Chaos in a Near{Integrable Hamiltonian Lattice
y
Vassilios M. Rothos
Nonlinear and Complex Systems Group
Department of Mathematical Sciences
Loughborough University
Loughborough, LE11 3TU U.K
z
Chris Antonopoulos and Lambros Drossos
Department of Mathematics
Centre for Research and Applications of Nonlinear Systems
University of Patras
Patras GR 261 10, Greece
August 14, 2001
Abstract
We study the chaotic dynamics of a near{integrable Hamiltonian Ablowitz-Ladik lattice, which
is N + 2{dimensional if N is even (N + 1 if N is odd) and possesses, for all N , a circle of unstable
equilibria at " = 0, whose homoclinic orbits are shown to persist for " = 0 on whiskered tori.
The persistence of homoclinic orbits is established through Mel'nikov conditions, directly from the
Hamiltonian structure of the equations. Numerical experiments which combine space portraits and
Lyapunov exponents are performed for the perturbed Ablowitz-Ladik lattice and large scale chaotic
behavior is observed in the vicinity of the circle of unstable equilibria of the " = 0 case. We conjecture
that this large scale chaos is due to the occurrence of saddle{center type xed points in a perturbed
1 d.o.f Hamiltonian to which the original system can be reduced for all N . As " > 0 increases, the
transient character of this chaotic behavior becomes apparent as the positive Lyapunov exponents
steadily increase and the orbits escape to innity.
6
y
z
Int. J. Bifurc. & Chaos, 2001 (in press)
E-mail: V.M.Rothos@lboro.ac.uk
E-mail: antonop@math.upatras.gr
1
1
Introduction
Homoclinic orbits of dynamical systems are important in applications for a number of reasons, one of
them being that they often form 'organizing centers' for the dynamics in their neighborhood. From
their existence and intersection one may, under certain conditions, infer the existence of chaos nearby
e.g. using shift dynamics associated with Smale horseshoes [Shil'nikov, 1965 and Smale, 1967].
Mel'nikov's method is one of the few analytical tools which can be used to detect, in the context of
perturbation theory, the splitting and intersection of homoclinic manifolds. It has the nice geometrical
interpretation that it can provide a measure of the projection of the splitting distance onto a direction
normal to the unperturbed level sets. A discussion of the generalization of the original ideas of Mel'nikov
to many dimensions can be found e.g. in [Wiggins, 1988].
The present paper deals with a class of multi{degree-of{freedom dynamical systems which arise as
spatial discretizations of a partial dierential equation on the periodic domain. In particular, we consider
the integrable discretization of the nonlinear Schrodinger equation (NLS), Ablowitz-Ladik (AL) lattice,
with a conservative type of perturbation also considered by [Li, 1998]:
iq_ = qn+1 2qn + qn 1 + jq j2 (q + q ) 2!2 q
n
+
h2
n
n+1
n
1
n
" a1 (qn2 + jqn j2 ) + a2 (qn2 + qn2 )qn + (a1 + 2a2 qn) n2 lnn
h
(1)
where qn are complex variables, n := 1 + h2 jqnj2, h = 1=N , n = 0; 1; ; N 1, !, a1 , a2 are real
constants and " is a small parameter. We assume that system (1) satises the following even periodic
boundary conditions
qN +n(t) = qn(t); qN n (t) = qn(t)
and its phase space is dened as:
N = (q; r) : r = q; q = (q0 ; : : : ; qN 1 ); qN +n = qn; qN n = qn
(2)
Equation (1) is a 2(M + 1){dimensional system, where M = N=2, if N is even, M = (N 1)=2, if N is
odd which is known to be integrable for " = 0 with Hamiltonian formulation and a corresponding Lax
pair.
First, we establish necessary conditions for the persistence of homoclinic orbits in Ablowitz-Ladik
lattices with Hamiltonian perturbations. The Mel'nikov conditions are derived directly from the Hamiltonian structure of the equations via the averaging method. The unperturbed system has nite dimensional whisker tori (hyperbolic invariant tori), with coincident whiskers (invariant manifolds), while
Mel'nikov's method allows us to study the splitting of whiskers and the persistence of homoclinic orbits.
The homoclinic splitting between the whiskers of hyperbolic tori is very important for the study of
diusion of orbits in phase space.
