

(z,z)1 j

A(x,x)k j A(x,y)k j A

(x,z)k j

A(y,x)k j A(y,y)k j A

(y,z)k j

A(z,x)k j A(z,y)k j A

(z,z)k j

x̂ŷ

ẑ

= −x+1

A(z,x)11 cosβ1t∗ + A(z,y)11 sinβ1t∗A(x,x)k1 cosβ1t∗ + A(x,y)k1 sinβ1t∗A(y,x)k1 cosβ1t∗ + A(y,y)k1 sinβ1t∗A(z,x)k1 cosβ1t∗ + A(z,y)k1 sinβ1t∗

,(2.40)

where the matrix on the left of this expression isA1 and the submatrix,A(x,x)k j A

(x,y)k j A

(x,z)k j

A(y,x)k j A(y,y)k j A

(y,z)k j

A(z,x)k j A(z,y)k j A

(z,z)k j

(2.41)


is A2. Then, by the nonsingularity ofA2, there is a unique [2(m−m1) + n1 − 1] rowvectorb that satisfies the following equation:

row(A1)σ=z, k=1 = bA2. (2.42)

However, from (2.40), we have

row(A1)σ=z, k=1 = A(z,x)11 cosβ1t∗ + A(z,y)11 sinβ1t∗. (2.43)

Computing the right-hand side of (2.42) using (2.41) and equating the result to theright-hand side of (2.43) gives

A(z,x)11 cosβ1t∗ + A(z,y)11 sinβ1t∗ =

∑σ=x,y

m1


in system (2.34), and obtain the system

A2

x̂1ŷ1ẑ1

= f l , (2.46)wherex̂1, ŷ1, andẑ1 are defined in (2.36), andf l is a [2(m− m1) + n1 − 1] columnvector,

entry( f l ) = −x+1(

A(σ,x)k1 cosβ1tl∗ + A(σ,y)k1 cosβ1t l∗

),{

σ = x, y, m1 < k ≤ m,σ = z, 2≤ k ≤ n1.

SinceA2 is nonsingular, system (2.46) has a unique solution: x̂lŷlẑl

= A−12 f l . (2.47)We substitute each solution (2.45;2.47) into system (2.35),and obtain the solution,

σ̂ lk = x+1(

A(σ,x)k1 cosβ1tl∗ + A(σ,y)k1 sinβ1t l∗

)+

m∑j=m1+1

(A(σ,x)k j x̂

lj + A(σ,y)k j ŷlj

)+

n1∑j=2

A(σ,z)k j ẑlj , (2.48)

σ = x, 1≤ k ≤ m1,σ = y, 2≤ k ≤ m1,σ = z, n1 < k ≤ n.

2.6.3. Fixed Points.Starting from solutions obtained in the last section, we want tosolve system (2.34;2.35) in the limitt∗ → +∞. Since

C(σ )k → 0, ast∗ → +∞,by the implicit function theorem, we have the following theorem.

Theorem 2.1. There exists an integer l0, such that there are infinitely many solutions,labeled by l (l≥ l0), to system (2.34;2.35):

t∗ = Tl , x̂01 = x̂(0,l )1 ,(x̂0j = x̂(0,l )j , ŷ0j = ŷ(0,l )j

), 2≤ j ≤ m1,

ẑ0k = ẑ(0,l )k , n1 < k ≤ n,(x̂1j = x̂(1,l )j , ŷ1j = ŷ(1,l )j

), m1 < j ≤ m,

ẑ1k = ẑ(1,l )k , 2≤ k ≤ n1,


where, as l→∞,

Tl = 1β1(lπ − ϕ)+ o(1),

x̂(0,l )1 = x̂l1+ o(1),x̂(r,l )j = x̂lj + o(1),

ŷ(r,l )j = ŷlj + o(1),{r = 0, 2≤ j ≤ m1,r = 1, m1 < j ≤ m,

ẑ(r,l )k = ẑlk + o(1),{r = 0, n1 < k ≤ n,r = 1, 2≤ k ≤ n1,

in whichx̂lj , ŷlj , andẑ

lk are given in Eqs. (2.47;2.48).

Sketch of the proof.Let

t∗ = 1β1

[2sπ + τ ], τ ∈ [0, 2π ], s ∈ Z+,

and letv denote the rest of the variables in (2.34;2.35). Then, Eqs. (2.34;2.35) can bewritten as

f (τ, v) ≡ g(τ, v)+ C(s; τ, v) = 0, (2.49)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, (2.50)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→∞, (2.51)

sup(τ,v)∈D

‖∇C(s; τ, v)‖ → 0, ass→∞. (2.52)

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 ). (2.53)Whens is sufficiently large, (2.53) is equivalent to

(τ ′i , v′i ) = −[∇ f (τi , vi )]−1[C(s; τi , vi )+ R(s; τi , vi ; τ ′i , v′i )], (2.54)

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 [2][5] for (2.54) implies the theorem. This completes theproof of the theorem.

Remark 2.2. By this theorem, there are infinitely many periodic orbits, in a neighbor-hoodof the homoclinic orbith. Moreover, by the asymptotic representations of the fixedpoints given in the theorem, this sequence of periodic orbits approaches the homoclinicorbit h asl →+∞.

2.7. Smale Horseshoes

In this section, starting from Theorem 2.1, we construct Smale horseshoes. Our construc-tion will be geometrical in nature. We will use n-dimensional versions of theConley-Moser conditions(cf. Moser [13] and Wiggins [19]).

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

Definition 5. We defineslabs Sl (2l ≥ l0) in 60 as follows:

Sl ≡{

q ∈ 60∣∣∣∣ η exp{−γ1(T2(l+1) − π /2)} ≤ z̃01(q) ≤ η exp{−γ1(T2l − π /2)},

|x̃01(q)| ≤ η exp{−1

2α1T2l

}, |σ̃ 1k (P10 (q))| ≤ η exp

{−1

2α1T2l

},

σ = x, y, m1 < k ≤ m; σ = z, 2≤ k ≤ n1}.

Sl is defined so that it includes two fixed points ofP (see Theorem 2.1). We denote thesetwo fixed points byp+l and p

−l , wherep

+l corresponds toT2l , and p

−l corresponds to

T2l+1 in Theorem 2.1. It follows that

z̃01(p+l ) > z̃

01(p−l ).

If l is sufficiently large,P10 (Sl ) is included in a ball centered atq− on61, with radius of

order

O

(exp

{−1

2α1T2l

}).


Therefore,

P10 (Sl ) ⊂ U−1 .Thus,

Sl ⊂ U,whereU is the domain of definition ofP. (See Eq. (2.22).) Then, sincep+l and p

−l are

fixed points, there exist respectively neighborhoods ofp+l andp−l , V

+l andV

−l , that are

included in the intersection

P(Sl ) ∩ Sl .

2.7.2. Sl and P10 (Sl ). In this subsection we will describe the geometry of the image ofSl underP10 . We denote coordinates on60 and61 by(

x̃01, z̃01, ξ

0s , ξ

0u

)and (

x̃11, ỹ11, ξ

1s , ξ

1u

),

respectively, whereξ τs (τ = 0, 1) is the [2(m1− 1)+ (n− n1)] vector,entry(ξ τs ) = σ̃ τk ,{

σ = x, y, 2≤ k ≤ m1,σ = z, n1 < k ≤ n;

ξ τu (τ = 0, 1) is the [2(m−m1)+ n1− 1] vector,entry(ξ τu ) = σ̃ τk ,{

σ = x, y, m1 < k ≤ m,σ = z, 2≤ k ≤ n1.

