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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)
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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(
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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
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318 Y. Li and S. Wiggins
(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.
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Symbolic Dynamics 319
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 Poincar´e 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 Poincar´e map in a
neighborhood of the homoclinic orbit for thegeneral class of
systems under consideration. In Section 2.6 we compute fixed points
ofthe Poincar´e 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
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320 Y. Li and S. Wiggins
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.
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Symbolic Dynamics 321
(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
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322 Y. Li and S. Wiggins
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
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Symbolic Dynamics 323
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,
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324 Y. Li and S. Wiggins
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
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Symbolic Dynamics 325
Definition 2. The Poincar´e section61 is defined by the
constraints,
z1 = η,|zk| < η, k = 2, . . . ,n,x2j + y2j < η2, j = 1, .
. . ,m.
Definition 3. The Poincar´e 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 Poincar´e 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,
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326 Y. Li and S. Wiggins
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.
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Symbolic Dynamics 327
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.
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328 Y. Li and S. Wiggins
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.
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Symbolic Dynamics 329
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).
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330 Y. Li and S. Wiggins
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)
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Symbolic Dynamics 331
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,
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332 Y. Li and S. Wiggins
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∗}
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Symbolic Dynamics 333
+{
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
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334 Y. Li and S. Wiggins
the same as the coefficient multiplying sinβkt∗ in (2.31).
Hence, equivalent equationsthat the fixed points must satisfy are
given by
0 = x+1(
A(σ,x)k1 cosβ1t∗ + A(σ,y)k1 sinβ1t∗)
+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(
A(σ,x)k1 cosβ1t∗ + A(σ,y)k1 sinβ1t∗)
+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,
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Symbolic Dynamics 335
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)
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336 Y. Li and S. Wiggins
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)
-
Symbolic Dynamics 337
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
-
338 Y. Li and S. Wiggins
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,
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Symbolic Dynamics 339
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 ).
-
340 Y. Li and S. Wiggins
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
}).
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Symbolic Dynamics 341
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)} .
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342 Y. Li and S. Wiggins
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 ).
-
Symbolic Dynamics 343
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.
-
344 Y. Li and S. Wiggins
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)
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Symbolic Dynamics 345
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.
-
346 Y. Li and S. Wiggins
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
-
Symbolic Dynamics 347
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-
-
348 Y. Li and S. Wiggins
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).
-
Symbolic Dynamics 349
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.
-
350 Y. Li and S. Wiggins
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)|} ,
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Symbolic Dynamics 351
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.
-
352 Y. Li and S. Wiggins
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);
-
Symbolic Dynamics 353
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.
-
354 Y. Li and S. Wiggins
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.
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Symbolic Dynamics 355
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
}.
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356 Y. Li and S. Wiggins
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
ı0 = ±4²1/2C1/2ω3/2+ O(²), (3.1)
ı1 = ±2√(1− cos2 k1)(1/µ2+ ω2)
(ω2− N2 tan2 π
N
)+ O(²), (3.2)
ıj = −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),
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Symbolic Dynamics 357
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.
-
358 Y. Li and S. Wiggins
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.
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Symbolic Dynamics 359
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.
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360 Y. Li and S. Wiggins
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))
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Symbolic Dynamics 361
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,
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362 Y. Li and S. Wiggins
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.
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Symbolic Dynamics 363
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}},
-
364 Y. Li and S. Wiggins
∂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
}.
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Symbolic Dynamics 365
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),
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366 Y. Li and S. Wiggins
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
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Symbolic Dynamics 367
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
-
368 Y. Li and S. Wiggins
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].
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Symbolic Dynamics 369
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.
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