We have also carried out numerical calculations associated with the near{integrable Hamiltonian
AL-lattice (1) with periodic boundary conditions. It is well known that equation (1) (for " = 0),
as well as its continuum limit, have solutions which are linearly unstable and are associated with
interesting mathematical and physical properties. Our numerical experiments through the calculation
of Lyapunov exponents and phase portraits indicate the existence of large scale chaotic behavior near
certain resonances, when " 6= 0. We investigate this chaotic dynamics for the specic perturbation
given in (1) for special values of the external parameters satisfying Mel'nikov conditions and observe
the phenomenon of transient chaos, through which, for " large enough orbits escape to innity.
Other researchers, [Li & McLaughlin, 1997], [Haller, 1998] have also studied persistence questions
related to discrete nonlinear Schrodinger (DNLS) equations. Their type of perturbations, however, is
2
n
o
dissipative and hence dierent from the one adopted here. Recently, Li [1998] proved the existence of
transversal homoclinic tubes which are asymptotic to locally invariant center manifolds for the DNLS
with Hamiltonian perturbations. Ablowitz & Herbst [1990] were concerned with the question of numerical instability involving periodic NLS equations. They showed that initial data which are near
low-dimensional homoclinic manifolds trigger numerically induced joint spatial and temporal chaos in
non{integrable numerical schemes at intermediate values of the mesh size h. This type of chaos disappears as the mesh is rened [Hersbt & Ablowitz, 1989]. Furthermore, the presence of homoclinic orbits
in DNLS{type systems has been studied by [Kollman & Bountis, 1998] who used dierent methods,
based on the homoclinic bifurcations of nite dimensional reduced mappings.
The present paper is organized as follows: In section 2, we formulate the Hamiltonian structure of
the problem and the geometry of the perturbed system. In section 3, we derive necessary criteria for
the persistence of homoclinic orbits and in section 4, we present numerical evidence showing that the
perturbed system (1) indeed possesses large scale chaotic dynamics near some fundamental resonances
of the system, identied by the analysis of section 2 and 3.
2
2.1
Hamiltonian Structure and Geometry
The geometry of the unperturbed system
In this section, we give a brief overview of the integrable AL lattice and its homoclinic structure. Eq.
(1), for " = 0, together with the equation of its complex conjugate variables qn possess the Hamiltonian
formulation:
1 N 1[qn(qn+1 + qn 1) 2 (1 + !2h2 )log(1 + h2jqnj2 )];
H0 = 2
h
h2
X
n=1
with the deformed Poisson brackets
fqn; qm g = i(1 + h2 jqnj2)Ænm ; fqn; qmg = fqn; qmg = 0;
and the symplectic form
N 1
~ q = 2(1 + hi2 jq j2) Im(dqn ^ dqn);
X
n=0
where we have dened
fB; C g = i
NX1 @B @C
n=1 @qn @ qn
n
@B @C (1 + h2 jqnj2):
@ qn @qn
(3)
The integrability of the unperturbed equation is proven using the discretized Lax pair [Ablowitz &
Ladik, 1976]:
un+1 = Ln(z; q)un u_ n = Bn (z; q)un
(4)
where Ln, Bn are 2 2 matrices and z = exp(i h) is the spectral parameter of the problem.
In the case of periodic boundary conditions, let Y (1) , Y (2) be the fundamental solutions of eq. (4).
The associated Floquet discriminant is defined by
: C N ! C (z; q) = tracefMq (N; z; q)g
(5)
where N is the phase space dened in (2) and M (n; z; q) columnsfYn(1) ; Yn(2) g is the fundamental
solution matrix of eqs. (4).
3
Following Li [1992], the Floquet theory is not standard as can be seen from the Wronskian relation:
WN (u+; u
) = D2W0 (u+ ; u );
D2 =
NY1
n=0
(1 + h2 jqnj2 )
where u+ and u are any two solutions to the linear system (4). The nonstandard Floquet theory for
integrable lattices (discrete systems in space and time) is established by Rothos [2001]. The Floquet
spectrum is dened as the closure of the complex z for which there exists a bounded eigenfunction to
(4). In terms of the Floquet discriminant , this is given by
n
o
(Ln ) = z 2 C
: 2D (z; q) 2D
Periodic and antiperiodic points zs are dened by
(zs; q) = 2D
A critical point zc is dened by the condition
d
dz (zc;q) = 0
An important sequence of constants of motion Fj for the unperturbed system is given in terms of the
Floquet discriminant by Li [1998]
Fj : C N ! C with Fj = 1 (zc; q)
(6)
D
j
where zc be a simple critical point as dened above and fH0; Fj g = 0. These invariants Fj 's are
candidate for building Melnikov functions.
