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J. Nonlinear Sci. Vol. 7: pp. 211–269 (1997)
© 1997 Springer-Verlag New York Inc.
Homoclinic Orbits and Chaos in DiscretizedPerturbed NLS
Systems:Part I. Homoclinic Orbits
Y. Li 1∗ and D. W. McLaughlin21 Department of Mathematics,
University of California at Los Angeles, Los Angeles, CA 90024,
USA2 Courant Institute of Mathematical Sciences, 251 Mercer
Street, New York, NY 10012, USA
Received September 1, 1995; revised manuscript accepted for
publication September 11, 1996Communicated by Jerrold Marsden
Summary. The existence of homoclinic orbits, for a
finite-difference discretized form ofa damped and driven
perturbation of the focusing nonlinear Schroedinger equation
undereven periodic boundary conditions, is established. More
specifically, for external param-eters on a codimension 1
submanifold, the existence of homoclinic orbits is
establishedthrough an argument which combines Melnikov analysis
with a geometric singular per-turbation theory and a purely
geometric argument (called the “second measurement” inthe paper).
The geometric singular perturbation theory deals with persistence
of invariantmanifolds and fibration of the persistent invariant
manifolds. The approximate locationof the codimension 1 submanifold
of parameters is calculated. (This is the material inPart I.) Then,
in a neighborhood of these homoclinic orbits, the existence of
“Smalehorseshoes” and the corresponding symbolic dynamics are
established in Part II [21].
Key words. discrete nonlinear Schroedinger equation, spectral
theory, persistent invari-ant manifolds, Fenichel fibers, Melnikov
analysis, homoclinic orbits
MSC numbers. 34, 35, 39, 58
PAC numbers. 02, 03, 42, 63
∗Present address: Department of Mathematics 2-336, Massachusetts
Institute of Technology, Cambridge, MA02139, USA.
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212 Y. Li and D. W. McLaughlin
1. Introduction
In this paper, we prove the existence of orbits, homoclinic to a
saddle fixed point, of thefollowing N particle (any 2< N 3,
3 tanπ
3< ω 0), β (> 0), 0 (> 0) are constants.The restriction
onω ensures that, forN > 2, the uniform solution (|qn| = ω) to
theunperturbed (² = 0) form of (1.1) has a codimension 2 center
manifold, a codimension1 center-unstable manifold, and a
codimension 1 center-stable manifold. (By increasingω to other
intervals, one has codimensionk, for any finitek, center-unstable
and center-stable manifolds for|qn| = ω. Their study is parallel to
that given in this paper. Note alsothat in the caseN = 2, |qn| = ω
is neutrally stable for anyω, and there is no hyperbolicstructure;
hence, we will not study this case.)
This system (1.1) is a finite-difference discretization of the
following perturbed NLSPDE:
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. Thus,
system (1.1) is of interest both as a perturbation ofa completely
integrable Hamiltonian system of large, but finite, dimension; and
as anapproximation to a PDE.
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 213
The motivation for this study comes from the numerical
experiments on chaos innear-integrable systems as summarized in
[22], where many references can be found.These numerical
experiments show that perturbed nonlinear Schroedinger
equations(1.2), actually discretizations thereof such as (1.1),
possess solutions with beautifulregular spatial patterns which
evolve irregularly (chaotically) in time. These numericalstudies
also correlate this chaotic behavior in the perturbed system with
the presence ofa hyperbolic structure in the unperturbed(² = 0)
integrable NLS equation [20].
In order to begin an analytical study of the perturbed NLS
system, we introduced in ref-erence [3] an extremely crude
truncation of the perturbed NLS PDE to a four-dimensionaldynamical
system, which we then studied both numerically and analytically
[2]. In sev-eral papers [15] and [23], by ourselves and others, the
existence of homoclinic orbitsand a symbol dynamics was established
for the perturbed four-dimensional dynamicalsystem. In this paper,
we establish the existence of homoclinic orbits in the 2(M +
1)dimensional finite-difference approximation (1.1) of the
perturbed NLS PDE (1.2) forany finite M . A symbol dynamics for
(1.1) is studied in a companion paper [21], andhomoclinic behavior
for the PDE (1.2) is studied in [17].
There are some key differences of this study from the earlier
four-dimensional sys-tem study [23]: (i) This study is for a system
with an arbitrary number of N particles;therefore, when N is large
enough, the behavior of such a system will be very similarto the
limiting PDE. (ii) The integrable theory for the unperturbed N
particles system isnovel. (iii) The “second measurement” (see below
for definition) is more difficult anddelicate in this higher
dimensional case. (iv) In this study we present a detailed
derivationof the Melnikov distance estimate for this particular
high dimensional system. (v) In acompanion paper, we establish the
existence of a symbolic dynamics for this high di-mensional system
which for arbitrary N is quite different from the symbol dynamics
inthe four-dimensional case in that it has oscillatory motion
reminiscent of a Silnikov be-havior in high dimensions. (vi)
Finally, we view this work in arbitrarily large (but
finite)dimensions as a necessary step toward a dynamical system
analysis of the PDE (1.2).
Related studies on NLS systems can be found in the works [10]
[9].Now we state our results on the system (1.1): Denote by6N (N ≥
7) the external
parameter space,
6N ={(ω, α, β, 0)
∣∣∣∣ ω ∈ (N tan πN , N tan2πN),
0 ∈ (0, 1), α ∈ (0, α0), β ∈ (0, β0);whereα0 andβ0 are any fixed
positive numbers
}.
Theorem 1.1. For any N (7 ≤ N < ∞), there exists a positive
number²0, such thatfor any² ∈ (0, ²0), there exists a codimension1
submanifold E² in6N; for any externalparameters (ω, α, β, 0) on E²
, there exists a homoclinic orbit asymptotic to a fixed pointq² .
The submanifold E² is in an O(²ν) neighborhood of the hyperplaneβ =
κ α, whereκ = κ(ω; N) is shown in Figure 1.1,ν = 1/2− δ0, 0< δ0¿
1/2.
Remark 1.1. In the cases (3≤ N ≤ 6), κ is always negative as
shown in Figure 1.1.Since we require both dissipation parametersα
andβ to be positive, the relationβ = κα
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214 Y. Li and D. W. McLaughlin
shows that the existence of homoclinic orbits violates this
positivity. ForN ≥ 7, κ canbe positive as shown in Figure 1.1.
WhenN is even and≥ 7, there is in fact a pairof homoclinic orbits
asymptotic to a fixed pointq² at the same values of the
externalparameters; since for evenN, we have the symmetry: Ifqn = f
(n, t) solves (1.1), thenqn = f (n+N/2, t)also solves (1.1). WhenN
is odd and≥ 7, the study can not guaranteethat two homoclinic
orbits exist at the same value of the external parameters.
This theorem is established in several steps: First, integrable
theory in the form of Laxrepresentations and Backlund
transformations is used to construct homoclinic orbits inthe
integrable unperturbed (² = 0) case. In addition, through the
Floquet discriminant1(z;qn) of the isospectral difference
operatorLn, critical hyperbolic tori are identi-fied, together with
natural representations of their stable and unstable manifolds.
Thisbackground material from integrable theory is summarized in
Section 2.
Next, one notes that a plane “of constants” is invariant for
both the unperturbed(² = 0) and the perturbed (² > 0) flows.
When restricted to this plane5, the dynamicscan be analized
explicitly; and, in particular, a saddle fixed pointq² can be
identifiedwhich emerges for² > 0 from a circle of fixed pointsSω
when² = 0. A change ofcoordinates is then introduced which is
centered upon the plane5. In these coordinates,persistent invariant
manifolds are constructed in a neighborhood of the plane5.
Theseconstructions are given in Section 3.
The persistent homoclinic orbits are singular deformations of
the integrable (² =0) orbits. This singular nature is apparent in
numerical studies [22], and it has beenestablished in the four
dimensional truncated system [15] [23]. It arises because of
thepresence of motion on two distinct time scalesO(t)when the phase
point is far from theplane5 andO(
√²t) when the phase point is near a resonant circle on5. Because
of
these two distinct time scales, we use geometric singular
perturbation theory. Specifically,we represent the center-stable
and center-unstable manifolds of the plane5 through thefibers of
Fenichel [7]. These constructions are described in Section 4.
With these preliminaries in place, the proof of the persistent
homoclinic orbits pro-ceeds with two measurements. Consider the
saddle fixed pointq² on the invariant plane5. We seek an orbit, not
on5, which is homoclinic toq² . The unstable manifoldWu(q²)of q² is
two-dimensional. Because of the two distinct time scales,
trajectories inWu(q²)leaveq² and remain near5 for a longO(1/
√²) time, before they rapidlyO(t) fly away
from 5. With the Fenichel fibers, we can define a “take-off
angle”θT at which thetrajectory “flies away from5” and use this
angle to label trajectories inWu(q²),
q(u,²)(t; θT ) ∈ Wu(q²).We determine ifq(u,²)(t; θT ) ∈ Wu(q²)
with two measurements.
The first measurement is described in Section 5. We note that
the center-stable man-ifold Wcs0 for the unperturbed (² = 0)
integrable system is codimension 1, and that itpersists as a
codimension 1 locally invariant manifoldWcs² for the perturbed (²
> 0)system. Moreover,Ws(q²) ⊂ Wcs² . Thus, we first ask
ifq(u,²)(t; θT ) ∈ Wcs² ? SinceWcs²is codimension 1, the answer
requires only one measurement, which is accomplishedwith a
“Melnikov integral,”
dist{q(u,²)(t; θT );Wcs²
} = ²M̂F (θT ;α, β, 0)+ o(²),
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 215
where
M̂F (θT ;α, β, 0) =∫ ∞−∞(gradF, gpert)|h(t;θT ) dt,
and wheregpert denotes the perturbation to the NLS equation
which depends upon theexternal parameters (α, β, 0). The
trajectoryh(t; θT ) denotes the homoclinic orbit forthe integrable
system which is labeled by the take-off angleθT on the circle of
fixedpoints Sω. A simple zero (inθT ) of the Melnikov function
establishes, by an implicitfunction theorem argument, the existence
of a trajectoryq(u,²)(t; θ∗∗T )which approachesq² as t → −∞, and
resides inWcs² (provided that the behavior of this trajectory canbe
controlled ast → +∞, control is required becauseWcs² is only a
locally invariantmanifold). The latter control will be provided by
the “ second measurement.”
The valueθ∗∗T of the “take-off” angle is a small perturbation of
the simple zeroθ∗T
of the Melnikov functionM̂F . This “take-off” angleθ∗T is a
function of the externalparameters (α, β, 0), and the Melnikov
measurement can be interpreted as determining(approximately) that
take-off angleθ∗∗T (α, β, 0) for which the trajectoryq
(u,²)(t; θ∗∗T ) ∈Wcs² .
The final step in the argument, which we call the “second
measurement,” establishesthat q(u,²)(t; θ∗∗T ) ∈ Ws(q²). For θT =
θ∗∗T , it is known thatq(u,²)(t; θ∗∗T ) ∈ Wcs² , andwhile Ws(q²) is
codimension 2,within Wcs² , the stable manifoldW
s(q²) is codimension1. Thus, only one additional measurement is
required to determine ifq(u,²)(t; θ∗∗T ) ∈Ws(q²). To set up this
measurement successfully, certain terms which are
quadraticallynonlinear must first be eliminated by a normal form
transformation. The measurementitself is carried out using
“energy-coordinates.” The technical details are described insection
6. This last measurement places one constraint on the external
parameters; thus,the homoclinic orbit persists on a codimension 1
set in parameter space.
The two measurements control the behavior ofq(u,²)(t; θ∗∗T )ast
→+∞and establishthe existence of an orbit homoclinic toq² . In a
companion paper [21], the existence of“Smale horseshoes” and a
corresponding symbolic dynamics is established, genericallyin a
neighborhood of such homoclinic orbits.
2. Integrable Background
In this section, we give all the integrable preliminaries needed
for studies in later chapters.Of particular importance is (i) the
simple family of integrable homoclinic orbits givenby (2.12); (ii)
the constant of motioñF1 given by equation (2.8); and (iii) the
definitionsand representations of the integrable center-stable abd
center-unstable manifoldsWcs,Wcu given in Corollary 2.
Setting² = 0 in the discretized perturbed NLS system (1.1), we
get the followingHamiltonian system:
i q̇n = 1h2
[qn+1− 2qn + qn−1
]+ |qn|2(qn+1+ qn−1)− 2ω2qn, (2.1)
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Fig. 1.1. The κ = κ(ω, N)curves for different values ofN.In
particular, note its positivity.