Definition 6. We define the diameter ofSl along theξ0u directions as follows:

du1 (Sl ) ≡ supC

{‖ξ0u (q1)− ξ0u (q2)‖} ,whereC stands for

C ≡ {q1,q2 ∈ Sl ; x̃01(q1) = x̃01(q2), z̃01(q1) = z̃01(q2), ξ0s (q1) = ξ0s (q2)} .We define the diameter ofP10 (Sl ) along theξ

1s directions as follows:

ds1(P10 (Sl )) ≡ sup

C

{‖ξ1s (q1)− ξ1s (q2)‖} ,whereC stands for

C ≡ {q1,q2 ∈ P10 (Sl ); x̃11(q1) = x̃11(q2), ỹ11(q1) = ỹ11(q2), ξ1u (q1) = ξ1u (q2)} .


Fig. 3. (a) Geometry of the slabs. (b) Ge-ometry of the image of a slab underP10 .

By the definition ofSl and the representation (2.17) ofP10 , we have

du1 (Sl ) ∼ o (exp{−(γ1+ α1/2)T2l }) , asl →+∞, (2.55)ds1(P

10 (Sl )) ∼ o (exp{−α1T2l }) , asl →+∞. (2.56)

Sl and P10 (Sl ) have the product representations as shown in Figure 3. The width of theintersection ofSl with the (x̃01, z̃

01)-plane, along thẽx

01-direction, is

2η exp

{−1

2α1T2l

}, (2.57)

and along̃z01-direction it is of the order

O(exp{−γ1T2l }), asl →+∞. (2.58)

As shown in Figure 3,P10 (Sl ) intersects the (̃x11, ỹ

11)-plane in the shape of an annulus

which has width of order

O

(exp

{−3

2α1T2l

}), asl →+∞, (2.59)

and radius of order

O(exp{−α1T2l }), asl →+∞. (2.60)

On the annulus, we have marked the relative coordinate-positions betweenP10 (p+l ) and

P10 (p−l ).


Definition 7. Let S̄l be the closure ofSl in 6̄0. The connected components of thestableboundaryof Sl are defined as

∂+s Sl ≡{

q ∈ S̄l∣∣ x̃01(q) = +η exp{−12α1T2l

}},

∂−s Sl ≡{

q ∈ S̄l∣∣ x̃01(q) = −η exp{−12α1T2l

}},

∂σs Sl ≡{

q ∈ S̄l∣∣ |σ(q)| = η, for someσ = √(x0j )2+ (y0j )2 (2≤ j ≤ m1),

or someσ = z0k (n1 < k ≤ n)}.

The union of all the connected components of the stable boundary ofSl is referred to asthe stable boundary of Sl , denoted by∂sSl .

Similarly, the connected components of theunstable boundaryof Sl are defined as

∂+u Sl ≡{q ∈ S̄l

∣∣ z̃01(q) = η exp{−γ1(T2l − π /2)}} ,∂−u Sl ≡

{q ∈ S̄l

∣∣ z̃01(q) = η exp{−γ1(T2(l+1) − π /2)}} ,∂σu Sl ≡

{q ∈ S̄l

∣∣∣∣ |σ(P10 (q))| = η exp{−12α1T2l},

for someσ =√(x0j )

2+ (y0j )2 (m1 < j ≤ m),

or someσ = z0k (1≤ k < n1)}.

The union of all the connected components of the unstable boundary ofSl is referred toasthe unstable boundary of Sl , denoted by∂uSl . The stable and unstable boundaries ofP10 (Sl ) andP(Sl ) are defined, respectively, as

∂τ P10 (Sl ) ≡ P10 (∂τSl ) (τ = s, u),

∂τ P(Sl ) ≡ P(∂τSl ) (τ = s, u).

In Figure 3, we have marked two pieces of∂sSl and∂uSl by (1, 2) and (3, 4), respec-tively. We also marked the corresponding boundaries ofP10 (Sl ) by the same letters. Notethe contraction, expansion, and bending in the deformation process from the rectanglein the (x̃01, z̃

01)-plane to the annulus in the (x̃

11, ỹ

11)-plane.


2.7.3. P(Sl ). In this subsection we will describe some features of the geometry of theimage ofSl under P. The Poincar´e mapP01 has the representation (2.20). Under thelinear approximation ofP01 , the coordinate frame(

x̃11, ỹ11, ξ

1s , ξ

1u

)on61, is mapped into an affine coordinate frame(

x̄11, ȳ11, ξ̄

1s , ξ̄

1u

)(2.61)

on60 with origin atq+, where thēx11− ȳ11 plane is the image of thẽx11− ỹ11 plane underthe linear approximation ofP01 . Notice that60 is already equipped with a Cartesiancoordinate frame (

x̃01, z̃01, ξ

0s , ξ

0u

), (2.62)

also with origin atq+. SinceP01 is a diffeomorphism, the representation (2.20) ofP01

can be rewritten as x̃0ỹ0z̃0

= (A+ 01) x̃1ỹ1

z̃1

, (2.63) δx̃0δ ỹ0δz̃0

= (A+ 02) δx̃1δ ỹ1δz̃1

, (2.64)where

01

x̃1ỹ1z̃1

= B,and thus,

01→ 0, asx̃1→ 0, ỹ1→ 0, andz̃1→ 0.Based upon (2.63;2.64), we will approximateP(Sl ) by

A(P10 (Sl ))

in the later construction. ThenP(Sl ) has the product representation as shown in Figure 4.In Figure 4 we also marked corresponding pieces of∂sP(Sl ) and∂u P(Sl ), as in Figure 3.ds1(P(Sl )) can be defined similarly as ford

s1(P

10 (Sl )).

ds1(P(Sl )) ∼ o (exp{−α1T2l }) , asl →+∞. (2.65)Since we are approximatingP01 by a linear transformation, and thex̄

11 − ȳ11 plane is the

image of thex̃11 − ỹ11 plane under this linear transformation, the annulus formed by theintersection ofP10 (Sl ) with the x̄

11 − ȳ11 plane is mapped to an annulus on thex̃11 − ỹ11

plane. This annulus has width of order

O

(exp

{−3

2α1T2l

}), asl →+∞, (2.66)

and radius of order

O (exp{−α1T2l }) , asl →+∞. (2.67)


Fig. 4.Geometry of the image of a slab underP.

2.7.4. Definition of Slices.We now define “slices of slabs.”

Definition 8. A stable sliceV in Sl is a subset ofSl defined as the regionswept 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 thestable boundaryof V , which is denoted by∂sV . The rest of the boundary ofV is called itsunstable boundary, which is denoted by∂uV . Unstable slices ofSl , denoted byH , are defined similarly.

By definition,

∂uV ⊂ ∂uSl ,∂sH ⊂ ∂sSl .

In the coordinates (x̃01, z̃

01, ξ

0s , ξ

0u

)on60, let G be a (2m1 − 1+ n − n1)-dimensional hyperplane inSl , specified by thecondition, {

z̃01 = const., ξ0u = const.}.

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

G ∩ V =K⋃

k=1Gk.