We do not intend to discuss the homoclinic structure too deeply here (see Li [1992, 1998] for details).
Due to the boundary conditions, the unperturbed integrable system (1) admits an invariant plane
n
o
= (q; q) 2 N : qn = q2 = : : : = qN ;
of spatially independent solutions. Consider the uniform solution qn = q; 8n of the form
qc(t) = aexp[ i2[(a2 !2 )t ' ];
(7)
where ' is a constant. We choose the amplitude a in the range: for N > 3 a 2 ( N tan N ; N tan 2N ) and
for N = 3 a > 3tan 3 . The invariant plane contains the resonance circle
n
N! = (q; q) : jqcj = !
o
entirely consists of xed points under the unperturbed ow.
The hyperbolic structure and homoclinic orbits for the unperturbed system have been studied by Li
[1992], using Backlund transformations, showing that, along the unstable and stable directions, solutions
leave and return to the circle of equilibria N! . This implies the existence of heteroclinic orbits in the
phase space N which connect different equilibria in N! . From any point on the resonant circle N! there
are precisely two homoclinic connections to another point of the circle. For the uniform solution (7),
there is only one set of quadruplets of double points which are not on the unit circle and denote one of
4
them by z = z1d = z1c. For this reason, we consider in the following analysis only the rst member of the
sequence of invariants Fj . The heteroclinic solutions are of the form (Li, 1992)
Qn (t)
where
= qc
E
Kn
1
(8)
1 sin sech cos2n#;
cos#
p
cos2 # 1
= 4N 2 sin# cos2 # 1t t0 ; = tan 1 p
; # = =N; = 1 + jqj2 =N 2 :
sin#
The index refers to two distinct families, which in turn are parameterized by the phase variable ',
and the initial time t0 . These orbit families form two, 2{dimensional homoclinic manifolds W (N! ) to
N! .
Formula (8) also gives the two limit points of the heteroclinic connections
' ]
lim
Q
(
t
)
=
a
exp[
i
(9)
n
t!1
2
where ' is a constant phase shift between the limit points of every heteroclinic orbit
2
' = 4tan 1 pcossin## 1
E = 1 + cos2
isin2 tanh;
q
Kn = 1
p
p
2.2
Geometry of the perturbed system
Usually, when we study a completely integrable Hamiltonian system with n degrees of freedom (d.o.f)
and Hamiltonian perturbation we use the action-angle formulation to derive the corresponding equations
of motion. From the properties of the unperturbed system, we know that its phase space is foliated by
a k n-parameter family of k-tori. The tori can be either resonant or nonresonant. From the point of
view of perturbation theory, we are interested in what happens to this k-parameter family of tori under
perturbation. Classical KAM theory is concerned with the persistence of k-dimensional diophantine
tori under perturbation. Broadly speaking, there are two approaches for proving the persistence of
k-dimensional tori on which the ow is quasiperiodic with diophantine frequencies: The global approach
of Arnol'd and the local approach of Kolmogorov.
The crucial problem lies with the fate of the lower (i.e k n) dimensional tori. The KAM tori are
elliptic in terms of their stability type. That is, they are neutrally stable in the linear approximation.
For the low dimensional tori, there is also the possibility that they may have exponentially growing and
contracting directions in the linear approximation, whence they are called whiskered tori. Gra [1974]
has considered a \KAM-type" situation where the unperturbed problem has a submanifold foliated by
invariant tori. Restricted to this submanifold, the dynamics is preserved and the preserved tori have
stable and unstable manifolds. In our problem the situation is different: The unperturbed system has
a circle of fixed points on a 2{dimensional manifold and each point on the circle of fixed points is
connected to another fixed point on the circle by a heteroclinic orbit.