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 217
which can be written in (noncanonical) Hamiltonian form,
q̇n = ρn ∂H∂q̄n
,
˙̄qn = −ρn ∂H∂qn
,
where
ρn ≡ 1+ h2|qn|2,
H ≡ − ih2
N−1∑n=0
{q̄n(qn+1+ qn−1)− 2
h2(1+ ω2h2) ln(1+ h2|qn|2)
}.
Moreover,∑N−1
n=0 {q̄n(qn+1+qn−1)} itself is also a constant of motion of the
system (2.1).This invariant, together withH , implies that
∑N−1n=0 ln ρn is a constant of motion too.
Therefore,
D2 ≡N−1∏n=0
ρn (2.2)
is a constant of motion.
2.1. Lax Pair Representation
Additionally, system (2.1) is an integrable Hamiltonian system
[1], with(M+1)constantsof motion. This integrability is proven with
the use of the discretized Lax pair [1]:
ϕn+1 = L(z)n ϕn, (2.3)ϕ̇n = B(z)n ϕn, (2.4)
where
L (z)n ≡(
z ihqn
ihq̄n 1/z
),
B(z)n ≡i
h2
(1− z2+ 2iλh− h2qnq̄n−1+ ω2h2 −zihqn + (1/z)ihqn−1
−i zhq̄n−1+ (1/z)ihq̄n 1/z2− 1+ 2iλh+ h2q̄nqn−1− ω2h2),
and wherez≡ exp(iλh). Compatibility of the over-determined
system (2.3), (2.4) givesthe “Lax representation”
L̇n = Bn+1Ln − Ln Bnof the discrete NLS (2.1).
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218 Y. Li and D. W. McLaughlin
2.2. Spectral Theory ofLn
Focusing attention upon the discrete spatial flow (2.3), we
letY(1),Y(2) be the funda-mental solutions of the ODE (2.3), i.e.,
solutions with the initial conditions,
Y(1)0 =(
10
), Y(2)0 =
(01
).
TheFloquet discriminant,
1: C× S 7→ C, (2.5)is defined by
1(z; Eq) ≡ tr{M(N; z; Eq)}, (2.6)whereS is the phase space
defined as follows:
S ≡{Eq =
(qr
) ∣∣∣∣ r = −q̄, q = (q0,q1, . . . ,qN−1)T ,qn+N = qn, qN−n =
qn
},
andM(n; z; Eq) ≡ columns{Y(1)n ,Y(2)n } is the fundamental
solution matrix of (2.3). InS(viewed as a vector space over the
reals), we define the inner product, for any two pointsEq and Ep,
as follows:
〈Eq, Ep〉 = 2 Re{ N−1∑
n=0q̄n pn
}.
And the norm ofEq is defined as‖Eq ‖2 ≡ 〈Eq, Eq〉.
Remark 2.1.1(z; Eq) is a constant of motion for the integrable
system (2.1) for anyz ∈ C. Since1(z; Eq) is a meromorphic function
inz of degree (+N,−N), the Floquetdiscriminant1(z; Eq) acts as a
generating function for(M+1) functionally independentconstants of
motion and is the key to the complete integrability of the system
(2.1).
The Floquet theory here is not standard, as can be seen from the
Wronskian relation,
WN(ψ+, ψ−) = D2 W0(ψ+, ψ−),
whereD is defined in (2.2),
Wn(ψ+, ψ−) ≡ ψ(+,1)n ψ(−,2)n − ψ(+,2)n ψ(−,1)n ,
ψ+ andψ− are any two solutions to the linear system (2.3). In
fact,Wn+1(ψ+, ψ−) =ρnWn(ψ+, ψ−). Due to this nonstandardness,
modification of the usual definitions ofspectral quantities [16] is
required.
Periodic and antiperiodic pointszs are defined by
1(zs; Eq) = ±2D.
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 219
A critical point zc is defined by the condition
d1
dz|(zc;Eq) = 0.
A multiple pointzm is a critical point which is also a periodic
or antiperiodic point. Thealgebraic multiplicityof zm is defined as
the order of the zero of1(z)± 2D. Usually itis 2, but it can exceed
2; when it does equal 2, we call the multiple point adouble
point,and denote it byzd. Thegeometric multiplicityof zm is defined
as the dimension of theperiodic (or antiperiodic) eigenspace of
(2.3) atzm, and is either 1 or 2.
An important sequence of constants of motionF̃j [20]
F̃j : Ä ⊂ S 7→ C (2.7)
is defined by
F̃j (Eq) = 1D1(zcj (Eq); Eq), (2.8)
whereÄ ⊂ S is the domain ofF̃j . We are going to usẽF1 to build
a Melnikov integralin Section 5.
Remark 2.2(Continuum Limit). In the continuum limit (i.e.,h→ 0),
the Hamiltonianhas a limit in the mannerHh → Hc, whereHc is the
Hamiltonian for NLS PDE,Hc =i∫ 1
0 {qxq̄x+2ω2|q|2−|q|4} dx. The Lax pair (2.3), (2.4) also tends
to the correspondingLax pair for NLS PDE with spectral parameterλ
(z = eiλh) [20]. If Q ≡ maxn{|qn|}is finite, thenρn → 1 ash → 0.
Therefore,D2 ≡ {
∏N−1n=0 ρn} → 1 ash → 0. The
nonstandard Floquet theory for the spatial part of the Lax pair
(2.3) becomes standardFloquet theory in the continuum limit.
Some useful symmetries of the eigen-functions of the Lax pair
(2.3), (2.4) (which areused in the explicit calculation of Section
2.3) may be found in Appendix A.
2.3. Hyperbolic Structure and Homoclinic Orbits
The hyperbolic structure and homoclinic orbits for (2.1) are
constructed through theBacklund-Darboux transformations, which were
built in [16]. First, we present theBacklund-Darboux
transformations. Then, we show how to construct homoclinic
or-bits.
In fact, we will present a form of Backlund-Darboux
transformations which is mostrelevant to hyperbolic structure. Fix
a solutionqn(t) of the system (2.1), for which thelinear operatorLn
has a double pointzd of geometric multiplicity 2, which is not
onthe unit circle. We denote two linearly independent solutions
(Bloch functions) of thediscrete Lax pair (2.3), (2.4) atz = zd by
(φ+n , φ−n ). Thus, a general solution of thediscrete Lax pair
(2.3), (2.4) at(qn(t), zd) is given by
φn(t; zd, c) = φ+n + cφ−n ,
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220 Y. Li and D. W. McLaughlin
wherec is a complex parameter called a Backlund parameter. We
useφn to define atransformation matrix0n by
0n =(
z+ (1/z)an bncn −1/z+ zdn
),
where
an = zd
(z̄d)21n
[|φn2|2+ |zd|2|φn1|2
],
dn = − 1zd1n
[|φn2|2+ |zd|2|φn1|2
],
bn = |zd|4− 1(z̄d)21n
φn1φ̄n2,
cn = |zd|4− 1
zdz̄d1nφ̄n1φn2,
1n = − 1z̄d
[|φn1|2+ |zd|2|φn2|2
].
From these formulae, we see that
ān = −dn, b̄n = cn.
Then we defineQn and9n by
Qn ≡ ih
bn+1− an+1qn, (2.9)
and
9n(t; z) ≡ 0n(z; zd;φn)ψn(t; z), (2.10)whereψn solves the
discrete Lax pair (2.3), (2.4) at(qn(t), z). Formulas (2.9)
and(2.10) are the Backlund-Darboux transformations for the
potential and eigenfunctions,respectively. We have the following
theorem [16].
Theorem 2.1(Backlund-Darboux Transformations).Let qn(t) denote a
solution of thesystem (2.1), for which the linear operator Ln has a
double point zd of geometric mul-tiplicity 2, which is not on the
unit circle and which is associated with an instability.We denote
two linearly independent solutions (Bloch functions) of the
discrete Lax pair(2.3), (2.4) at(qn, zd) by (φ+n , φ
−n ). We define Qn(t) and9n(t; z) by (2.9) and (2.10).
Then
1. Qn(t) is also a solution of the system (2.1). (The evenness
of Qn can be guaranteedby choosing the complex Backlund parameter c
to lie on a certain curve, as shown inthe example below.)
2. 9n(t; z) solves the discrete Lax pair (2.3), (2.4) at(Qn(t),
z).
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 221
3. 1(z; Qn) = 1(z;qn), for all z ∈ C.4. Qn(t) is homoclinic to
qn(t) in the sense that Qn(t)→ ei θ± qn(t), exponentially as
exp(−σ |t |) as t→ ±∞. Hereθ± are the phase shifts,σ is a
nonvanishing growthrate associated with the double point zd, and
explicit formulas can be developed forthis growth rate and for the
phase shiftsθ±.
This theorem is quite general, constructing homoclinic solutions
from a wide class ofstarting solutionsqn(t). Its proof is by direct
verification [16], [20].
Next, we study specifically the most important
example—Backlund-Darboux trans-formations for the uniform solutions
of (2.1). Let
qn = q, ∀n; q = a exp{−2i [(a2− ω2)t ] + i γ }, (2.11)whereN tan
πN < a < N tan
2πN for N > 3, 3 tan
π3 < a 3, (2.14)
3 tanπ
3< |q0|
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222 Y. Li and D. W. McLaughlin
Fig. 2.1. Geometric illustration of the singular level sets in
“figure 8⊗ A,” and theircorresponding spectral identification.
Define the “resonance circle” inA,
Sω ≡{Eq∣∣∣∣ Eq ∈ 5, |q0| = ω}. (2.15)
Sω entirely consists of fixed points under the integrable flow
(2.1).
The formula in the above corollary represents the singular level
set connecting to acircle in the annulusA. See Figure 2.1 for its
geometric illustration and the correspondingspectral
identification. For more detail, see [20] and [22].
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 223
In fact, we can spell out the following theorem from the
Backlund-Darboux transfor-mation theorem 2.1.
Corollary 2. In S, under the integrable flow (2.1), there exist
a codimension 1 “center-unstable manifold of Sω,” denoted by Wcu,
and a codimension 1 “center-stable manifoldof Sω,” denoted by Wcs.
There also exists a codimension 2 “center manifold of Sω,”denoted
by Wc; Wc ∩5 = A. Moreover, Wcu = Wcs =(figure 8)⊗Wc.
Proof. This is a corollary of Theorem 2.1.
Remark 2.3. In principle, “(figure 8)⊗Wc” can be calculated
through the Backlundtransformations given in theorem 2.1; the
special case “(figure 8)⊗A” has been calculatedin corollary 1. We
can represent “(figure 8)⊗A” by the heteroclinic orbit formulae
inCorollary 1 as follows:
“(figure 8)⊗ A” =⋃
t∈(−∞,∞),γ∈[0,2π ],ω∈Iω,σ=±Qn(t; N, ω, γ, r = 0, σ ),
whereIω denotes the restriction interval onω given in (1.1),
andQn is given in Corol-lary 1.
3. Persistent Invariant Manifolds
In this section, we are going to discuss persistence of the
center-unstable, center-stable,and center manifolds ofSω: Wcu, Wcs,
andWc, respectively. Here, “persistence” meansthat there exists a
positive number²2, such that, for any² ∈ (−²2, ²2), there
existcodimension 1 locally invariant manifoldsWcu² andW
cs² , and a codimension 2 locally
invariant manifoldWc² . Moreover,Wcu0 = Wcu, Wcs0 = Wcs, andWc0
= Wc. In a certain
neighborhood ofSω, Wcu² , Wcs² , andW
c² are smooth in² for ² ∈ (−²2, ²2). We will give
precise definition of “local invariancy.” The simple example in
the next subsection willshow the necessity of the introduction of
the concept oflocal invariancy.
3.1. Persistent Invariant Plane
It is easily seen that the plane5 remains an invariant manifold
under the perturbed flow(1.1); however, motion on this plane is
very different in the perturbed and integrablecases, either of
which can be analyzed with phase plane methods.
On the invariant plane5, the dynamics is governed by the
following two-dimensionalODE:
iqt = 2[|q|2− ω2
]q + i ²
[− αq + 0
]. (3.1)
Changing variables to an amplitude-phase representation (q = I
ei γ , I = |q|) yields the
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224 Y. Li and D. W. McLaughlin
following pair of equations:
İ = ²(−α I + 0 cosγ ), (3.2)γ̇ = −2(I 2− ω2)− ²(0/I ) sinγ.
(3.3)
The phase plane diagram of this system is shown in Figure 3.1.