See Figure 5 for an illustration.


Fig. 5. Components ofG ∩ V , the inter-section of a(2m1−1+n−n1)-dimensionalhyperplane inSl with a stable slice.

Definition 9. The diameter of the stable sliceV is defined as

d(V) ≡ supG

{sup

k

{sup

q1,q2∈Gk

{|x̃01(q1)− x̃01(q2)| + ‖ξ0s (q1)− ξ0s (q2)‖}}}

.

The diameter of an unstable sliceH is defined similarly.

2.7.5. Generic Intersection and Horseshoes.Introducing polar coordinates (r̃ 11, θ̃11 )

on the (̃x11, ỹ11)-plane,P

10 restricted to this plane has the following representation,

r̃ 11 = e−α1t∗(x+1 + x̃01),θ̃11 = β1t∗.

From Theorem 2.1, the “time of flight” of a fixed point from60 to61 is given by

Tl = 1β1(lπ − ϕ)+ o(1), asl →+∞.

Consequently, for any small positiveθ0, there existsl1, such that, for anyl ≥ l1,θ̃11(P

10 (p

+l1))− θ0 ≤ θ̃11(P10 (p+l )) ≤ θ̃11(P10 (p+l1 ))+ θ0 (s+), (2.68)

θ̃11(P10 (p

−l1))− θ0 ≤ θ̃11(P10 (p−l )) ≤ θ̃11(P10 (p−l1 ))+ θ0 (s−). (2.69)

That is, the (̃x11, ỹ11) components ofP

10 (p

+l ) and P

10 (p

−l ) are “in the two sectors on the

(x̃11, ỹ11)-planes+ ands− of angles 2θ0,” as shown in Figure 6. Under the restriction of

P01 to the (̃x11, ỹ

11)-plane, the sectorss+ ands− map to two sectors,

s̄+ = P01 (s+), (2.70)s̄− = P01 (s−), (2.71)

on the (̄x11, ȳ11)-plane. See Figure 7. The boundaries (1, 2) include parts of the boundaries


Fig. 6. The (̃x11, ỹ11) components ofP

10 (p

+l ) andP

10 (p

−l ) shown

in the two sectors, denoteds+ ands−, on the (̃x11, ỹ11)-plane. The

angular width of each sector is 2θ0.

x

_

1

y

_

1

1

1

s

+

_

_

s _

p

l

1

+

p

l

1

_

E

s

+

E

u

+

P

(

)

S

l

u

P

(

)

P

s

S

l

)

(

Fig. 7. The image of the sectorss+ ands− under thePoincaré map on the(x̄1, ȳ1)-plane.

of s̄+ ands̄−. For anyl ≥ l1, we introduce a system of curvilinear coordinates (ēu, ēs)on the (̄x11, ȳ

11)-plane such that

{ēu = 0}, {ēu = bu(constant)},{ēs = 0}, {ēs = bs(constant)},

correspond to the boundaries 3, 4; 1, 2 of P(Sl ) restricted to the (̄x11, ȳ11)-plane, respec-


Fig. 8. Curvilinear coordinates (ēu, ēs) on the(x̄11, ȳ

11)-plane.

tively. Cf. Figure 8. From now on, we restrict the coordinates (ēu, ēs) to the two sectorss̄+ ands̄−.

Definition 10. Define two subsets ofP(Sl ) as follows:

S̄+l ≡{q ∈ P(Sl )

∣∣ (ēu, ēs)(q) ∈ s̄+} ,S̄−l ≡

{q ∈ P(Sl )

∣∣ (ēu, ēs)(q) ∈ s̄−} .Let (E+u , E

+s ) be the tangent vectors

E+u ≡ Tp+l1 ēu, E+s ≡ Tp+l1 ēs.

We make the generic assumption that

(G5a) Span{ex̃01 , E

+u , eξ0s , eξ̄1u

}= 60,

(G5b) Span{ex̃01 , E

+s , eξ0s , eξ̄1u

}= 60,

whereex̃01 , eξ̄1u , etc., represent unit vectors corresponding to the respective coordinatedirections. The assumption says that, for example, genericallyE+u is not parallel to thecodimension [2(m−m1)+ n1] hyperplane

Span{ex̃01 , eξ0s

}.

Then, {ex̃01 , E

+u , eξ0s

}spans a codimension [2(m−m1)+ n1− 1] hyperplane. Similarly, for (G5b).


Next, without loss of generality, focusing attention atp+l , we discuss the intersection

P(Sl ) ∩ Slin the coordinates {

x̃01, ēu, ξ0s , ξ̄

1u

}.

Definition 11. On S̄+l andS̄−l we define the diameters ofSl andP(Sl ) as follows:

du(Sl ) ≡ supC

{|ēu(q1)− ēu(q2)| + ‖ξ̄1u (q1)− ξ̄1u (q2)‖} ,where

C ≡ {q1,q2 ∈ Sl ; x̃01(q1) = x̃01(q2), ξ0s (q1) = ξ0s (q2)} .ds(P(Sl )) ≡ sup

C

{|x̃01(q1)− x̃01(q2)| + ‖ξ0s (q1)− ξ0s (q2)‖} ,where

C ≡ {q1,q2 ∈ S̄+l ⊂ P(Sl ); ēu(q1) = ēu(q2), ξ̄1u (q1) = ξ̄1u (q2)} .By assumption (G5a) and Eqs. (2.55;2.58),

du(Sl ) ∼ O (exp{−γ1T2l }) , asl →+∞. (2.72)By assumption (G5a) and Eq. (2.65),

ds(P(Sl )) ∼ o (exp{−α1T2l }) , asl →+∞. (2.73)

Definition 12. Define two sections atp+l as follows:

5s(Sl ) ≡{q ∈ Sl

∣∣ ēu(q) = ēu(p+l ), ξ̄1u (q) = ξ̄1u (p+l )} ,5u(P(Sl )) ≡

{q ∈ S̄+l ⊂ P(Sl )

∣∣ x̃01(q) = x̃01(p+l ), ξ0s (q) = ξ0s (p+l )} .Then there exists an order

O

(exp

{−1

2α1T2l

}), asl →+∞, (2.74)

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

O(exp{−α1T2l }) , asl →+∞, (2.75)in the neighborhood ofp+l in 5

u(P(Sl )). Recall from (2.1) that

α1 < γ1. (2.76)

The geometry behind definitions 10–12 is illustrated in Figure 7. We now use the previ-ously developed geometry and estimates to prove the following proposition.


Fig. 9.Geometry of the intersection ofP(Sl ) with Sl .

Proposition 1. Under assumption (G5a), there exists a sufficiently large l0, such thatfor all l ≥ l0, P(Sl ) intersects Sl in two disjointconnected components, V+l and V−l . V+land V−l intersect both∂

+u Sl and∂

−u Sl and they donot intersect∂sSl . Moreover, V

+l and

V−l are stable slices in Sl with

∂sV+

l ⊂ ∂sP(Sl ), (2.77)and

∂sV−

l ⊂ ∂sP(Sl ). (2.78)See Figure 9.