As we mentioned earlier, there exists a 2{dimensional manifold N which is invariant under the
flow of (1) for " = 0 and is symplectic with the restricted nondegenerate two-form ~ = ~ j. Thus, for
" = 0, system (1) restricted to becomes a 1 d.o.f completely integrable Hamiltonian system and the
Liouville{Arnold{Jost theorem [Arnol'd, 1978] guarantees the existence of an open set on which we
can introduce canonical action{angle variables (I; ) 2 R S1.
5
By restricting the ow to the invariant plane , one obtains a dimensional reduction of the system
(1) for all N . This means that on the equation of motion takes the following form:
iq_ = 2(jqj2
n
o
!2 )q + " [ a1 (q + q) + a2 (q2 + q2 )]q + [ a1 + 2a2 q] 2 ln
h
This shows that for " = 0, contains a circle of equilibria N! given by
Figure 1: The phase portrait of the integrable slow Hamiltonian H of (15) for N = 4; ! = 4:0 and initial
conditions near the origin.
N! = fq : jqj = !g
which is surrounded by periodic solutions (in the space ) of the form (7), where the constant ! satises:
2
N tan < ! < N tan ; if N > 3; ! > 3tan ; if N = 3
N
N
3
For any point q 2 , there always exists an open neighborhood with local regular coordinates I; , such
that the symplectic form is written in the canonical form ~ j = n dIn ^ dn. Introducing now the
amplitude{phase representation (I; ) 2 R S1 in a neighborhood of by writing q = Ie i, we obtain
the equations of motion
dI = "sin(a1 + 4a2 I cos)WI
dt
d = 2(I 2 !2 ) " 2I (a1cos + 2 I cos2) + a1 cos + 2a2 cos2 WI
(10)
dt
I
P
where
WI =
1 + h2I 2 ln(1 + h2I 2 )
h2
Thus, for " = 0, system (1) restricted to becomes a 1 d.o.f completely integrable Hamiltonian system.
6
Introducing the new scaling I = ! + p"J (which is just the standard blow-up transformation
used in
classical mechanics) to study the dynamics near a resonant action value and setting p" = 0, we obtain
a Hamiltonian system with
H(J; ) = 2!J 2 + W! (a1 cos + a2 !cos2)
:= 12 hJ; DI2 h0 (!)J i + h1 ()
(11)
where h0 := H0j is the restriction of H0 to the manifold and h1 = H1jN! .
For a1 ; a2 > 0 and ja1 =4a2 !j < 1, the xed points of the reduced system are (0; 0); (0; ) (saddletype) and (0; cos 1 ( a1=4a2 !)) (center-type). For a1 > 0; ja1 =4a2 !j > 1, (0; 0) is a saddle point and
(0; ) center xed point (see Fig. 1) below some of the orbits starting very close to the origin.
In order to provide a clearer geometric formulation for the problem, it is convenient to introduce
local coordinates r = (rs; ru) 2 R 2 , z 2 R 2(M 1) , (I; ) 2 R S1 and rc = (z; I; ), in a neighborhood
of the set N^! (see Li & McLaughlin [1997], Haller [1998]. Rothos [1999]) and the perturbed AL-lattice
can be rewritten in the following form dened on the compact domain:
Y = (r; z; J; ) : jrj C; jzj Cz ; N tan N < J < N tan 2N ; 2 S1
with C; Cz are xed positive constants,
r_ = B r + R(rs; ru ; z; J; ; pp")
z_ = Az
+ Z (r ; r ; z; I; ; ")
_J = p"E (rs; rsu; z;u J; p; p")
p
_ = F0 (rs ; ru ; z; ) + "F (rs ; ru ; z; J; ; ")
(12)
where R; Z; E; F are nonlinear functions of class C l, the matrix B is dened as B = diag( a; a) where
a = 2 (1 cos2 2 )(!2 + h 2)(!2 N 2 tan2 )
o
n
r
N
and the matrix A has purely imaginary eigenvalues ia1 ; : : : ; ia2(M
such that
At
N
1)
and there exists a constant CA
je zj CAjzj:
In the above local coordinates, the manifold satises the equations rs = 0; ru = 0, z = 0 and thus