On the plane5 and inanO(
√²) neighborhood of the resonance circleI = ω, we let I ≡ ω+√²J,
τ ≡ √²t ,
and obtain
J ′ = −∂H1∂γ−√²αJ, (3.4)
γ ′ = ∂H1∂ J−√²
[2J2+ 0
ω +√²J sinγ], (3.5)
where′ ≡ ∂∂τ
,H1 ≡ αγω−0 sinγ − 2ωJ2. A simple phase plane analysis shows
that,to the first orderO(²0), the dynamics of (3.4) and (3.5) is
thefishdynamics as shownin Figure 3.2, and to the orderO(
√²), the dynamics of (3.4) and (3.5) is the brokenfish
dynamics as shown in Figure 3.3. Under the perturbed dynamics,
there are three fixedpoints on the plane5: a sinko² near the
origin; a sinkp² near the resonance circle; anda saddleq² which is
also near the resonance circle. In the coordinates of thefish,
thesaddle fixed point (denoted bỹq²) is located at(J, γs) = (0,
arccos{χαω}); thus, we seethat the condition for the existence of
thefishstructure is
χα ≤ 1ω, (3.6)
while thenoseof thefish is located at(J, γn) = (0, γn). Hereγn
satisfies the equationχαωγn − sinγn = χαωγs − sinγs. (3.7)
From the above phase plane analysis, we see that, although5
(2.13) is still an invariantplane,A defined in (2.14) is not an
invariant subset any more. Nevertheless,A is locallyinvariant,
according to the definition given in the next subsection.
3.2. Persistent Invariant Manifold Theorem
Normally hyperbolic invariant manifold theorems have been
established by Fenichel [4],Kelley [13], Hirsch, Pugh, and Shub
[11], Sacker [24], and others. Here, we mainly followFenichel [4],
[7] for a persistence theorem of normally hyperbolic invariant
manifolds,under small perturbation of the flow, which he proved
through a combination of a graphtransform method first introduced
by Hadamard [8] for two-dimensional maps and ageometric singular
perturbation theory developed in [7].
Definition 2 (Local Invariancy). LetFt be a flow (a solution
operator) defined onS, Vbe a submanifold ofS, with boundary∂V (V̄ ≡
V ∪ ∂V). We sayV is locally invariantunder the flowFt in S, if
there exists a neighborhoodU of V in S, such that, for allQ ∈ V ,
if τ ≥ 0 and⋃t∈[0,τ ] Ft (Q) ⊂ U , then⋃t∈[0,τ ] Ft (Q) ⊂ V , and
ifτ ≤ 0 and
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 225
Fig. 3.1.Phase plane diagram of the dynamics on5.
⋃t∈[τ,0] F
t (Q) ⊂ U , then⋃t∈[τ,0] Ft (Q) ⊂ V . Intuitively speaking, the
orbitFt (Q),starting from the pointQ in V , can leaveV in forward
or backward time; nevertheless,it can only leaveV through the
boundary∂V .
Definition 3 (The Maximal Invariant Set). LetFt be a flow (a
solution operator) de-fined onS, andV be a submanifold ofS.
Define
A+(V) ≡{
Q ∈ V∣∣∣∣ ⋃
t∈[0,∞]Ft (Q) ⊂ V
},
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226 Y. Li and D. W. McLaughlin
A−(V) ≡{
Q ∈ V∣∣∣∣ ⋃
t∈[−∞,0]Ft (Q) ⊂ V
},
I (V) ≡{
Q ∈ V∣∣∣∣ ⋃
t∈[−∞,∞]Ft (Q) ⊂ V
}.
We callA+(V), A−(V), andI (V) the maximal positively invariant
set inV , the maximalnegatively invariant set inV , and the maximal
invariant set inV , respectively.
Fig. 3.2.The “fish” dynamics.
Fig. 3.3.The broken “fish” dynamics.
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 227
The persistent invariant manifold theorem stated specifically
for our system (1.1) isas follows [7].
Theorem 3.1(Local Persistent Invariant Manifolds).In the phase
spaceS there existsa neighborhoodU of Sω, inside of which, for
every fixed k (2 ≤ k < ∞), there existsa positive number²2 =
²2(U , k), such that for any² ∈ (−²2, ²2), there exist a
Ckcodimension 1 locally invariant manifold Wcu² indexed by², a
C
k codimension 1 locallyinvariant manifold Wcs² indexed by², and
a C
k codimension 2 locally invariant manifoldWc² indexed by², under
the perturbed flow F
t² given by (1.1). These manifolds are C
k
smooth in² for ² ∈ (−²2, ²2). Moreover, Wcu0 = Wcu, Wcs0 = Wcs,
and Wc0 = Wc,where Wcu, Wcs, and Wc are the “invariant manifolds of
Sω” identified in Corollary 2for the integrable flow (2.1).
Furthermore, A² ≡ 5 ∩ U ⊂ Wc² for any² ∈ (−²2, ²2),and
A+(U) ⊂ Wcs² ,A−(U) ⊂ Wcu² ,
I (U) ⊂ Wc² ,
where A+(U), A−(U), and I(U) denote the maximal positively
invariant set inU , themaximal negatively invariant set inU , and
the maximal invariant set inU , respectively.
Proof. The setup for the proof of this theorem following
Fenichel [7] is given in thenext subsection.
In this paper, we use “•” to denote the action of an operator
upon a set.
Theorem 3.2(Global Persistent Invariant Manifolds).The global
persistent center-un-stable, center-stable, and center manifolds
are defined to be
⋃t≥0 F
t² •Wcu² ,
⋃t≤0 F
t² •
Wcs² , and⋃−∞
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228 Y. Li and D. W. McLaughlin
3.3.1. An Enlarged Phase Space.In order to study smoothness in²
of persistent in-variant manifolds, it is convenient to treat² as a
variable, and to consider the followingenlarged system(EDPNLS):
i q̇n = 1h2
[qn+1− 2qn + qn−1
]+ |qn|2(qn+1+ qn−1)− 2ω2qn
+i ²[− αqn + β
h2(qn+1− 2qn + qn−1)+ 0
], (3.8)
²̇ = 0.
The correspondingenlarged function spacebecomesŜ:
Ŝ = S × (−²3, ²3). (3.9)
We can define a family of invariant planes parametrized by², 5̂²
,
5̂² ≡{(Eq, ²) ∈ Ŝ
∣∣∣∣ Eq ∈ 5}.It is easily seen that̂5² is invariant under the
EDPNLS flow (3.8). Restricted to theinvariant planeŝ5² , the
EDPNLS flow (3.8) becomes
i q̇ = 2[|q|2− ω2]q + i ² [−αq + 0],²̇ = 0.
In the enlarged function space, the resonance circleŜω is
defined as follows:
Ŝω ≡ {(Eq, 0) | (Eq, 0) ∈ 50, |q| = ω}.
It is easily seen that̂Sω is a set of equilibria of (3.8).
3.3.2. A Neighborhood of the CircleŜω of Fixed Points. In order
to study dynamicsnear the circlêSω of fixed points we writeqn
=
[(ω + δr )+ δ fn(t)
]expi θ(t),
² = δ2²′,(3.10)
whereδ > 0 is a small parameter,r is real, and fn has spatial
mean 0 (i.e.,〈 fn〉 ≡∑N−1n=0 fn = 0).
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 229
Definition 4. Define the mean zero subspace ofS as follows:
S0 ≡{Eq ∈ S
∣∣∣∣ 〈qn〉 = 0}.Definition 5. Define the phase space in terms of
the new variables (r, θ, fn, ²′),
E ≡{(r, θ, fn, ²
′)∣∣∣∣ (r, ²′) ∈ R2; θ ∈ [0, 2π ], Ef ∈ S0}.
Then, through the scaling transformation (3.10), there is a
one-to-one correspondencebetweenE andŜ.
Definition 6. Define the compact domainEc in the phase spaceE as
follows:
Ec ≡{(r, θ, fn, ²
′) ∈ E∣∣∣∣ |r | ≤ 1, |²′| ≤ 1; θ ∈ [0, 2π ], ‖ Ef ‖ ≤ 1}.
3.3.3. The Equations in Their Final Setting. Substituting
representation (3.10) intothe enlarged discrete perturbed NLS
system (3.8) yields the following systemdefined onEc, with δ as
thenew perturbation parameter, and with² as avariable.
i ḟn =[
1
h2( fn+1− 2 fn + fn−1)+ (ω + δr )2( fn+1+ fn−1+ 2 f̄n)
]+ δ
[2(ω + δr )( fn f̄n − 〈 fn f̄n〉)+ (ω + δr )
(( fn + f̄n)( fn+1+ fn−1)
− 〈( fn + f̄n)( fn+1+ fn−1)〉)]
+ δ2[
fn f̄n( fn+1+ fn−1)− 〈 fn f̄n( fn+1+ fn−1)〉 − fn(
2〈 fn f̄n〉
+ 12
[〈( fn + f̄n)( fn+1+ fn−1)〉 + 〈( fn + f̄n)( f̄n+1+ f̄n−1)〉])
− ²′(
0
(ω + δr ) fn sinθ + i [α fn −β
h2( fn+1− 2 fn + fn−1)]
)]− δ3
[fn
2(ω + δr )(〈 fn f̄n( fn+1+ fn−1)〉 + 〈 fn f̄n( f̄n+1+ f̄n−1)〉
)],
ṙ = δ[− ² ′
(α(ω + δr )− 0 cosθ
)− i (ω + δr )/2
(〈( fn + f̄n)( fn+1+ fn−1)〉
− 〈( fn + f̄n)( f̄n+1+ f̄n−1)〉)]
− i δ2/2[〈 fn f̄n( fn+1+ fn−1)〉 − 〈 fn f̄n( f̄n+1+ f̄n−1)〉
],
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230 Y. Li and D. W. McLaughlin
θ̇ = −2δr (2ω + δr )−δ2
[²′0
(ω + δr ) sinθ + 2〈 fn f̄n〉 + 1/2(〈( fn + f̄n)( fn+1+ fn−1)〉
+ 〈( fn + f̄n)( f̄n+1+ f̄n−1)〉)]
− δ3
2(ω + δr )[〈 fn f̄n( fn+1+ fn−1)〉 + 〈 fn f̄n( f̄n+1+ f̄n−1)〉
],
²̇′ = 0.We refer to this system as the (δ 6= 0) system. Settingδ
= 0 in this system, we get the(δ = 0) system,
i ḟn =[
1
h2( fn+1− 2 fn + fn−1)+ ω2( fn+1+ fn−1+ 2 f̄n)
],
ṙ = 0,θ̇ = 0,²̇′ = 0.
For the (δ = 0) system, defined onEc, we know that the subset
ofEc,
Âc ≡{(r, θ, fn, ²
′) ∈ Ec∣∣∣∣ Ef = 0},
entirely consists of fixed points. At any point in̂Ac, its
stable space is one-dimensional,its unstable space is also
one-dimensional, and its center space has codimension 2. Thelinear
growth (= decay) rate is
Ä = 2√(1− cos2 k1)(1/h2+ ω2)
(ω2− N2 tan2 π
N
),
which is the same at any point in̂Ac, k1 = 2π /N. The
center-stable, center-unstable, andcenter manifolds ofÂc are the
embeddings of the corresponding center-stable, center-unstable, and
center linear subbundles, respectively. We denote them byL̂cs,
L̂cu, andL̂c,respectively. BotĥLcs andL̂cu have codimension 1,
and̂Lc has codimension 2. Viewingδ as a perturbation parameter, and
viewing the (δ 6= 0) system as a perturbed system ofthe (δ = 0)
system, we can study persistence of these invariant manifolds,
under thisδ-perturbation. This was done by Fenichel [7]. After
completing the whole argument,we get the claim: For anyδ ∈ [0, δ0],
whereδ0 is a positive number, there exist locallyinvariant
manifoldsŴcsδ , Ŵ
cuδ , andŴ
cδ , which can be represented as graphs overL̂
cs,L̂cu, and L̂c, respectively; moreover,̂Wcsδ=0 = L̂cs, Ŵcuδ=0
= L̂cu, andŴcδ=0 = L̂c. Thecrucial point is thatδ0 is independent
of²′. If we setδ = δ0, and transformEc into asubsetDc of Ŝ through
the scaling transformation (3.10), then we have a claim for
theregionDc; moreover,Dc is independent of². As ²′ → 0; thereby,² →
0, Dc is of orderO(²0).