Proof. We begin by showing thatV+l andV−

l are disjoint. The proof is by contradiction.Assume they are not disjoint; then there exists a curveg connectingp+l and p

−l , such

that

g ⊂ Sl , (2.79)and

g ⊂ P(Sl ). (2.80)In the same coordinates {

x̃01, ēu, ξ0s , ξ̄

1u

},

define

w(g) ≡ supq1,q2∈g

{|ēu(q1)− ēu(q2)|} ,


and then by (2.79), there is a constantD1, such that

w(g) < D1 exp{−γ1T2l },and, by (2.80) and the definitions ofp+l and p

−l , there is a constantD2, such that

w(g) > D2 exp{−α1T2l }.By (2.76), forl sufficiently large, this is a contradiction. ThusV+l andV

−l are disjoint.

For l sufficiently large, it follows from (2.73) and the definition ofSl (Definition 5)thatV+l andV

−l do not intersect∂sSl . Similarly, for l sufficiently large, it follows from

(2.72), (2.74), (2.75), (2.76), and the definition ofSl (Definition 5) thatV+

l and V−

lintersect∂+u Sl and∂

−u Sl .(2.77) and (2.78) follow from Definition 7 and the fact thatV

+l

andV−l intersect∂+u Sl and∂

−u Sl .

We now let

H+l = P−1(V+l ),H−l = P−1(V−l ).

It follows from (2.77;2.78) and Definition 7 thatH+l and H−l are unstable slices. The

unstable boundaries ofH+l andH−l are

∂u H+l = P−1(∂uV+l ), (2.81)

∂u H−l = P−1(∂uV−l ). (2.82)

In summary, for sufficiently largel , on eachSl , we have defined two unstable slices

H+l and H−l ,

and two stable slices

V+l and V−

l ,

such that Vσl = P(Hσl ),∂sVσl = P(∂sHσl ),∂uVσl = P(∂u Hσl ), σ = +,−.

(2.83)

In the next subsection we will establish shift dynamics on eachSl . Consequently, oneachSl , we have a Smale horseshoe. Thus, we have infinitely many Smale horseshoeslabeled byl on60.

2.8. Symbolic Dynamics

In this section, we will construct an invariant Cantor set3 in Sl and show that thePoincaré mapP restricted to3 is topologically conjugate to the shift automorphism ontwo symbols 0 and 1.


2.8.1. The Shift Automorphism. Let 4 be a set which consists of elements of thedouble infinite sequence form,

a = (. . .a−2a−1a0,a1a2 . . .),

whereak = 0 or 1,k ∈ Z. We introduce a topology in4 by taking as neighborhoodbasis of

a∗ = (. . .a∗−2a∗−1a∗0,a∗1a∗2 . . .),the set

Nj ={a ∈ 4 ∣∣ ak = a∗k (|k| < j )} ,

for j = 1, 2, . . . .This makes4 a topological space. The shift automorphismχ is definedon4 by

b ≡ χ(a), bk = ak+1.It is well-known that the shift automorphism has a countable infinity of periodic orbits ofall periods, an uncountable infinity of nonperiodic orbits, and a dense orbit. Moreover,it also exhibitssensitive dependence on initial conditions, which is a hallmark of chaos.

2.8.2. Conley-Moser Conditions.The Conley-Moser conditions are sufficient condi-tions for establishing the topological conjugacy between the Poincar´e mapP restrictedto a Cantor set3, and the shift automorphism on symbols; see [13] or [19].

Denote

H+l , H−l ;V+l ,V−l

by

H0, H1;V0,V1,respectively. Then we have

Conley-Moser condition (i):Vj = P(Hj ),∂sVj = P(∂sHj ), ( j = 0, 1),∂uVj = P(∂u Hj ).

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

d(Ṽ) ≤ νd(V),

where

Ṽ = P(V ∩ Hk) (k = 0, 1);


for any unstable sliceH ⊂ Hj ( j = 0, 1),d(H̃) ≤ νd(H),

where

H̃ = P−1(H ∩ Vk) (k = 0, 1).

Remark 2.3. In Conley-Moser condition (i), we have dropped the Lipschitz conditionfor the boundaries of the slices given in [13] [19]. In our case, a stable slice

V ⊂ Vj ( j = 0, 1),and an unstable slice,

H ⊂ Hk (k = 0, 1),can possibly intersect into more than one point. In this case we only choose one of themfor the invariant set that we construct.

The above Conley-Moser condition (i) has been verified in the last section (cf. Rela-tion (2.83)). We next discuss Conley-Moser condition (ii). By the representation (2.17)of P10 and the representation (2.63;2.64) ofP

01 , we have

d(Ṽ) ≤ ν1d(V),where

ν1 ∼ O (exp{−α1T2l }) , asl →+∞;d(H̃) ≤ ν2d(H),

where

ν2 ∼ O (exp{−γ1T2l }) , asl →+∞.

Lemma 2.4. If

· · · ⊂ H (k) ⊂ · · · ⊂ H (2) ⊂ H (1)is an infinite sequence of unstable slices, and, moreover,

d(H (k))→ 0, as k→∞,then

∞⋂k=1

H (k) ≡ H (∞)

is a (2m1− 1+ n− n1)-dimensional connected surface; moreover,∂H (∞) ⊂ ∂sH (1).

Similarly, for stable slices.

Proof. By definition∂sH (k) ⊂ ∂sH (1) for eachk. Hence∂H (∞) ⊂ ∂sH (1). The dimen-sion of H (∞) follows from the fact thatd(H (k)) → 0, ask → ∞ implies that the2(m−m1)+ n1 unstable dimensions shrink to zero.


2.8.3. Topological Conjugacy.Let

a = (. . .a−2a−1a0,a1a2 . . .)be any element of4. 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 2.4,

V(a) =∞⋂

k=1Va0a−1...a−k

defines a codimension (2m1− 1+ n− n1) connected surface; moreover,∂V(a) ⊂ ∂uSl . (2.84)

Similarly, define inductively fork ≥ 1 the unstable slicesHa0a1 = 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 2.4,

H(a) =∞⋂

k=0Ha0a1...ak

defines a (2m1− 1+ n− n1)-dimensional connected surface; moreover,∂H(a) ⊂ ∂sSl . (2.85)

By (2.84), (2.85), and the dimensions ofH(a) andV(a),

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 : 4 7→ Sl ,φ(a) = p.


By the above construction,

P(p) = φ(χ(a)).That is,

P ◦ φ = φ ◦ χ.Let

3 ≡ φ(4);then3 is a compact invariant (underP) Cantor subset ofSl . Moreover,φ is a homeomor-phism from4 to3 (with the topology inherited fromSl ). For more detailed discussion,see [13] [19]. Thus we have the theorem:

Theorem 2.2. There exists a compact invariant Cantor subset3 of Sl , such that Prestricted to3 is topologically conjugate to the shift automorphismχ on two symbols0and1. That is, there exists a homeomorphism

φ: 4 7→ 3,such that the following diagram commutes.

4φ−→ 3

χ

y yP4 −→

φ3

(2.86)

Remark 2.4. Topological conjugacy ofP to the shift or subshift dynamics on manysymbols can also be established. But we omit that construction here. For a relevantdiscussion on such topics, see [19].

3. Application to Discrete NLS Systems

In this section, we apply the theory developed in the last section to the discretizedperturbed NLS systems (1.1) studied in [9].

3.1. Transformation of (1.1) to the Form (2.1)

Here we state the results from [9] that are used in the transformation of (1.1) to the form(2.1).