system (12) coincides with (10) and describes the dynamics on . For " = 0 the torus N! admits a
unique codimension 2 center manifold
M0 = f(rs ; ru; rc ) : rs = hcs (rc ;0); ru = hcu (rc ;0); rc 2 V R 2M g
where the functions hcs; hcu of class C l and rc = (z; J; ) . By the uniqueness of M0, M0 holds
implying hcs (0; J; ) = 0; hcu (0; J; ) = 0. Then, there exists a unique, codimension 2 locally invariant
manifold M" of class C l which depends on "
M" = fr 2 N : rs = hcs (rc; "); ru = hcu (rc; "); rc 2 V R 2M g
and M". The manifold M" admits codimension 1 local stable-unstable manifolds Wlocsu (M") that
are of class C l in (rs; ru ; rc) and ", such that
s (M" ) = r 2 N : ru = hu (rs ; rc ; "); (rs ; rc ) 2 R R 2(M 1) R S1
Wloc
n
u (M" ) =
Wloc
n
o
r 2 N : rs = hs(ru ; rc; "); (ru ; rc ) 2 R R 2(M
7
o
1) R S1
The full Hamiltonian H = H0 + "H1 restricts as
H" = HM" = H0jN! + "H + O(jzj2; "jzj; "3=2 )
with the slow Hamiltonian H (near the resonance) dened in (11). From the system (12), we can write
H" as
H"(z; J; ) = "H(J; ) + 21 hAz; zi + O(jzj3 ; p"jJ jjzj2; "jzj; "3=2 )
(13)
with A = JA, here J is the symplectic matrix. As we mention above H generates a completely integrable
ow on with a family of invariant 1-tori (circles). The corresponding Hamiltonian equations can be
derived from H" through the restricted symplectic form:
~ M = dr"1(z; ! + p"J; ) ^ dr"2(z; ! + p"J; ) + dz1 ^ dz2 + p" d ^ dJ
Since the matrix A has 2(M 1) purely imaginary eigenvalues the rst two terms in H", indicate the
presence of (2M 1)-dimensional tori on the slow manifold. However, the noncanonical structure of the
symplectic form prevents us from using the Hamiltonian vector eld on M" in a near{integrable form
to which existing versions of the KAM theorem can be applied (see Gra [1974]). The other diÆculty
is the presence of dierent time scales in the problem, which is due to the fact that we work close to a
resonance, in a domain explicitly excluded by KAM{type methods. KAM tori can be constructed with
"resonance islands" in the vicinity of the slow manifold .
3
Mel'nikov criteria and Homoclinic orbits
Our next goal is to develop criteria for the persistence of homoclinic solutions through the Hamiltonian
structure of the system (1) with " 6= 0. The perturbed AL{lattice can be put in the abstract form:
i q_n =
"
@H0
n
@ qn
1
+ " @H
@ qn
#
(14)
where
H (qn ; a1 ; a2 ; ")
=
H0 (qn ) + "H1 (qn ; a1 ; a2 )
N 1
2
= h12
qn(qn+1 + qn 1 ) 2 (1 + !2 h2 )lnn
h
n=1
N 1
1 f a1(qn + qn) + a2 (q2 + q2 ) glnn
H1 (qn ; a1 ; a2 ) = 2
n
n
h
X n
H0 (qn )
X
n=1
o
(15)
The evolution of any real{valued functional S under the ow governed by eq. (14) for " = 0, formally
obeys:
dS = f S ; H g
0
dt
where f ; g the Poisson bracket on any two functional dened in eq. (3). The AL-lattice conserves the
quantities
N 1
I = ln(1 + h2 jqnj2); F1 = D1 (zc; q)
X
n=0
8
We consider real{valued functionals S , which are smooth functions on qn:
S (qn) =
NX1
n=0
s (qn ; qn ; qn1 ; qn1 );
(16)
where s is an analytic function of its arguments. The evolution of any such functional S that Poisson
commutes with H0 when evaluated along a solution of the perturbed AL-lattice (14), gives:
dS = N 1 @ S q_ + @ S q_n = S ; H0 " i N 1 @ S @H1 @ S @H1 n
dt n=0 @ qn n @qn
@ qn @qn
n=0 @qn @ qn
N 1
1
= 2" Im @@qSn n @H
@qn
X h
i
X h
i
X
n=0
then
dS (qn) = "S (qn(t))
dt
where for any qn(t), the functional S is dened by:
S (qn(t)) = 2
NX1
n=0
(17)
@ S @H1
@ qn n @qn
Im
!