Generalization to infinite-dimensional systems is done in the
book [18]. In this book,we study persistent invariant manifolds for
certain PDEs (with Eq. (1.2) as a special
-
Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 231
example), their fibrations, smoothness of the manifolds and the
fibers, using graph trans-form approaches in detail.
3.4. Stable and Unstable Manifolds of the Saddleq²
We know, from the previous section on the persistent invariant
plane, that on the plane5the perturbed system has exactly three
fixed points:(o²,q², p²). Of these,o² is a defor-mation of the
origin, while the other two fixed pointsq² andp² deform from two
pointson the circleSω. In the plane5, o² andp² are sinks whileq² is
a saddle. In the full phasespaceS, linear stability analysis shows
thato² is a sink, while bothp² andq² are saddles.Moreover, the
unstable linear subspace ofp² is one-dimensional and the stable
linearsubspace ofp² has codimension 1; the unstable linear subspace
ofq² is two-dimensionaland the stable linear subspace ofq² has
codimension 2. By the invariant manifold theo-rems (see, e.g.,
Kelley [13]),Wu(p²) exists and is one-dimensional,Ws(p²) also
existsand has codimension 1,Wu(q²) exists and is
two-dimensional,Ws(q²) also exists andhas codimension 2.
Remark 3.2. In this paper, we only studyq² . In fact, we are
going to locate homoclinicorbits asymptotic toq² . The size of the
growth and decay rates with respect to the smallperturbation
parameter² is very important. Consider the saddleq² : One growth
rate islargeO(²0) and is associated with the integrable
instability. Its eigendirection points offthe invariant plane5 into
the “cosk1n” (k1 = 2π /N) direction in the phase space. Theother
growth rate isO(²
12 ), with its eigendirection in the plane5. The fractional
power
of ² arises because of the resonanceγ̇ = 0 (when I = ω, ² = 0 in
(3.3)). Similarly,there is one decay rateO(²0) and a secondO(²
12 ). All other decay rates areO(²). This
ordering for small² introduces a dichotomy of time scales which
is central throughoutour analysis.
In Appendix B, we summarize the growth rates and eigenvectors of
the linear stabilityanalysis ofq² in S.
4. Fenichel Fibers
Although the center-stable and center-unstable manifoldsWcs²
andWcu² of Theorem 3.1
areCk in ², trajectories insideWcs² or Wcu² are not evenC
0 in ², due to their singularnature in time. The central object
we are looking for is a homoclinic orbit connectingto q² . For
this, we need better coordinates in the neighborhood ofWc² in W
cs² or W
cu² .
Fenichel [5], [6], [7] introduced such coordinates (called
Fenichel fibers) by using thedichotomy of time scales and the graph
transform technique of Hadamard [8]. Thesefibers, in our setting,
are one-dimensional curves. The fibers have several nice
properties.In particular, they are smooth in². We shall
representWcs² andW
cu² as unions over these
Fenichel fibers rather than as unions over orbits.
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232 Y. Li and D. W. McLaughlin
4.1. A Simple Example Showing Fenichel Fibers
Consider the simple two-dimensional system{ẋ = −²x,ẏ = −y,
where 0≤ ² ¿ 1. For more detailed explanation of this example,
see [15]. Below wegive a brief summary. Figure 4.1 is an
illustration of the orbits which perturb singularly.The Fenichel
fibers are defined as follows: Define a family of one-dimensional
curvesindexed by pointsq0 = (x0, 0) ∈ Rx,
F (s,²)(q0) ={
q = (x0, y)∣∣∣∣ y ∈ R}.
Hereq0 is called “the base point,” and for eachq0, F (s,²)(q0)
is a stable Fenichel fiberwith base pointq0. See Figure 4.2 for an
illustration of the fibers. In particular, we canread off the
following nice properties of the fibers:
1. The “(² 6= 0) fibers” are identical with the “(² = 0)
fibers.”2. Denote byFt² the solution operator. Letq0 ∈ Rx be a base
point,q0 = (x0, 0);
thenFt² • q0 = (x0e−²t , 0). ∀q1 ∈ F (s,²)(q0), thenq1 = (x0,
y1), for somey1 ∈ R.Ft² • q1 = (x0e−²t , y1e−t ). Therefore, we
have
‖Ft² • q1− Ft² • q0‖ = e−t‖q1− q0‖,where‖ ‖ is the Cartesian
norm inR2. Thus, points on a fiber suffer the same forwardtime fate
as the base point.
3. Although a fiber itself is not invariant under the flow,
fibers as a family are invariantunder the flow, i.e., fibers
commute with the solution operator:
Ft² • F (s,²)(q0) = F (s,²)(Ft² • q0).More importantly, similar
properties hold quite generally.
4.2. Fiber Theorem
Before we state the fiber theorem, we need some definitions.
Definition 7 (Locally Positively (or Negatively) Invariant
Family of Submanifolds).SupposeV is a locally invariant submanifold
inS, under the flowFt² . Let {M(Q): Q ∈V} be a family of
submanifolds inS parametrized byQ ∈ V . We say that{M(Q): Q ∈V} is
locally positively invariant under the flowFt² if Ft² (M(Q)) ⊂M(Ft²
(Q)) for allQ ∈ V and allt ≥ 0 such that⋃t ′∈[0,t ] Ft ′² (Q) ⊂ V .
We say that{M(Q): Q ∈ V} islocally negatively invariant underFt² if
F
t² (M(Q)) ⊂M(Ft² (Q)) for all Q ∈ V and all
t ≤ 0 such that⋃t ′∈[t,0] Ft ′² (Q) ⊂ V .Definition 8 (Cr1
Family ofCr2 Manifolds). SupposeV is a locally invariant
subman-ifold in S, and let{M(Q): Q ∈ V} be a family ofCr2
submanifolds inS parametrized
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 233
Fig. 4.1.Orbits for the simple example.
Fig. 4.2.Fibers for the simple example.
by Q ∈ V . LetM = {(Q, Q′) | Q ∈ V, Q′ ∈M(Q)}.
We say that{M(Q): Q ∈ V} is a Cr1 family of Cr2 submanifolds ifM
is a Cr1submanifold ofS × S.
Fibration theorems of hyperbolic invariant manifolds were proved
by Fenichel [5],[6], and Hirsch, Pugh, and Shub [11]. In the
setting of geometric singular perturbationsof differential systems,
Fenichel [5] [6] [7] proved the fibration theorem using the
graphtransform method of Hadamard [8]. Infinite-dimensional version
of the fibration theoremis proved in the book [18]. The theorem
stated specifically for our system (1.1) is asfollows.
-
234 Y. Li and D. W. McLaughlin
Theorem 4.1. Assume the conditions in the “Local Persistent
Invariant Manifold The-orem” 3.1, then inside the local persistent
center-stable manifold Wcs² , there is a “C
k−1
family of Ck one-dimensional manifolds”{F (s,²)(Eq): Eq ∈ Wc² },
called stable Fenichelfibers.
• Wcs² can be represented as a union of these fibers,
Wcs² =⋃Eq∈Wc²F (s,²)(Eq).
• Each fiberF (s,²)(Eq) intersects the persistent center
manifold Wc² transversally inexactly one point which is the base
pointEq of the fiber.• Two fibersF (s,²)(Eq1) andF (s,²)(Eq2) are
either disjoint (in which caseEq1 6= Eq2) or
identical (in which caseEq1 = Eq2), for any Eq1 and Eq2 in Wc²
.Moreover, these stable fibers have the following properties:
1. {F (s,²)(Eq): Eq ∈ Wc² } is Ck−1 smooth in² for ² ∈ (−²2,
²2). The precise meaning ofthis statement is as follows: Let̂S ≡ S
× (−²2, ²2), then{(F (s,²)(Eq), ²): (Eq, ²) ∈Wc² × (−²2, ²2)} is a
“C k−1 family of Ck one-dimensional manifolds in̂S.”
2. {F (s,²)(Eq): Eq ∈ Wc² } is a “locally positively invariant
family of submanifolds.”3. Let κs be the positive constantκs ≡
√(1− cos2 k1)(1/h2+ ω2)(ω2− N2 tan2 πN ).
There is a positive constant Cs such that ifEq ∈ Wc² and Eq1 ∈ F
(s,²)(Eq), then
‖F τ² (Eq1)− F τ² (Eq)‖ ≤ Cse−κsτ‖Eq1− Eq‖,
for all τ ≥ 0 such that Ft² (Eq) ∈ Wc² , t ∈ [0, τ ].
Furthermore, ifEq belongs to themaximally positive invariant set
A+(U), then
F (s,²)(Eq) ={Eq1 ∈ U : ‖Fτ² (Eq1)− F τ² (Eq)‖ ≤ Cse−κsτ‖Eq1−
Eq‖, for all τ ≥ 0
}.
4. For any Eq, Ep ∈ Wc² ; Eq 6= Ep; any Eq1 ∈ F (s,²)(Eq), and
anyEp1 ∈ F (s,²)( Ep); if
Ft² (Eq), Ft² ( Ep) ∈ Wc² , ∀t ∈ [0,∞);
moreover,
‖Ft² ( Ep1)− Ft² (Eq)‖ → 0, as t→∞;then, { ‖Ft² (Eq1)− Ft²
(Eq)‖
‖Ft² ( Ep1)− Ft² (Eq)‖}/
e−12κst → 0, as t→∞.
Similarly, for Wcu² .
Remark 4.1(Uniqueness of Fenichel Fibers). By “uniqueness of
Fenichel fibers,” wemean the uniqueness of a family of fibers
having all the properties stated in Theorem 4.1.If Eq ∈ A+(U), then
the Fenichel fiberF (s,²)(Eq) is unique from item 4 in Theorem
4.1.
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 235
The argument is as follows: AssumeF (s,²)1 (Eq),F (s,²)2 (Eq)
are two different stable Fenichelfibers with the same base pointEq
∈ Wc² ; moreover,Eq ∈ A+(U), then there are two pointsEp1 ∈ F
(s,²)1 (Eq) andEq1 ∈ F (s,²)2 (Eq), such thatEp1 does not belong
toF (s,²)2 (Eq), andEq1 doesnot belong toF (s,²)1 (Eq). By item 4
in Theorem 4.1, we have{ ‖F τ² (Eq1)− F τ² (Eq)‖
‖F τ² ( Ep1)− F τ² (Eq)‖}→ 0, ast →∞,
and {‖Fτ² ( Ep1)− F τ² (Eq)‖‖F τ² (Eq1)− F τ² (Eq)‖
}→ 0, ast →∞.
This contradiction implies the “uniqueness.”
We can view these fibers as “equivalence classes” that are
one-dimensional manifoldsof “initial conditions” which, under the
time flow, approach each other at the “fastestrate.” Consider for
example the persistent center-stable manifoldWcs² . On the
persistentcenter manifoldWc² ⊂ Wcs² , all expansion and contraction
rates are slow when comparedto the contraction rates off ofWc² .
Two pointsP1 and P2, each on the manifoldW
cs² ,
are said tolie on the same fiberif the two solution
trajectoriesq(t; P1) andq(t; P2),initialized atP1 andP2
respectively, approach each other at the fastest rate ast →+∞.Thus,
a fiber for the manifoldWcs² is an equivalence class of points on
the manifoldW
cs² ,
where the equivalence relation in effect “factors out” the
fastest contraction.The primary use of these fibers is to “relate
or connect” the fast dynamics with slow
dynamics over semi-infinite time intervals. For this connection,
each fiber is labeled bythe point at which it intersects the slow
manifoldWc² , i.e., the “base point” of the fiber. Alltrajectories
with initial points on the fiber approach that trajectory on the
slow manifoldwhich is initialized at the basepoint of the
fiber.
4.3. The Unique Explicit Fenichel Fibers for “(figure 8)⊗ A”From
integrable theory, we knowF (s,0)(Eq) andF (u,0)(Eq) quite well; in
some cases, wecan represent them explicitly. For example, we can
read off the explicit representationsof the (unique) Fenichel
Fibers for “(figure 8)⊗ A” from Corollary 1.
Theorem 4.2. For the integrable discretized NLS equation (2.1),
we have
1. The stable fiber in “(figure 8)⊗ A” with the base pointEq ∈
A: (qn ≡ q, ∀n;q = a exp{i γ }).
F (s,0)(Eq) = qei 2P[
G
Hn− 1
], (4.1)
where
G = 1+ cos 2P − i sin 2P tanhr,
Hn = 1± 1cosϑ
sin P sechr cos 2nϑ,
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236 Y. Li and D. W. McLaughlin
P = arctan√ρ cos2 ϑ − 1√ρ sinϑ
,
ϑ = πN, ρ = 1+ |q|
2
N2,
where r∈ (−∞,∞) parametrizes the one-dimensional fiber. As
r→+∞,F (s,0)(Eq)→ q. Moreover, in this case,{F (s,0)(Eq)} are the
unique stable Fenichel fibers in“(figure 8) ⊗ A” with the base
points in A.