The phase space for the discrete NLS systems (1.1) was defined as follows:

S ≡{Eq ≡

(qr

) ∣∣∣∣ r = −q̄, q = (q0,q1, . . . ,qN−1)T ,qn+N = qn, qN−n = qn

}.


It is easily verified that (1.1) has a saddle-type equilibrium point (the approximate an-alytical form can be found in [9]). The eigenvalues associated with the linearization of(1.1) aboutq² are given by

Ä±0 = ±4²1/2C1/2ω3/2+ O(²), (3.1)

Ä±1 = ±2√(1− cos2 k1)(1/µ2+ ω2)

(ω2− N2 tan2 π

N

)+ O(²), (3.2)

Ä±j = −²[α + 2βµ−2(1− coskj )

]± 2i ∣∣∣√|Ej Fj |∣∣∣ ( j = 2, . . . ,M), (3.3)wherekj = 2 jπ /N andC, Ej , andFj are constants which can be found in [9] (the exactexpressions are not important here, and so we omit them).

From these formulae, for² sufficiently small, we see that

0< Ä+0 < Ä+1 ,

0< − Re{Ä+2 } = − Re{Ä−2 } < · · · < − Re{Ä+M} = − Re{Ä−M} < −Ä−0 < −Ä−1 .Therefore,Ä+0 is the weakest growth rate, and− Re{Ä+2 } is the weakest decay rate.Moreover, we also have

− Re{Ä+2 } < Ä+0 . (3.4)Hence, the linearized system can be transformed to (real) Jordan canonical form. Wethen transformq² in (1.1) to the origin, and express (1.1) in the coordinates which putthe linear part in (real) Jordan canonical form. In this manner (1.1) is reduced to the formof system (2.1): ẋj = −αj xj − βj yj + Xj (x, y, z),ẏj = βj xj − αj yj + Yj (x, y, z),żk = δkγkzk + Zk(x, y, z) (3.5)

( j = 1, . . . ,M − 1, k = 1, . . . ,4),where

δk ={

1, k = 1, 2,−1, k = 3, 4,

x ≡ (x1, . . . , xM−1)T ,

y ≡ (y1, . . . , yM−1)T ,

z≡ (z1, . . . , z4)T ,

Xj (0, 0, 0) = Yj (0, 0, 0) = 0,gradXj (0, 0, 0) = gradYj (0, 0, 0) = 0 ( j = 1, . . . ,M − 1),

Zk(0, 0, 0) = gradZk(0, 0, 0) = 0 (k = 1, . . . ,4),


and

αj =[α + 2βµ−2(1− coskj+1)

],

βj = 2∣∣√|Ej+1Fj+1|∣∣ ( j = 1, . . . ,M − 1),

γ1 = γ3 = 4²1/2C1/2ω3/2+ O(²),

γ2 = γ4 = 2√(1− cos2 k1)(1/µ2+ ω2)(ω2− N2 tan2 πN )+ O(²).

Thus, the problem of constructing Smale horseshoes for (1.1) has been reformulated asfor the system (3.5), with the following two known facts:

• The system (3.5) possesses a homoclinic orbith asymptotic to(x, y, z) = (0, 0, 0).• The smallest repelling rateγ1 is larger than the smallest attracting rateα1.This problem is reduced to a special case of (2.1), with

m= m1 = M − 1, n1 = 2, n = 4.

Hence, under the generic assumptions (G1, . . . ,G5), the homoclinic orbith gives riseto Smale horseshoes, as described in Theorem 2.2, for system (1.1).

In the next section, we will discuss the generic assumptions.

3.2. The Generic Assumptions

The generic assumptionsG2,G4, andG5 will still stand as assumptions since we do notknow how to analytically verify them for our system (numerical verification is possible).

G1 can be realized by choosing appropriate values for the parameters

(², ω, α, β, 0)

on E² defined in Theorem 1.1. As we argued in the introduction, the nonresonanceconditions are not a serious restriction. From Theorem 1.1 it follows that a homoclinicorbit exists on a codimension-one surface in the parameter space. Each nonresonancecondition also defines a codimension-one surface in the parameter space, and it intersectsthe surface on which the homoclinic orbits exist in a set of measure zero. There are acountable number of such resonance conditions, so the set of parameter values on thissurface where nonresonance fails is still a set of measure zero. Hence, on a codimension-one surface in parameter space there is a set of full measure where the nonresonanceconditions hold. This is another example of the usefulness of the Conley-Moser approach.Since the horseshoes constructed by this method are structurally stable, they also persistfor this set of measure zero where the nonresonance conditions break down, since theset of full measure where nonresonance holds is dense.


G3 can be realized as follows. By the Melnikov argument in [9], for any appropriatefixed

(², ω, α, β, 0) ,

the distance on the homoclinic section6

d ≡ Dist. {Wu(q²),Ws(m²)}∣∣6 = d(γ ),where(γ, t) parametrizeWu(q²). There existsγ0, such that

d(γ0) = 0, d′(γ0) 6= 0.This is also true for

(², ω, α, β, 0)

on E² . Thus at anyq ∈ h,

dim{

TqWu(q²)

⋂TqW

s(m²)}= 1.

But

Ws(q²) ⊂ Ws(m²).Therefore,G3 is true.

3.3. Smale Horseshoes and Chaos Created by a Symmetric Pair of Homoclinic Orbitsin the Discrete NLS Systems

We now show how the construction of horseshoes can be carried out near the symmetricpair of homoclinic orbits. In the discrete NLS systems (1.1), ifN is even and> 7, thereis, in fact, a symmetric pair of homoclinic orbits asymptotic to the fixed pointq² . Thisfollows from the fact that ifqn = f (n, t) solves (1.1), thenqn = f (n + N/2, t) alsosolves (1.1). That is, the system (1.1) is equivariant with respect to the symmetry group,

Ĝ ≡ {1, σ }, (3.6)whereσ ◦ f (n, t) = f (n+ N/2, t); i.e.,σ is a translation over half period,σ−1 = σ ,and 1 denotes the identity. In terms of the coordinates used in (3.5),

σ ◦ {xj , yj } = {(−1) j+1xj , (−1) j+1yj },j = 1, 2, . . . ,M − 1,

σ ◦ zk = (−1)k+1zk,k = 1, . . . ,4.

(3.7)

Since system (3.5) and system (1.1) are equivalent, system (3.5) is equivariant withrespect to the symmetry group (3.6). Letĥ1 be the homoclinic orbit (asymptotic toq²)proved in Theorem (1.1); then̂h2 ≡ σ ◦ ĥ1 is another homoclinic orbit also asymptoticto q² . (ĥ1, ĥ2) constitute the symmetric pair of homoclinic orbits.


3.3.1. Equivariant Smooth Normal Form Reduction. In Section 2.2 we used a smoothnormal form reduction to linearize the system in a neighborhood of the origin. However,it may not be true that the normalized system is equivariant with respect to the symmetrygroupĜ. In this section we show how this issue can be treated.

First we expand on our treatment of the normal form reduction in Section 2.2. Let

R: (x, y, z) 7→ (x′, y′, z′)be theC2 diffeomorphism that reduces the system (3.5) to linear normal form in aneighborhood of (0, 0, 0), as discussed in Section 2.2.R is the identity map outside aneighborhood of (0, 0, 0); moreover,

R(0, 0, 0) = (0, 0, 0),gradR(0, 0, 0) = identity map.