2. The unstable fiber in “(figure 8)⊗ A” with the base pointEq ∈
A: (qn ≡ q, ∀n;q = a exp{i γ }).
F (u,0)(Eq) = qe−i 2P[
G
Hn− 1
]. (4.2)
As r → −∞, F (u,0)(Eq) → q. Moreover, in this case,{F (u,0)(Eq)}
are the uniqueunstable Fenichel fibers in “(figure 8)⊗ A” with the
base points in A.
Remark 4.2. As for the perturbed fibersF (s,²)(Eq) andF
(u,²)(Eq), we do not have explicitrepresentations. However, we know
that they are smooth in² which will be enough forour later use.
5. Melnikov Measurement:Wu(q²) ∩Wcs²In this section, we present
a Melnikov measurement (which we often call the “firstmeasurement”)
through which we answer the question: Is there any intersection
betweenWu(q²) andWcs² other thanW
u(q²)|5 (≡ Wu(q²) ∩5)?We know thatWcs² is a codimension 1
submanifold inS, and thatWu(q²) is a
two-dimensional submanifold. Generically, an intersectionWu(q²)
∩Wcs² will be one-dimensional. Since we also know that the trivial
intersectionWu(q²)|5 is one-dimensional,another question arises: Is
there a “nontrivial” intersection?
5.1. Main Argument
First, we rewrite equation (1.1) in the vector form:
d
dtEqn = ρn
{J gradEqn H
}+ ² Egn, (5.1)
where
Eqn ≡ (qn, rn)T , rn ≡ −q̄n, J =(
0 1−1 0
),
H ≡ − ih2
N−1∑n=0
{q̄n(qn+1+ qn−1)− 2
h2(1+ ω2h2) ln(1+ h2|qn|2)
},
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 237
Egn ≡ (gn,1, gn,2)T ,gn,1 ≡ −αqn + 0 + βN2(qn+1− 2qn +
qn−1),gn,2 ≡ −αrn − 0 + βN2(rn+1− 2rn + rn−1).
Because we are working in a high-dimensional phase space, the
first natural question is:How manymeasurements must be made to
determine ifWu(q²) intersects withWcs² ? Theanswer is: one
measurement is enough. Intuitively,Wcs² has codimension 1
(coordinatized
by grad F̃1). In order to measure the distance betweenWu(q²)
andWcs² , we need only
to measure along the transverse directiongrad F̃1, which
requires only one Melnikovintegral. See Figure 5.1 for the detailed
geometry.
The rigorous argument begins by considering, withinS, theO(²0)
neighborhoodUof the resonance circleSω, with boundary∂U , of the
“local persistent invariant manifoldtheorem 3.1” and the “fiber
theorem 4.1.” In this neighborhoodU , we have the followingfiber
representations:
Wu(q²) =⋃{
F (u,²)(Eq)∣∣∣∣ Eq ∈ Wu(q²)|5}, (5.2)
Wu(q²)|5 ={(γ, I )
∣∣∣∣ I = ω +√²Ju(γ )}, (5.3)whereJu(γ ) denotes the curve in the
plane5which represents the unstable manifold ofq² restricted to
this plane; see Eq. (3.4), (3.5). Similarly,Wcs² has the fiber
representation,
Wcs² =⋃{
F (s,²)(Eq)∣∣∣∣ Eq ∈ Wc² }. (5.4)
For convenience of argument, letM² be aO(²0) neighborhood of the
resonance circleSω in Wc² , chosen so thatM² is properly included
inU , i.e.,M² never touches theboundary ofU . Notice thatWu(q²)|5
⊂M² ; in fact, we chooseM² large enough sothat the wholefish lies
insideM² . DefineWu(M²) andWs(M²) as follows:
Wu(M²) ≡⋃{
F (u,²)(Eq)∣∣∣∣ Eq ∈M²}, (5.5)
Ws(M²) ≡⋃{
F (s,²)(Eq)∣∣∣∣ Eq ∈M²}. (5.6)
Define∂Uu² ≡ ∂U∩Wu(M²),∂Us² ≡ ∂U∩Ws(M²). The following lemma is
a corollaryof the “fiber theorem 4.1”:
Lemma 5.1. In S, every point in∂Uu² is on a unique unstable
fiber explicitly expressedby (5.5), with a unique base point inM² .
Similarly, every point in∂Us² is on a uniquestable fiber explicitly
expressed by (5.6), with a unique base point inM² .
Similarly, inS, define∂Uu(q²) ≡ ∂U ∩Wu(q²). The following lemma
is also a corollaryof the “fiber theorem 4.1”:
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238 Y. Li and D. W. McLaughlin
Lemma 5.2. In S, every point in∂Uu(q²) is on a unique unstable
fiber explicitly ex-pressed by (5.2), (5.3), with a unique base
point in Wu(q²)|5.
We begin to study the possibility of an intersection
betweenWu(q²)andWs(M²) ⊂ Wcs² ,by considering all base points
inWu(q²)|5 of the unstable fibers inWu(q²). (Cf. Fig-ure 5.1.)
Choose a base point(γ, I u) ∈ Wu(q²)|5; then there is a
uniqueperturbedunstable fiber,F (u,²)(γ, I u) with the base
point,(γ, I u). There is also a uniqueun-perturbedunstable fiber,F
(u,0)(γ, I u) with the samebase point,(γ, I u). In the un-perturbed
case, this fiber is explicitly represented in Theorem 4.2. By
“fiber theorem4.1,” F (u,²)(γ, I u) is Ck−1 smooth in²; therefore,
the fiberF (u,²)(γ, I u) is in a O(²)neighborhood ofF (u,0)(γ, I
u). We know thatF (u,0)(γ, I u) intersects the codimension1
boundary∂U transversally. (If necessary, we may shrink the regionU
to accomplishthis.) Thus,F (u,²)(γ, I u) also intersects the
codimension 1 boundary∂U transversally.Moreover, letEq (u,²) ≡ F
(u,²)(γ, I u) ∩ ∂U , Eq (u,0) ≡ F (u,0)(γ, I u) ∩ ∂U . Then we
have
Eq (u,²) = Eq (u,0) + O(²). (5.7)
Starting fromEq (u,0), theunperturbeddiscrete NLS equation (2.1)
produces an orbithhomoclinic to the invariant plane5, with the
explicit representation given in Corollary1. After a finite time,h
will intersect with∂U a second time. This unique intersectionon the
“landing side” is denoted byEq (s,0). We emphasize thatEq (s,0) is
on a uniqueunperturbedstable fiber with base point(γ∗, I (s,0)) ∈
5; moreover,I (s,0) = I u. Startingfrom Eq (u,²),
theperturbeddiscrete NLS equation (1.1) will also produce an
orbithu² .From the smoothness property of the solution operator, in
a finite time interval,hu² willstay in anO(²) neighborhood ofh.
Thus,hu² will also intersect with∂U at Eq (s,²), andEq (s,²) is in
O(²) neighborhood ofEq (s,0). The Melnikov measurement determines
ifEq (s,²)(and thereforehu² ) lies in W
s(M²).The next step is to set up a homoclinic coordinate system
at a point onh in between
Eq (u,0) andEq (s,0). For any such pointEqh, let6 be a
codimension 1 hyperplane atEqh whichis transversal to the
homoclinic orbith and which contains the vectorgrad F̃1. (Thismakes
sense because the constant of motionF̃1 Poisson commutes with the
integrableHamiltonian which defines the homoclinic orbith.) Wcs0 ∩
6 has codimension 1 in6,andgrad F̃1 is transversal toWcs0 ∩6. Let
{Er j } denote a coordinate of the tangent spaceof Wcs0 ∩6 at Eqh.
Then{grad F̃1, Er j } is a coordinate system for6 with the origin
atEqh.In this coordinate frame, the intersection pointEq (u,²)h ≡
hu² ∩6 has the representation
Eq (u,²)h = ²au grad F̃1+∑
j
²auj Er j , (5.8)
which follows immediately from the relation (5.7) and the
regularity of the solutionoperatorFt² . (For any fixedT , 0≤ T
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 239
where{asj } parametrize the codimension 1 hypersurfaceWs(M²)∩6
in6, and f s({asj })is a smooth function. Now we choose the special
values forasj : a
sj = ²auj , for all j . This
special choice determines a unique pointEq s∗ on6s, with the
representation
Eq s∗ = f s({²auj }) grad F̃1+∑
j
²auj Er j . (5.10)
By the “global persistent invariant manifold theorem 3.2,” the
global persistent center-stable manifoldWs(M²) ⊂ Wcs² is Ck in ²;
then Ws(M²) ∩ 6 stays in anO(²)neighborhood ofWcs0 ∩6 in 6 which
passes through the origin. Thus, we have
f s({²auj }) = ²as, (5.11)for some real numberas. By Eqs. (5.8),
(5.10), (5.11), we define the following signeddistance betweenEq
(u,²)h andEq s∗ :
d1 = Signed Distance{Eq (u,²)h , Eq s∗
}≡ ²{au − as}
∥∥∥∥grad F̃1∥∥∥∥2≡ 〈grad F̃1, Eq (u,²)h − Eq s∗ 〉. (5.12)
Now the problem is reduced to measuring the distance betweenEq
(u,²)h andEq s∗ . Moreover,only one measurement, ingrad F̃1
direction, is needed. See Figure 5.1 for the detailedgeometry.
5.2. Derivation of a Melnikov Integral
In this subsection, we are going to derive a Melnikov integral
as a measure of the signeddistance defined above. See Figure 5.1
for the detailed geometry.
We give the following parametrization for the orbitsh andhu²
:
h ≡ h(t), −∞ < t < +∞,h(0) = Eqh;
hu² ≡ hu² (t), −∞ < t ≤ 0,hu² (0) = Eq (u,²)h .
Let hs² denote the orbitFt² (Eq s∗ ), 0≤ t < +∞, with the
parametrization
hs² ≡ hs²(t), 0≤ t < +∞,hs²(0) = Eq s∗ .
Notice that
d1 = Signed Distance{Eq (u,²)h , Eq s∗
}≡ 〈grad F̃1, Eq (u,²)h − Eq s∗ 〉;
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240 Y. Li and D. W. McLaughlin
then we can decomposed1 as follows:
d1 = 〈∇ F̃1(Eqh), Eq (u,²)h − Eqh〉 − 〈∇ F̃1(Eqh), Eq s∗ −
Eqh〉,
where “∇” denotes “grad”. Let
1+(t) ≡ 〈∇ F̃1(h(t)), hs²(t)− h(t)〉; 0≤ t < +∞,
1−(t) ≡ 〈∇ F̃1(h(t)), hu² (t)− h(t)〉; −∞ < t ≤ 0.
Then,
d1 = 1−(0)−1+(0). (5.13)See Figure 5.1 for an illustration. We
will prove the following:
Theorem 5.1.
1−(0) = ²∫ 0−∞〈∇ F̃1(h(t)), Eg(h(t))〉 dt + O(²2),
1+(0) = −²∫ +∞
0〈∇ F̃1(h(t)), Eg(h(t))〉 dt + O(²2).
Thus,
d1 = Signed Distance{Eq (u,²)∗ , Eq s∗
}= ²MF̃1 + O(²2),
where
MF̃1 =∫ +∞−∞〈∇ F̃1(h(t)), Eg(h(t))〉 dt (5.14)
=∫ ∞−∞
{ N−1∑n=0
(δ F̃1δqn
gn,1+ δ F̃1δrn
gn,2
)∣∣∣∣(evaluated at h(t))} dt.Proof. To prove this theorem, we
note thath(t), hs²(t), andh
u² (t) solve the following
equations:
ḣ(t) = J∇H(h(t)),ḣs²(t) = J∇H(hs²(t))+ ² Eg(hs²(t)),ḣu² (t) =
J∇H(hu² (t))+ ² Eg(hu² (t)),
whereJ is a symplectic matrix which can be written in the block
form:J = diag{J0, . . . ,JN−1}, in which,
Jn =(
0 ρn−ρn 0
),
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 241
whereρn is given in Eq. (2.1). Thus,
ḣs²(t)− ḣ(t) = J∇H(hs²(t))− J∇H(h(t))+ ² Eg(hs²(t))
= ∇(J∇H
)(h(t)) •
(hs²(t)− h(t)
)+ ² Eg(h(t))+ R1+ R2,
where
R1 ≡ J∇H(hs²(t))− J∇H(h(t))−∇(J∇H
)(h(t)) •
(hs²(t)− h(t)
),
R2 ≡ ²(Eg(hs²(t))− Eg(h(t))
).