More importantly, in certain small neighborhoodÄ of (x′, y′, z′) = (0, 0, 0), system(3.5) is reduced to the linear system,

ẋ′j = −αj x′j − βj y′j ,ẏ′j = βj x′j − αj y′j ,ż′k = δkγkz′k,

(3.8)

( j = 1, . . . ,M − 1, k = 1, . . . ,4).where

δk ={

1, k = 1, 2,−1, k = 3, 4.

In the entire phase spaceS,the system (3.5) is transformed intoẋ′j = −αj x′j − βj y′j + X′j (x′, y′, z′),ẏ′j = βj x′j − αj y′j + Y′j (x′, y′, z′),ż′k = δkγkz′k + Z′k(x′, y′, z′)

(3.9)

( j = 1, . . . ,M − 1, k = 1, . . . ,4).where

δk ={

1, k = 1, 2,−1, k = 3, 4,

x′ ≡ (x′1, . . . , x′M−1)T ,y′ ≡ (y′1, . . . , y′M−1)T ,z′ ≡ (z′1, . . . , z′4)T .

Moreover,X′j , Y′j , Z

′k vanish identically insideÄ.


It is not clear that system (3.9) is equivariant with respect to the symmetry groupĜ(3.6). LetRσ be the average ofR over the groupĜ:

Rσ = 12(R+ σ−1Rσ), (3.10)

Rσ : (x, y, z) 7→ (x′′, y′′, z′′).Rσ is also aC2 diffeomorphism, and outside a neighborhood of (0, 0, 0), Rσ is theidentity map. Moreover,

Rσ (0, 0, 0) = (0, 0, 0),gradRσ (0, 0, 0) = identity map.

It is easy to verify that inside the neighborhood of the originÄ of (x′′, y′′, z′′) = (0, 0, 0)the system (3.5) is also reduced to the linear system byRσ :

ẋ′′j = −αj x′′j − βj y′′j ,ẏ′′j = βj x′′j − αj y′′j ,ż′′k = δkγkz′′k

(3.11)

( j = 1, . . . ,M − 1, k = 1, . . . ,4).where

δk ={

1, k = 1, 2,−1, k = 3, 4.

In the whole spaceS, system (3.5) is transformed intoẋ′′j = −αj x′′j − βj y′′j + X′′j (x′′, y′′, z′′),ẏ′′j = βj x′′j − αj y′′j + Y′′j (x′′, y′′, z′′),ż′′k = δkγkz′′k + Z′′k (x′′, y′′, z′′)

(3.12)

( j = 1, . . . ,M − 1, k = 1, . . . ,4),where

δk ={

1, k = 1, 2,−1, k = 3, 4,

x′′ ≡ (x′′1, . . . , x′′M−1)T ,y′′ ≡ (y′′1, . . . , y′′M−1)T ,z′′ ≡ (z′′1, . . . , z′′4)T .

Moreover,X′′j , Y′′j , Z

′′k vanish identically insideÄ. More importantly, system (3.12) is

equivariant with respect to the symmetry groupĜ (3.6): If (x(t), y(t), z(t)) solves (3.5),thenσ ◦ (x(t), y(t), z(t)) also solves (3.5). Therefore, both

(x′′(t), y′′(t), z′′(t)) = Rσ ◦ (x(t), y(t), z(t))


and

Rσ ◦ [σ ◦ (x(t), y(t), z(t))] ≡ σ ◦ (x′′(t), y′′(t), z′′(t))solve (3.12).

If we let

h1 ≡ Rσ ◦ ĥ1, h2 ≡ Rσ ◦ ĥ2,then,

h2 = σ ◦ h1. (3.13)Thus, the problem of constructing horseshoes for system (3.5) is transformed to con-structing horseshoes for system (3.12), with the following two known facts:

• System (3.12) possesses a symmetric pair of homoclinic orbitsh1 andh2 (h2 = σ ◦h1)asymptotic to (0, 0, 0).• The weakest repelling rateγ1 is larger than the weakest attracting rateα1.From now on, we will drop the double primes in system (3.12).

3.3.2. Chaos Created by the Symmetric Pair of Homoclinic Orbits.Leth±i (i = 1, 2)be the forward and backward time segments of the pair of homoclinic orbitshi (i = 1, 2),h±2 = σ ◦ h±1 . The definitions of the Poincar´e sections60 and61, and the Poincar´emapsP10 , P

01 , andP ≡ P01 ◦ P10 are identical to those stated in the general theory. Let

q+i (i = 1, 2) be the intersection points ofh+i (i = 1, 2) with the (z1 = 0) boundaryof 60,

q+i ≡ h+i ∩ ∂60,whereq+i has the coordinates

x1 = x(+,i )1 , y1 = 0,xj = x(+,i )j , yj = y(+,i )j (2≤ j ≤ M − 1),

z1 = z2 = 0, z3 = z(+,i )3 , z4 = z(+,i )4 ,

and

q+2 = σq+1 ; (3.14)i.e.,

x(+,2)1 = x(+,1)1 , z(+,2)3 = z(+,1)3 , z(+,2)4 = −z(+,1)4 ;{x(+,2)j , y

(+,2)j

}={(−1) j+1x(+,1)j , (−1) j+1y(+,1)j

}(2≤ j ≤ M − 1).

Similarly, letq−i (i = 1, 2) be the intersection points ofh−i (i = 1, 2) with 61:

q−i ≡ h−i ∩61,


whereq−i has the coordinates

z1 = η, z2 = z(−,i )2 , z3 = z4 = 0,xj = 0, yj = 0 (1≤ j ≤ M − 1),

and

q−2 = σq−1 ; (3.15)i.e.,

z(−,2)2 = −z(−,1)2 .Finally, we have the fact

P01 (q−i ) = q+i (i = 1, 2).

As in the general theory, we denote coordinates on60 by(x01; {x0j , y0j }, 2≤ j ≤ M − 1; z0k, 1≤ k ≤ 4

),

and coordinates on61 by({x1j , y1j }, 1≤ j ≤ M − 1; z1k, 2≤ k ≤ 4) .Applying the fixed point Theorem 2.1 to neighborhoods ofq+1 andq

+2 , respectively, we

have two sequences of fixed points,{q(i )l , l = l0, . . . ,∞

}(i = 1, 2).

Alternatively, we can apply the fixed point Theorem 2.1 to a neighborhood ofq+1 toobtain a sequence of fixed points. Then we apply the group elementσ to this sequenceof fixed points and obtain a different sequence of fixed points in the neighborhood ofq+2 . Thus,

q(2)l = σq(1)l , l = l0, . . . ,∞. (3.16)We collect these results in

Corollary 1. In each neighborhood of q+i (i = 1, 2), there exists a sequence of fixedpoints, {

q(i )l , l = l0, . . . ,∞}

(i = 1, 2);moreover,

q(2)l = σq(1)l , l = l0, . . . ,∞.The sequences of fixed points{q(i )l } have the asymptotic behavior described in Theorem2.1.


By (3.16),

z01(q(2)l ) = z01(q(1)l ).

Then,q(2)l andq(1)l correspond to the sameTl as described in Theorem 2.1. For conve-

nience of later construction, as in the general theory, we denote by

p(+,i )l ≡ q(i )2l , p(−,i )l ≡ q(i )2l+1, 2l ≥ l0.