Differentiating1+(t), we have
1̇+(t) = 〈∇ F̃1(h(t)), ḣs²(t)− ḣ(t)〉 + 〈∇2F̃1(h(t)) • ḣ(t),
hs²(t)− h(t)〉
= ²〈∇ F̃1(h(t)), Eg(h(t))〉 + 〈∇ F̃1(h(t)), R1〉 + 〈∇ F̃1(h(t)),
R2〉 + E,where
E =〈∇ F̃1(h(t)),∇
(J∇H
)(h(t)) •
(hs²(t)− h(t)
)〉
+〈∇2F̃1(h(t)) •
(J∇H(h(t))
), hs²(t)− h(t)
〉.
We know that the Poisson bracket
{F̃1, H} = 〈∇ F̃1,J∇H〉 = 0,then, by differentiation, we
have〈
∇ F̃1(Eq),∇(J∇H
)(Eq) • δEq
〉+ 〈∇2F̃1(Eq) • δEq,J∇H(Eq)〉 = 0.
We also know that∇2F̃1(Eq) is a symmetric bilinear operator;
therefore,
〈∇2F̃1(Eq) • δEq,J∇H(Eq)〉 = 〈∇2F̃1(Eq) • J∇H(Eq), δEq〉.SettingEq
= h(t), δEq = hs²(t)− h(t), we see thatE = 0. Thus,
1̇+(t) = ²〈∇ F̃1(h(t)), Eg(h(t))〉+ 〈∇ F̃1(h(t)), R1〉 + 〈∇
F̃1(h(t)), R2〉. (5.15)
Similarly,
1̇−(t) = ²〈∇ F̃1(h(t)), Eg(h(t))〉+ 〈∇ F̃1(h(t)), R3〉 + 〈∇
F̃1(h(t)), R4〉, (5.16)
-
242 Y. Li and D. W. McLaughlin
where
R3 ≡ J∇H(hu² (t))− J∇H(h(t))−∇(J∇H
)(h(t)) •
(hu² (t)− h(t)
),
R4 ≡ ²(Eg(hu² (t))− Eg(h(t))
).
Let Eq (s,²)∗ denote the point at whichhs²(t) intersects∂U ,
then Eq (s,²)∗ is on a uniqueperturbed stable fiber with base
pointb² ∈ M² ⊂ Wc² . The proof of the theorem iscompleted with the
following steps:
1. In Appendix C, we establish the following:
Lemma 5.3. • limt→−∞1−(t) = 0.• If F τ² (b²) ∈ Wc² for all 0 ≤ τ
< +∞, thenlimt→+∞1+(t) = 0. If F τ² (b²) does not
stay in Wc² for all 0 ≤ τ < +∞, then it stays in Wc² for all
0 ≤ τ ≤ T(²), whereT(²) ∼ O( 2
κ1ln 1/²); moreover,1+(T(²)) ∼ O(²2).
2. Finally, in Appendix D, we establish the following:
Lemma 5.4. ∫ 0−∞〈∇ F̃1(h(t)), R3〉 dt ∼ O(²2),∫ 0
−∞〈∇ F̃1(h(t)), R4〉 dt ∼ O(²2),∫ +∞(or T(²))
0〈∇ F̃1(h(t)), R1〉 dt ∼ O(²2),∫ +∞(or T(²))
0〈∇ F̃1(h(t)), R2〉 dt ∼ O(²2),
∫ +∞T(²)〈∇ F̃1(h(t)), Eg(h(t))〉 dt ∼ O(²2). (5.17)
By Lemmas 5.3 and 5.4 and Eqs. (5.15), (5.16), we have the
following representationsfor 1−(0) and1+(0):
1−(0) = ²∫ 0−∞〈∇ F̃1(h(t)), Eg(h(t))〉 dt + O(²2),
1+(0) = −²∫ +∞(or T(²))
0〈∇ F̃1(h(t)), Eg(h(t))〉 dt + O(²2).
Moreover, by the estimate (5.17), we can always represent1+(0)
as
1+(0) = −²∫ +∞
0〈∇ F̃1(h(t)), Eg(h(t))〉 dt + O(²2).
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 243
Fig. 5.1. Geometric illustration of the Melnikov measurement.
For the definition of no-tation, see the text. Note that the
Melnikov function approximates the distance betweenEq (u,²)∗ andEq
s∗ .
This completes the proof of the theorem.
Figure 5.1 is the geometric illustration of the argument in this
section.
Remark 5.1. The integral defined in (5.14) is called the
Melnikov integral, and in thenext section, we will further
approximateMF̃1.
5.3. An Approximation
Next we are going to approximate the unperturbed orbith,
homoclinic to a circle in theannulusA, by another unperturbed orbit
which is homoclinic to the resonance circleSω,
-
244 Y. Li and D. W. McLaughlin
and obtain the following approximation for the Melnikov
integral:
MF̃1 = M̂F̃1 + O(√
² ln21√²
),
where
M̂F̃1 =∫ ∞−∞
{ N−1∑n=0
(δ F̃1δqn
gn,1+ δ F̃1δrn
gn,2
)∣∣∣∣(evaluated athω)} dt. (5.18)We know, by the above
construction, thath intersects∂U in backward time atEq (u,0);
thus, Eq (u,0) is on a unique unperturbed unstable fiberF
(u,0)(γ, I u) with base point(γ, I u) ∈ Wu(q²)|5. We know, from the
representation (5.3) ofWu(q²)|5, that I u =ω +√²Ju(γ ); thus,(γ, I
= ω) is in anO(√²) neighborhood of(γ, I u). (Notice thathere we
take the two base points with the same phaseγ .) Moreover, the
unstable fiberF (u,0)(γ, ω) is also an orbit. (Denote the orbit
byhω, which is an orbit homoclinic tothe resonance circleSω;
moreover, it is also an unstable fiber in backward time, and
astable fiber in forward time.) LetEq (u,0)ω ≡ F (u,0)(γ, ω)∩∂U ;
then, by the “fiber theorem4.1,” Eq (u,0)ω is in an O(
√²) neighborhood ofEq (u,0). By the regularity of the
solution
operatorFt0, in the regionS/U (i.e., the complement ofU in S),
hω stays in anO(√²)
neighborhood ofh. Recall thatEq (s,0) is the intersection point
ofh to ∂U in forward time.Let Eq (s,0)ω be the intersection point
ofhω to ∂U in forward time. Then,Eq (s,0)ω is in anO(√²)
neighborhood ofEq (s,0). Now we are going to estimate the deviation
betweenhω
andh insideU and the change in Melnikov integrals. Since the
estimates are parallel forthe backward time (t → −∞) part and the
forward time (t → +∞) part, we just takethe backward time (t → −∞)
part as the example. By the “fiber theorem 4.1,” a pointon an
unstable fiber approaches the base point exponentially under
backward time flow;thus, letT ² ≡ 1/κu ln 1√² , then for allt ∈
(−∞,−T ² ],∥∥∥∥Ft0(Eq (u,0)ω )− Ft0(γ, ω)∥∥∥∥ ≤ Ĉu√²,whereĈu =
Cu‖Eq (u,0)ω − (γ, ω)‖; similarly,∥∥∥∥Ft0(Eq (u,0))− Ft0(γ, I
u)∥∥∥∥ ≤ Ĉ′u√²,whereĈ′u = Cu‖Eq (u,0) − (γ, I u)‖. Equivalently,
for allt ∈ (−∞,−T ² ],
Ft0(Eq (u,0)ω ) = Ft0(γ, ω)+ O(√²),
Ft0(Eq (u,0)) = Ft0(γ, I u)+ O(√²).
We also know that
grad F̃1
∣∣∣∣(evaluated atFt0(γ, ω) or Ft0(γ, I u)) ≡ 0,
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 245
for anyt ∈ (−∞,∞). Moreover, from the explicit formula forgrad
F̃1 below, we have∥∥∥∥grad F̃1∥∥∥∥(evaluated ath or hω) ≤ C1eκut
,for all t ∈ (−∞,−T ²), whereC1 is a constant independent of².
Thus,∫ −T ²
−∞
{ N−1∑n=0
(δ F̃1δqn
gn,1+ δ F̃1δrn
gn,2
)∣∣∣∣(evaluated ath or hω)} dt ∼ O(√²).Next, we estimate the
deviation betweenFt0(Eq (u,0)ω ) and Ft0(Eq (u,0)) for t ∈ [−T ²,
0].From the explicit formulae in Corollary 1, and Theorem 4.2, we
have the explicit repre-sentation ofFt0(Eq (u,0)ω ) andFt0(Eq
(u,0)),
Ft0(Eq (u,0)ω ) = q0[
G
Hn− 1
]∣∣∣∣(given by the corollary atq = q0),Ft0(Eq (u,0)) = q1
[G
Hn− 1
]∣∣∣∣(given by the corollary atq = q1),whereq0 = ω exp{i (γ −
2P0)}, q1 = I u exp{−2i [((I u)2 − ω2)t ] + i (γ − 2P1)}, P0and P1
are given in Corollary 1 according to|q| = ω and |q| = I u,
respectively. Byexplicitly checking the above representations and
noticing that
2[(I u)2− ω2]T ² ∼ O(√
² ln1√²
),
we have, for allt ∈ [−T ², 0],∥∥∥∥Ft0(Eq (u,0)ω )− Ft0(Eq
(u,0))∥∥∥∥ ≤ C̃√² ln 1√² ,whereC̃ is independent of². Therefore, we
get∫ 0
−T ²
{ N−1∑n=0
(δ F̃1δqn
gn,1+ δ F̃1δrn
gn,2
)∣∣∣∣(evaluated ath)} dt=∫ 0−T ²
{ N−1∑n=0
(δ F̃1δqn
gn,1+ δ F̃1δrn
gn,2
)∣∣∣∣(evaluated athω)} dt+ O
(√² ln2
1√²
).
Finally, we have
MF̃1 = M̂F̃1 + O(√
² ln21√²
),
where
M̂F̃1 =∫ ∞−∞
{ N−1∑n=0
(δ F̃1δqn
gn,1+ δ F̃1δrn
gn,2
)∣∣∣∣(evaluated athω)} dt. (5.19)
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246 Y. Li and D. W. McLaughlin
5.4. Calculation ofM̂F̃1
In this subsection we calculatêMF̃1. Let−γ1 ≡ γ − 2P, where
P = arctan√ρ cos2 ϑ − 1√ρ sinϑ
,
ϑ = πN, ρ = 1+ ω
2
N2.
Then,hω has the representation, given in Corollary 1,
Qn ≡ Qn(t, r ; N, ω, γ1,±) = ωe−i γ1[
G
Hn− 1
], (5.20)
where
G = 1+ cos 2P − i sin 2P tanhτ,
Hn = 1± 1cosϑ
sin P sechτ cos 2nϑ,
τ = 4N2√ρ sinϑ√ρ cos2 ϑ − 1 t + r.
At qn ≡ ωe−i γ1, let z+ be the real double point of the Lax pair
(2.3), (2.4),z+ = √ρ cosϑ +
√ρ cos2 ϑ − 1.
We have the following two linearly independent Bloch functions
atz+:
ψ+n =(√
ρeiϑ)n
eÄ+t((1/z+ −√ρeiϑ)e−i (γ1/2)−i (ω/N)ei (γ1/2)
),
ψ−n =(√
ρe−iϑ)n
eÄ−t( −i (ω/N)e−i (γ1/2)(z+ −√ρe−iϑ)ei (γ1/2)
),
where
Ä+ = i N2[√ρ(1/z+ − z+)eiϑ + 2 lnz+
],
Ä− = i N2[√ρ(1/z+ − z+)e−iϑ + 2 lnz+
].
Let φn ≡ c+ψ+n + c−ψ−n ; then, following the calculation in
[20], we haveδ F̃1δEqn
∣∣∣∣(evaluated athω) = Cz+ Wn1̂n An+1
((1/z2+)φ̄n1φ̄(n+1)1(1/z̄2+)φ̄n2φ̄(n+1)2
),
whereCz+ is a complex constant and
Wn = ψ+n1ψ−n2− ψ+n2ψ−n1,1̂n = |φn1|2+ |z+|2|φn2|2,An = |φn2|2+
|z+|2|φn1|2.
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 247
Let Gn ≡ δ F̃1δqn | (evaluated athω), and decomposeGn as
follows:
Gn = Gen + Gon,whereGen is the even part ofGn, i.e., G
en = GeN−n, Gon is the odd part ofGn, i.e.,
Gon = −GoN−n. Gon gives no contribution to the Melnikov
integral. Up to multiplicationby a real constant, we have
Gen =sechτ
3n
{cosϑ sechτ + (sin P − i cosP tanhτ) cos 2nϑ
},
where
3n = 1+ 2(1/ cosϑ) sin P sechτ cos 2nϑ cos 2ϑ+ 1/2(1/ cos2 ϑ)
sin2 P sech2 τ(cos 4nϑ + cos 4ϑ).
Finally, we obtain an explicit formula for the Melnikov
function̂MF̃1.
Proposition 1.
M̂F̃1 = 0{
M0 − χαMα + χβMβ}, (5.21)
whereχα ≡ α/0, χβ ≡ β/0, M0 ≡ cosγ1M̂0; moreover,
M̂0 ≡ M̂0(ω, N) =∫ ∞−∞
sechτN−1∑n=0
1
3n
{cosϑ sechτ + sin P cos 2nϑ
}dτ,
Mα ≡ Mα(ω, N) =∫ ∞−∞
N−1∑n=0
Re{GenQn} dτ,
Mβ ≡ Mβ(ω, N) = N2∫ ∞−∞
N−1∑n=0
Re{Gen(Qn+1− 2Qn + Qn−1)} dτ,
where Qn is given in (5.20).
5.5. The Intersection BetweenWu(q²) and Ws(M²) ⊂ Wcs²From the
arguments in the previous subsections, we get the following
characterizationof the signed distance betweenWu(q²) andWs(M²) ⊂
Wcs² :
d1 = Signed Distance{Eq (u,²)h , Eq s∗
}= ²
{M̂F̃1 + O
(√² ln2
1√²
)}, (5.22)
whereM̂F̃1 is given in Eq. (5.21). Moreover,
d1 = Signed Distance{Eq (u,²)h , Eq s∗
}= Function(²;ω, α, β, 0; γ1),
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248 Y. Li and D. W. McLaughlin
in which ² is a small positive parameter² ∈ (0, ²2), ²2 is a
positive number given inTheorem 3.1,{ω, α, β, 0} are external
parameters defined in the region6N defined inthe introduction,
andγ1 is an internal parameter. By the relation−γ1 = γ − 2P at
thebeginning of the last subsection, we can determine the
“take-off”γ wheneverγ1 is given,and vice versa. From the explicit
formula (5.21) forM̂F̃1, we see thatM̂F̃1 depends on
{ω, α, β, 0} andγ1 in a very simple way. SettinĝMF̃1 = 0, we
have
cosγ1M̂0(ω, N)− χαMα(ω, N)+ χβMβ(ω, N) = 0. (5.23)
The following lemma is an immediate consequence of the above
equation (5.23).
Lemma 5.5. For N tanπ /N < ω < N tan 2π /N, there exists a
regionDω in the firstquadrant of R2, such that, for any(χα, χβ) ∈
Dω, there are two values ofγ1: γ±1 ≡γ±1 (ω, χα, χβ) = ±arccos{(χαMα
− χβMβ)/M̂0}, at whichM̂F̃1 = 0. Moreover, whenM̂F̃1 is viewed as a
function ofγ1, these zeros are simple.
By this lemma, Eq. (5.22), and the implicit function theorem, we
have the followingtheorem:
Theorem 5.2(Intersection betweenWu(q²) andWs(M²) ⊂ Wcs² ). There
exists a sub-region 6sN of 6N and a positive number²4, such that,
for any fixed parameters{ω, α, β, 0; ²} ∈ 6sN × (0, ²4), there are
two values ofγ1 = γ (±,²)1 which areO(√² ln2 1√
²) close toγ±1 , at which there are transversal intersections
between W
u(q²)and Ws(M²) ⊂ Wcs² ; moreover, the intersections are
generic, i.e., the two intersectionsets are one-dimensional curves.
These two curves h(u,±)² can be determined as follows:From γ (±,²)1
, we can findγ
(±,²) = −γ (±,²)1 + 2P, as discussed above; thenγ = γ (±,²)will
identify the “take-off” base points of the perturbed unstable
fibers in Wu(q²) onwhich the two points h(u,±)² ∩ ∂U sit.
Remark 5.2. The distinct anglesγ = γ (±,²) can be interpreted as
follows: The two-dimensional unstable manifoldWu(q²) consists in a
one parameter family of orbitshu² ,indexed by the “take-off” angleγ
. The two distinct members of this family of “take-off”orbits which
are labeled byγ = γ (±,²) will approach, in forward time, the slow
manifoldM² ; other “take-off” orbits labeled by nearby “take-off”
anglesγ will not approach theslow manifoldM² in forward time. Thus,
the Melnikov measurement selects thosedistinct “take-off” angles
for which the orbit returns to the slow manifold.
6. Existence of Orbits Homoclinic toq² : The Second
Measurement
In the last chapter, we have proved that there are generic
intersections betweenWu(q²)andWs(M²) ⊂ Wcs² , so that for external
parameters in a fixed open set, there are orbitswhich tend toq² in
backward time, and approachM² in forward time. Denote one ofthese
orbits byh² . In this chapter, we are going to show that, by
restricting the externalparameters to a codimension 1 submanifold
in external parameter space, these orbits are
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 249
in fact homoclinic toq² , i.e., in forward time they also tend
toq² . The detailed geometryfor the argument below is shown in
Figure 6.1.
Definition 9. Define the following restricted objects:
Ws(q²)|5 ≡ 5 ∩Ws(q²),Ws(q²)|M² ≡ M² ∩Ws(q²),Wu(q²)|M² ≡ M²
∩Wu(q²).
From the discussion in previous chapters, the following lemma is
obvious.
Lemma 6.1. Wu(q²)|M² = Wu(q²)|5, which is one-dimensional.
Ws(q²)|5 is alsoone-dimensional. Ws(q²)|M² has codimension 1 inM²
.
Following the notations in the last chapter, letEq (s,²) ≡ ∂U ∩
h² (called the “landingpoint”) denote the intersection point of the
orbith² with the boundary∂U . Notice, fromthe last chapter, thatEq
(s,²) ∈ ∂Us² ≡ ∂U ∩Ws(M²). Then,Eq (s,²) is on a unique stablefiber
in Ws(M²) with base pointEq (s,²)b ∈M² . By the “fiber theorem
4.1,”Eq (s,²) andEq (s,²)b have the same fate in forward time. Then
the problem of following the orbith²in forward time is reduced to
following the orbit inM² starting from the base pointEq (s,²)b .
Therefore, the question: Ast → +∞, doesh² → q²? is reduced to:
DoesEq (s,²)b ∈ Ws(q²)|M²? Next, we are going to answer this
question affirmatively.
From the argument in the last chapter, we know thatEq (s,²) is
O(²) close toEq (s,0)which is the intersection point of the
unperturbed orbith with ∂U in forward time;moreover,Eq (s,0) is on
an unperturbed stable fiber whose base pointEq (s,0)b certainly
lieson the invariant plane5. Thus, by the “fiber theorem 4.1,”Eq
(s,²)b is O(²) close to theinvariant plane5, i.e.,
Distance{Eq (s,²)b , Eq (s,0)b
}= O(²); Distance
{Eq (s,²)b ,5
}= O(²). (6.1)
Now we define an important object—the cylindrical neighborhoodVµ
of 5 ∩M² inM²—as follows:
Vµ ≡{Eq∣∣∣∣ Eq ∈M², Distance{Eq,5} ≤ ²µ, 0< 1− µ¿ 1}.
(6.2)
Then by Eq. (6.1), we have the lemma:
Lemma 6.2. The “landing base point”Eq (s,²)b ∈ Vµ.
We know, from Lemma 6.1, thatWs(q²)|M² has codimension 1 inM² .
Here we makean assumption about the “height of the wall,”
W ≡(
Ws(q²)|M²)∩ Vµ.
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250 Y. Li and D. W. McLaughlin
Assumption 1. The height of the “wall”W shown in Figure 6.1 is
large enough so thatW separates Vµ into two disconnected regions.
Moreover,W intersects5 transversallyfor any² ∈ [0, ²2).
Remark 6.1. This assumption can be verified through the use of a
normal form trans-formation, which can be adapted from the work of
[17]. Here we omit this adaption, asthe normal form procedure is
described in detail in the above reference.
Next, we consider a signed distance betweenEq (s,²)b andW, and
characterize the signeddistance by a certain signed distance
defined on the invariant plane5. (See Figure 6.1for the detailed
geometry.) The geometric picture is as follows:W provides a
sufficientlyhigh codimension 1 middlewall in Vµ. We can specify its
positive and negative sides asin Figure 6.1. The pointEq (s,²)b
always lies insideVµ. If, through adjustment of externalparameters,
the pointEq (s,²)b can be made to sit in both regions, then by
continuity, therewill exist parameters for which the pointEq (s,²)b
sits on the wallW. Hence, for this specialchoice of external
parameters, the orbith² must return toq² in forward time.
Let qb5 ∈M² ∩5 be a point on5 that realizes the distance
betweenEq (s,²)b and5;then, from Eq. (6.1), we have
Eq (s,²)b = qb5 + ²g, (6.3)whereg is bounded. By the “height”
assumption 1, we can characterize the wallW asfollows: For anyEq
∈W, there existsq5 ∈ Ws(q²)|5 that realizes the distance betweenEq
andWs(q²)|5,
Eq = q5 + ²µ f, (6.4)where f is bounded. Therefore,
W ={Eq = q5 + ²µ f
∣∣∣∣ q5 ∈ Ws(q²)|5, f is bounded}.See Figure 6.1 for a geometric
illustration.
Let d5 be a signed distance betweenqb5 andWs(q²)|5, inside5,
andd2 be a signed
distance betweenEq (s,²)b andW in Vµ. We have
|d2| = infq5∈Ws(q² )|5, f ∈S(q5)
{‖Eq (s,²)b − (q5 + ²µ f )‖
},
= infq5∈Ws(q² )|5, f ∈S(q5)
{‖qb5 − q5 + (²g− ²µ f )‖
}, (6.5)
|d5| = infq5∈Ws(q² )|5
{‖qb5 − q5‖
},
= minq5∈Ws(q² )|5
{|qb5 − q5|
}. (6.6)
Thus, by (6.3) and (6.4), we have
d2 = d5 + O(²µ). (6.7)
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 251
Fig. 6.1. Geometry for the second measurement. For definition of
notation, see thetext. The second measurement approximates the
distance betweenEq (s,²)b andW.
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252 Y. Li and D. W. McLaughlin
Fig. 6.2.Geometry for the reduction.
Here,d5 is O(√²) and can be both positive and negative (which
will be shown below);
hence,d2 is alsoO(√²) and can be both positive and negative.
Thus, the calculation
of the signed distanced2 is reduced to the calculation of the
signed distanced5 on theinvariant plane5. This calculation, on5, is
identical with that described in detail in[23]. Denote byd̂5 the
signed distance betweenqb5 and theunbrokenfish boundaryWs(q̃²)|5,
then
d5 = d̂5 + O(²). (6.8)The detailed geometry for this reduction
is shown in Figure 6.2. In Figure 6.3, we showγs andγn as functions
ofχα andω. For appropriate values ofχα andω, γs− γn >
4P1.Therefore, the length of the fish parametrized byγ is big
enough to capture the phaseshift 4P1. See Figure 6.4 for an
illustration. Therefore, in the original coordinate (γ, I ),the
signed distanced5 betweenqb5 andW
s(q²)|5, and the signed distancêd5 betweenqb5 andW
s(q̃²)|5 are bothO(√²), and can be both positive and negative.
We have thefollowing:
Theorem 6.1.
d2 = d̂5 + O(²µ), (6.9)whered̂5 ≡ Signed Distance{qb5,Ws(q̃²)|5}
which takes values in(−C
√²,C√²) for
some constant C. The roots of the equationd̂5 = 0 can be
approximated by the roots ofthe equation,
4αωP − 20 sin 2P cosγ1 = 0.
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 253
Fig. 6.3.The functionsαs = αs(χα, ω) andαn = αn(χα, ω).
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254 Y. Li and D. W. McLaughlin
Fig. 6.4.Phase shifts of heteroclinic orbits vs. “fish”
geometry.