Similarly, as in the general theory, we define the slabsS(i )l (i = 1, 2),

S(i )l ≡{

q ∈ 60∣∣∣∣ η exp{−γ1(T2(l+1) − π /2)} ≤ z01(q) ≤ η exp{−γ1(T2l − π /2)},

|x01(q)− x(+,i )1 | ≤ η exp{−1

2α1T2l

},

|z12(P10 (q))− z(−,i )2 | ≤ η exp{−1

2α1T2l

}},

and we note that

x(+,1)1 = x(+,2)1 , z(−,1)2 = −z(−,2)2 .We define a new slab̂Sl as follows:

Ŝl ≡{

q ∈ 60∣∣∣∣ η exp{−γ1(T2(l+1) − π /2)} ≤ z01(q) ≤ η exp{−γ1(T2l − π /2)},

|x01(q)− x(+,1)1 | ≤ η exp{−1

2α1T2l

},

|z12(P10 (q))| ≤ |z(−,1)2 | + η exp{−1

2α1T2l

}}.

Then

S(1)l ∪ S(2)l ⊂ Ŝl .By the above definitions,

p(±,i )l ⊂ P(S(i )l ) ∩ S(i )l (i = 1, 2). (3.17)

The stable and unstable boundaries ofS(i )l (i = 1, 2) and Ŝl can be defined similarly tothe way we definedSl in the general theory.

Definition 13. Let S̄(i )l (i = 1, 2) and ¯̂Sl be, respectively, the closures ofS(i )l (i = 1, 2)and Ŝl in 6̄0. The components of the stable boundary ofS

(i )l (i = 1, 2) and Ŝl are,

respectively, defined as

∂sS+,(i )l ≡

{q ∈ S̄(i )l

∣∣∣∣ x01(q)− x(+,i )1 = +η exp{−12α1T2l}},


∂sS−,(i )l ≡

{q ∈ S̄(i )l

∣∣∣∣ x01(q)− x(+,i )1 = −η exp{−12α1T2l}},

∂sSσ,(i )l ≡

{q ∈ S̄(i )l

∣∣∣∣ |σ(q)| = η,for someσ =

√(x0j (q))

2+ (y0j (q)) (2≤ j ≤ M − 1);

or, σ = |z0k| (k = 3, 4)},

∂sŜ+l ≡

{q ∈ ¯̂Sl

∣∣∣∣ x01(q)− x(+,i )1 = +η exp{−12α1T2l}},

∂sŜ−l ≡

{q ∈ ¯̂Sl

∣∣∣∣ x01(q)− x(+,i )1 = −η exp{−12α1T2l}},

∂sŜσl ≡

{q ∈ ¯̂Sl

∣∣∣∣ |σ(q)| = η, for someσ = √(x0j (q))2+ (y0j (q)) (2≤ j ≤ M − 1);or, σ = |z0k| (k = 3, 4)

}.

Similarly, the components of the unstable boundaries, denoted by∂uS(i )l (i = 1, 2) and

∂uŜl are, respectively, defined as

∂uS+,(i )l ≡

{q ∈ S̄(i )l

∣∣∣∣ z01(q) = η exp{−γ1(T2l − π /2)}},∂uS−,(i )l ≡

{q ∈ S̄(i )l

∣∣∣∣ z01(q) = η exp{−γ1(T2(l+1) − π /2)}},∂uS

σ,(i )l ≡

{q ∈ S̄(i )l

∣∣∣∣ |σ(q)| = η exp{−12α1T2l}, for σ = z12(P10 (q))− z(−,i )2

},

∂uŜ+l ≡

{q ∈ ¯̂Sl

∣∣∣∣ z01(q) = η exp{−γ1(T2l − π /2)}},∂uŜ−l ≡

{q ∈ ¯̂Sl

∣∣∣∣ z01(q) = η exp{−γ1(T2(l+1) − π /2)}},∂uŜ

σl ≡

{q ∈ ¯̂Sl

∣∣∣∣ |σ(q)| = η exp{−12α1T2l}, for σ = z12(P10 (q))− z(−,i )2

}.


Fig. 10. Geometry of the intersection ofP(S(1)l ) andP(S(2)l ) with Ŝl .

A stable slice and an unstable slice ofŜl can be defined in the same way as forSl in thegeneral theory. Then the following corollary follows immediately from the definition:

Corollary 2. S(i )l (i = 1, 2) are unstable slices of̂Sl .

The diameters of different objects can be defined in the same way as in the general theory.Then, the same estimates as in the general theory show that

du(Ŝl ) ∼ O(exp{−γ1T2l }), asl →+∞,ds(P(S(i )l )) ∼ o(exp{−α1T2l }), asl →+∞.

By (3.17),P(S(i )l )(i = 1, 2) intersectŜl into four regionsV (±,i )l (i = 1, 2), such that

p(±,i )l ∈ V (±,i )l (i = 1, 2).The same argument as forSl versusP(Sl ), in the general theory, implies the following.

Proposition 2. Under the generic assumptions above, there exists a sufficiently largel0, such that for all l≥ l0, P(S(i )l ), (i = 1, 2), intersectsŜl in two disjoint connectedcomponents, V(±,i )l , (i = 1, 2).

V (±,i )l intersect both∂+u Ŝl and∂

−u Ŝl and they donot intersect∂sŜl . Moreover, V

(±,i )l

are stable slices in̂Sl with

∂sV(±,i )

l ⊂ ∂sP(Ŝl ). (3.18)See Figure 10.

Define

H (±,i )l ≡ P−1(V (±,i )l ) (i = 1, 2),


thenH (±,i )l (i = 1, 2) are unstable slices. More importantly,

H (±,i )l ⊂ S(i )l (i = 1, 2).If we denote

H (+,1)l , H(−,1)l ; H (+,2)l , H (−,2)l ,

respectively, by

H1, H2; H−1, H−2.The same argument as in the general theory shows the following.

Theorem 3.1. There exists a compact invariant Cantor set3(1,2) ⊂ (S(1)l ∪ S(2)l ) ⊂ Ŝl ,such that P restricted to3(1,2) is topologically conjugate to the shift automorphismχ4on four symbols−2,−1, 1, and2. That is, there exists a homeomorphism

φ4: 44 7→ 3(1,2),(where44 is defined similarly as4 in the general theory) such that the following diagramcommutes.

44φ4−→ 3(1,2)

χ4

y yP44 −→

φ43(1,2)

(3.19)

3.4. Interpretation of Numerical Observation on the Discrete NLS Systems: TheChaotic Center-Wing Jumping

In the chaotic regime typical numerical output of the discrete NLS system (1.1) is shownin Figure 1. Notice that there are two typical profiles at a fixed time: One is a breathertype profile with its hump located at the center of the spatial period interval and the otheris also a breather type profile, but with its hump located at the boundaries (wing) ofthe spatial period interval. More importantly, these two types of profiles are half spatialperiod translates of each other. If we label the profiles with their humps at the center ofthe spatial period interval by “C” and those profiles with their humps at the wing of thespatial period interval by “W;” then,

“W” = σ ◦ “C” , (3.20)whereσ is the symmetry group element. More importantly, the time series of the output inFigure 1 is a chaotic jumping between “C” and “W,” which we refer to aschaotic center-wing jumping. By the relation (3.20), and the study in the last subsection, we interpretthe chaotic center-wing jumping as the numerical realization of the shift automorphismχ4 on four symbols−2,−1; 1, and 2.

We make this more precise in terms of the phase space geometry. In [10] [8] the hy-perbolic structure for the (² = 0) integrable case of (1.1) was identified. This hyperbolic


Fig. 11. Heuristic illustration of hyperbolic structure and ho-moclinic orbits. Note the (z2, z4)-plane are only local coordi-nates valid near the hyperbolic saddle.

Fig. 12. The homoclinic solution of the IDNLSfor 0≤ t ≤ 50 with initial conditionq(x, 0) =12 + ²0(1+ i ) cospx. The surfaceq(x, t). Thefigure is from [14].

structure projected onto (z2, z4)-plane is illustrated as in Figure 11. Keep in mind thatthe (z2, z4)-plane are only local coordinates valid near the hyperbolic saddle. From thesymmetry we know that

Loop 2= σ ◦ Loop 1.

Loop 1 (for example) has the spatial-temporal profile realization as in Figure 12. Anorbit inside Loop 1,Lin has a spatial-temporal profile realization as in Figure 13. Anorbit outside Loop 1 and Loop 2,Lout has the spatial-temporal profile realization as inFigure 14. The two slabs (unstable slices ofŜl ) S

(i )l (i = 1, 2), projected onto (z2, z4)-

plane, are also illustrated in Figure 11. Then, the chaotic center-wing jumping (Figure 1)as the realization of the shift dynamics inS(1)l andS

(2)l becomes more apparent.

4. Conclusion

In Part I of this study ([9], this journal), Li and McLaughlin proved the existence ofhomoclinic orbits in the discrete NLS systems (1.1). In this paper (Part II of our study ondiscrete NLS systems), we construct Smale horseshoes and symbolic dynamics basedupon the existence of those homoclinic orbits. In particular, we study horseshoes created


Fig. 13. The solution of the IDNLS for 0≤ t ≤50 with initial conditionq(x, 0) = 12 + .1(1+i ) cospx. The surfaceq(x, t) shows an excita-tion locked in the center. Also, there exists a sec-ond case, which is not shown, with the excitationlocked in the wings. The figure is from [14].

Fig. 14. The solution of the IDNLS for 0≤ t ≤50 with initial conditionq(x, 0) = 12+.1i cospx.The surfaceq(x, t) with a standing wave whosemaximum travels between the center and thewings.

through a symmetric pair of homoclinic orbits in the discrete NLS systems. Such astudy leads to the interpretation of numerical observations on the discrete NLS systemin terms of achaotic center-wing jumping. The mechanism of chaos here is the Silnikov-type mechanism [18] [4] [19]. Nevertheless, in this paper, we took a different approachfrom Silnikov’s. That is, instead of using a fixed point argument on a product of spaces,as developed by Silnikov, we checked directly the Conley-Moser conditions to constructSmale horseshoes, which lead to symbolic dynamics.

This study generalizes the four-dimensional model study [12] to any finite dimensions,which gives a satisfactory interpretation for the numerical observations (Figure 1) of theperturbed cubic NLS pde (1.2) [11], since the numerical study on the pde (1.2) is, in fact,a study on the system (1.1), which is a finite-difference approximation of the pde (1.2).

The study of the discrete NLS system (1.1) itself is also of great interest [1].


5. Appendix

The following example shows that linearized dynamics cannot approximate the fullnonlinear dynamics in the neighborhood of (0, 0, 0):

ẋ = x,ẏ = − 12 y,ż= −z+ xy.

The solution isx(t) = x(t1)e(t−t1),y(t) = y(0)e− 12 t ,z(t) = z(0)e−t + 23x(t1)y(0)e−

12 t[e(t−t1)(1− e− 32 t )

].

If

x(t1) = y(0) = z(0) = η,then

z(t1) = ηe−t1 + 23η2e−

12 t1[1− e− 32 t1

].

As t1→+∞,z(t1) ∼ e− 12 t1.

Thus, for a long time, linearized dynamics is dominated by a nonlinear effect.

Acknowledgments

We would like to thank Marty Golubitsky and Mike Field for a discussion on equivariantsmooth linearization.

S. Wiggins would like to acknowledge partial research support by NSF Grant No.DMS-9403691.

References

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[2] M. S. Berger.Nonlinearity and Functional Analysis. Academic Press, New York, 1977,p. 113.

[3] B. Deng. The Silnikov’s Problem, Exponential Expansion, Strongλ-Lemma.C1 Lineariza-tion, and Homoclinic Bifurcation.J. Diff. Eq., 79:189–231, 1989.

[4] B. Deng. On Silnikov’s Homoclinic-saddle-focus Theorem.J. Diff. Eq., 102:305–329, 1993.[5] J. Dieudonne.Foundations of Modern Analysis. Academic Press, New York, 1960.[6] P. Hartman.Ordinary Differential Equations. John Wiley and Sons, Inc., New York, 1964.[7] P. J. Holmes. A Strange Family of Three-Dimensional Vector Fields Near a Degenerate

Singularity.J. Diff. Eq., 37:382–403, 1980.


[8] Y. Li. Backlund Transformations and Homoclinic Structures for the NLS Equation.Phys.Letts. A, 163:181–187, 1992.

[9] Y. Li and D. W. McLaughlin. Homoclinic Orbits and Chaos in Discretized Perturbed NLSSystems: Part I. Homoclinic Orbits.J. Nonlin. Sci., 7:211–269, 1997.

[10] Y. Li and D. W. McLaughlin. Morse and Melnikov Functions for NLS Pdes.Comm. Math.Phys., 162:175–214, 1994.

[11] D. W. McLaughlin and E. A. Overman. Whiskered Tori for Integrable Pdes and ChaoticBehavior in Near Integrable Pdes.Surv. Appl. Math. 1, 1993.

[12] D. W. McLaughlin, E. A. Overman, S. Wiggins, and C. Xiong. Homoclinic Orbits in aFour-Dimensional Model of a Perturbed NLS Equation: A Geometric Singular PerturbationStudy. Dynamics Reported 5, Springer-Verlag, New York, 1996.

[13] J. Moser. Stable and Random Motions in Dynamical Systems.Annals of MathematicsStudies, 77, Princeton University Press, Princeton, NJ, 1973.

[14] C. M. Schober.Numerical and Analytical Studies of the Discrete Nonlinear SchroedingerEquation. PhD thesis, University of Arizona, 1991.

[15] L. P. Silnikov. A Case of the Existence of a Countable Number of Periodic Motions.Sov.Math. Dokl., 6:163–166, 1965.

[16] L. P. Silnikov. The Existence of a Denumerable Set of Periodic Motions in Four-Dimensional Space in an Extended Neighborhood of a Saddle-Focus.Sov. Math. Dokl.,8:54–58, 1967.

[17] L. P. Silnikov. On a Poincare-Birkhoff Problem.Math. USSR Sb., 3:353–371, 1967.[18] L. P. Silnikov. A Contribution to the Problem of the Structure of an Extended Neighborhood

of a Rough Equilibrium State of Saddle-Focus Type.Math. USSR Sb., 10:91–102, 1970.[19] S. Wiggins.Global Bifurcations and Chaos: Analytical Methods. Springer-Verlag, Berlin,

1988.

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