Proof. By (6.7), (6.8), we have the relation (6.9). Notice
thatd̂5 = 0 if and only if
1H1 ≡ H1(qb5)−H1(q̃²) = 0,
whereq̃² ≡ (J, γs) ≡ (0, arccos{χαω}),H1 = αγω − 0 sinγ − 2ωJ2.
Moreover, wehave
H1(q̃²) = H1(γ (±,²), I u)+ O(√²),
H1(qb5) = H1(Eq (s,0)b )+ O(√²),
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 255
Fig. 6.5.Geometry for the approximation.
whereEq (s,0)b ≡ (−γ1 − 2P + O(√²), I u), γ (±,²) = −γ1 + 2P
(see Corollary 1 and
Theorem 5.2), andP is given in subsection “Calculation of̂MF̃1.”
Thus, we have
1H1 = 4αωP − 20 sin 2P cosγ1+ O(√²).
The detailed geometry is shown in Figure 6.5. This completes the
proof of the theorem.
Now, we can combine the Melnikov measurement and the second
measurement toget the approximate roots for the two distance
equationsd1 = 0 andd2 = 0, given by{
cosγ1M̂0(ω, N)− χαMα(ω, N)+ χβMβ(ω, N) = 0,4αωP − 20 sin 2P
cosγ1 = 0,
(6.10)
where M̂0(ω, N), Mα(ω, N), and Mβ(ω, N) are defined in
Proposition 1. SolvingEq. (6.10), we get the following relation
amongχα, χβ , andω:
χβ = κ(ω, N)χα,
κ(ω, N) = 1Mβ(ω, N)
{Mα(ω, N)− ωM̂0(ω, N) 2P
sin 2P
}. (6.11)
By the implicit function theorem, we have the following theorem
(cf. Introduction):
Theorem 6.2. For any N (7 ≤ N < ∞), there exists a positive
number²0, such thatfor any² ∈ (0, ²0), there exists a codimension1
submanifold E² in 6N, E² is an O(²ν)perturbation of the hyperplaneβ
= κ α, whereκ = κ(ω; N) is shown in Figure 1.1,
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256 Y. Li and D. W. McLaughlin
ν = 1/2− δ0, 0 < δ0 ¿ 1/2. For any external parameters (ω, α,
β, 0) on E² , thereexists a homoclinic orbit asymptotic to the
fixed point q² .
Remark 6.2. Note that the two measurementsd1 = 0 andd2 = 0 can
be satisfied asfollows: First, choose the “take-off” phaseγ = −γ1+
2P approximately by
cosγ1 = 1M̂0(ω, N)
[χαMα(ω, N)− χβMβ(ω, N)
],
and thenχα andχβ approximately by the linear relation
χβ = κ(ω, N)χα, κ(ω, N) = 1Mβ(ω, N)
{Mα(ω, N)− ωM̂0(ω, N) 2P
sin 2P
}.
Remark 6.3. For N ≥ 7, κ can be positive. (See Figure 1.1.)
7. Conclusion
For the discretized perturbed NLS system (1.1), we have proved
the existence of homo-clinic orbits. In part II, we will construct
Smale horseshoes in the neighborhood of thesehomoclinic orbits.
This is a study on homoclinic and chaotic dynamics on perturbed
soliton systems.The prerequisite for such study is that the
unperturbed soliton system, viewed as aHamiltonian system, should
have singular level sets (i.e., “Figure 8” structures).
Amongsoliton systems which fall into this class are the focusing
NLS equation, sine-Gordonequation, Davey-Stewartson equation [19],
etc. The Backlund-Darboux representationsof the singular level set
enable us to build explicit Melnikov integrals which measure
thedistance between certain perturbed unstable and stable
manifolds.
In the process of proving the existence of (or constructing)
homoclinic orbits, severalmathematical tools from dynamical system
theory are utilized. Among them are a per-sistence of invariant
manifold theorem combined with a geometric singular
perturbationtheory, and a Fenichel-Hadamard fibration theorem. For
more on geometric singularperturbation theory, see the recent
review by Jones [12].
In this paper we have built a general argument for locating
homoclinic orbits in per-turbed soliton systems. This argument
should be applicable to a wide class of perturbedsoliton
equations.
Numerically locating homoclinic orbits, for the perturbed (lower
dimensional) dis-crete NLS systems studied in this paper, is in
progress [14].
8. Appendix A: Symmetries of the Eigen-Functions
In this appendix, we discuss some symmetries of the
eigen-functions of the Lax pair(2.3), (2.4).
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 257
Lemma 8.1. If ψn solves the Lax pair (2.3), (2.4) at z, then
Jψ̄n solves the same Laxpair at z∗ ≡ 1/z̄. Here J= ( 0 1−1 0), z∗
is the symmetry point of z with respect to the unitcircle centered
at the origin. Applying this fact to the fundamental solutions, we
have1̄(z) = 1(z∗).
There is a continuum counterpart (h → 0, i.e., for the Lax pair
for NLS PDE) of thislemma in whichz∗ is replaced byλ∗ = λ̄, which
is the symmetry point ofλwith respectto the real axis.
Lemma 8.2. If qn is even (i.e., qN−n = qn), andψn solves the Lax
pair (2.3), (2.4) atz, then
φn ≡(φn,1φn,2
)= αn
(ψ̄−n+1,2ψ̄−n+1,1
),
whereαn+1αn= ρn, solves the same Lax pair atz̄. Applying this
fact to the Bloch functions,
we have1̄(z) = 1(z̄).
There is a continuum counterpart of this lemma in whichz̄ is
replaced by−λ̄.
Lemma 8.3. If ψn solves the Lax pair (2.3), (2.4) at (z,qn),
then einπψn solves the Laxpair at (−z,−qn). Applying this fact to
the fundamental solutions, we have ei Nπ1(z,qn)= 1(−z,−qn).
There is no continuum counterpart of this lemma. Due to this
lemma, the spectrum inthis discrete case is a double-copy of the
corresponding spectrum in the continuum case.
9. Appendix B: Linear Stability of q²
In this appendix, we study the linear stability ofq² . Consider
a harmonic perturbationof q² ,
qn = q² + η q̃n, 0< η ¿ 1,
q̃n =(
Aj eÄj t + Bj eǞj t
)coskj n,
wherekj = 2π jN , Aj and Bj are complex constants. We have the
followingdispersionrelation atkj :
Äj ≡ ıj = −²[α + 2βh−2(1− coskj )
]± 2√Ej Fj , (9.1)
where
Ej = 2|q² |2+ (|q² |2− ω2)− (1− coskj )(1/h2+ |q² |2),Fj = (1−
coskj )(1/h2+ |q² |2)− (|q² |2− ω2).
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258 Y. Li and D. W. McLaughlin
From Eqs. (3.4), (3.5), we can calculate that|q² | = ω − C² +
O(²2) for some positiveconstantC. We have the following asymptotic
formulae for attracting or repelling rates:
1. For j = 0; hencek0 = 0, E0 = 2|q² |2+ (|q² |2−ω2) > 0, F0
= −(|q² |2−ω2) > 0;thenE0F0 > 0,
E0F0 = 4²Cω3+ O(²2).Thus,ı0 has the aymptotic form:
ı0 = ±4²1/2C1/2ω3/2+ O(²).
2. For j = 1; hence k1 = 2πN , F1 > 0. SinceN tan πN < ω
< N tan2πN , E1 =(1+ cosk1)(ω2− N2 tan2 πN )+ O(²) > 0.
Therefore,E1F1 > 0,
E1F1 = (1− cos2 k1)(1/h2+ ω2)(ω2− N2 tan2 π
N
)+ O(²).
Thus,ı1 has the aymptotic form:
ı1 = ±2√(1− cos2 k1)(1/h2+ ω2)
(ω2− N2 tan2 π
N
)+ O(²).
3. For j = 2, 3, . . . ,M (M = N/2, N even; M = (N − 1)/2, N
odd); Fj > 0.SinceN tan πN < ω < N tan
2πN , Ej = (1+ coskj )(ω2 − N2 tan2 jπN ) + O(²) < 0.
Therefore,Ej Fj < 0. Thus,ıj has the form
ıj = −²[α + 2βh−2(1− coskj )
]± 2i√|Ej Fj |.
So we have three well-separated attracting and repelling rate
scales:O(²0) (at j = 1),O(²1/2) (at j = 0), andO(²) (for all other
j ). j = 2 is the slowest attracting direction.Next, we calculate
the eigenvectors (and linearized flow in the linear subspaces
atq²).The amplitudesAj andBj satisfy the equations[
iÄj − (2κ1(coskj − 1)+ κ2)]
Aj = 2|q² |2B̄j , (9.2)[i Ǟj − (2κ1(coskj − 1)+ κ2)
]Bj = 2|q² |2 Āj , (9.3)
whereκ1 = h−2+|q² |2+ i ²βh−2, κ2 = 4|q² |2−2ω2− i ²α. The
compatibility conditionof (9.2) and (9.3) gives the dispersion
relation (9.1).
1. For j = 0, 1; Äj is real, and then from the relations (9.2)
and (9.3), we have∣∣∣∣i Ǟj − (2κ1(coskj − 1)+ κ2)∣∣∣∣ = 2|q²
|2.Thus, we can write
i Ǟ±j − (2κ1(coskj − 1)+ κ2) = 2|q² |2eiϕ±j ,
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Homoclinic Orbits and Chaos in Discretized Perturbed NLS
Systems: Part I 259
where
ϕ±j = arctan{ıj − (2 Im{κ1}(coskj − 1)+ Im{κ2})
2 Re{κ1}(1− coskj )− Re{κ2}
}. (9.4)
Then,Bj = e−iϕ±j Āj . Thus, we have
q̃n = (Aj + e−iϕ±j Āj )e
ıj t coskj n.
Let Aj = r j ei θj , then
Aj + e−iϕ±j Āj = r j e−i
12ϕ±j
[ei (θj+
12ϕ±j ) + e−i (θj+ 12ϕ±j )
]= 2r j cos
(θj + 1
2ϕ±j
)e−i
12ϕ±j ≡ c±j e−i
12ϕ±j ,
wherec±j ≡ 2r j cos(θj + 12ϕ±j ) are real parameters.
Thereforeq̃n takes the final form,
q̃n = c±j e−i12ϕ±j eÄ
±j t coskj n,
in which ϕ±j are fixed phases given in (9.4) (which identify the
stable and unstabledirections) andc±j are real parameters.
2. For j = 2, . . . ,M ; Äj is complex.Bj = bj Āj , wherebj =
2|q² |2(i Ǟ+j −(2κ1(coskj−1)+ κ2))−1. Then,q̃n takes the form
q̃n =[
Aj ei Im{Ä+j }t + bj Āj e−i Im{Ä
+j }t]eRe{Ä
+j }t coskj n,
where Aj is a complex parameter. ReplacingÄ+j by Ä
−j gives the same linearized
motion.
10. Appendix C: Estimates of the Limits
In this appendix, we want to find out the limits limt→−∞1−(t)
and limt→+∞1+(t), inorder to represent1−(0) by
1−(−∞)+∫ 0−∞
1̇−(t) dt,
and1+(0) by
1+(+∞)−∫ +∞
01̇+(t) dt,
respectively. Unfortunately, in some case, calculating
limt→+∞1+(t) is not feasible.Nevertheless, since we are only
interested in the values of1−(0) and1+(0), thecrucial
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260 Y. Li and D. W. McLaughlin
point is to find computable formulas for them. Thus, in such
cases, we will represent1+(0) by
1+(T(²))−∫ T(²)
01̇+(t) dt,
for some largeT(²).First, we calculate limt→−∞1−(t), and show
that limt→−∞1−(t) = 0:
limt→−∞1
−(t) = limt→−∞〈∇ F̃1(h(t)), h
u² (t)− h(t)〉.
From explicit representation of∇ F̃1(h(t)) in a later section,
we see that there existT1(> 0) andC1 (> 0), such that∥∥∥∥∇
F̃1(h(t))∥∥∥∥ ≤ C1eκ1t , for all −∞ < t ≤ −T1, (10.1)whereκ1
> 0. By fiber theorem 4.1, we see that there existC2 (> 0)
andC3 (> 0), suchthat ∥∥∥∥hu² (τ − Tu² )− F τ² (γ, I u)∥∥∥∥ ≤
C2eκ2τ , (10.2)∥∥∥∥h(τ − Tu0 )− F τ0 (γ, I u)∥∥∥∥ ≤ C3eκ2τ ,
(10.3)
for all −∞ < τ ≤ 0;where
κ2 > 0, hu² (−Tu²