(18)
We will consider here only those functional S that satisfy
S ; H0 = S ; I = S ; F1 = 0
(19)
Thus, it follows that S is governed by the ows of F1 ; I:
S ; F1 = S ; I = 0
(20)
Denition 3.1 A solution Qn (t) of AL{lattice which is homoclinic to a circle of equilibria N! is said
to persist under the Hamiltonian perturbation H1 if and only if there exists an "-dependent family of
solutions Q"n (t) of the perturbed problem that satises two requirements:
1. For each " it is homoclinic to the saddle point q~" in the resonant annulus A! N! in the sense
lim S (Q"n(t)) = S (~q")
(21)
jtj!1
n
o
n
n
o
o
n
n
o
o
and
lim S (Q"n(t)) = S (~q") = 0
For any functional S that Poisson commutes with H0 ; I; F1 and
lim S (Q"n (t)) = S (Qn(t))
"!0
t!1
2.
(22)
(23)
Eq. (17) for the evolution of any conserved functional S evaluated at Q"n(t) may be expressed as
d
"
"
(24)
dt S (Qn(t)) S (~q") = "S (Qn(t))
By eq. (21) the integration of eq. (24) gives
+1
+1
0 = 1" S (Q"n(t)) S (~q")
= 1 S (Q"n(t))dt
(25)
Taking the limit " ! 0 in (25) we have the following:
9
Z
1
Proposition 3.1 A necessary condition for the persistence of a AL-lattice solution Qn (t) that is homoclinic to a circle of equilibria N! is that it satisfy the Mel'nikov condition:
Z 1
MS (t0 ) =
S (Qn(t))dt = 0
(26)
1
For S = I; F1 , the above eq takes the form:
with
MI (t0 )
=
MF1 (t0 )
=
=
1 NX1
1 (Q (t; t ))dt
2 1 Im @@qIn n @H
n
0
@qn
n=0
!
Z 1 N 1
X
@ F1 @H1
2 1 Im @ q n @q (Qn(t; t0 ))dt
n
n
n=0
a1 k1 (; ; '; !; N ) + a2 k2 (; ; '; !; N )
!
Z
= aexp[ i( 2' + ')]1 (Qn; #; n; t)
(28)
where 1 is a known function of the general solutions of the Lax pair and ki are converging integrals.
Remark From the symplectic structure of our system, we can prove that the Mel'nikov function
MI is identical zero.
Now, we are in a position to study the asymptotic behavior of the perturbed orbits near the center manifold M" after the intersection of stable and unstable manifolds. We recall that in the new
coordinates system (r; z; J; ) the homoclinic orbits can be written as:
Qn(t; t0 ) = (r; z; J; )(t t0 )
(cf. eq. (8) ). Application of the Mel'nikov formula in the resonant annulus A! for S = H0 provides
us with a suÆcient condition for the persistence of homoclinic solution Qn(t), i.e the existence of orbits
homoclinic to the periodic orbits inside the resonance.
Eq. (26) for S = H0 becomes:
1
1 d
H1 (Qn (t t0 ))dt
MH (t0 ) =
H0 ; H1 (Qn (t t0 ))dt =
1
1 dt
= H1(Qn(+1; I; )) H1(Qn( 1; I; ))
(29)
= H1(r0; z0 ; ! + p"J; ) H1(r0; z0 ; ! + p"J; )
where we have used the new coordinates (r; z; J; ) in the neighborhood of M" and the asymptotic
behavior of the homoclinic solutions Qn(t) as t ! 1, (cf. eq. (9)). For small " we Taylor{expand eq.
(29) to obtain:
p
MH = H(J; ) H(J; ) + O( ")
where H dened in eq.(11). A simple calculation gives:
) sin sin( )
MH = 2W! a1 + 4!a2 cos(
2
2
2
Setting MH = 0, it is easy to nd the following solutions
1
1
3
1
a1
1 = + ; 1 = + ; 3 = + cos 1
2
2
2
2
2
4!a2
10
rF 1 =
@ F1 @ F1
;
;
@q @ q
(27)
Z
n
@ F1
@q
Z
o
0
0
h
i
0
0
For = 0; 2, we have
D MH0 (a1 + 4!a2 cos)sin = 0
This means that DMH0 is zero at the values of the hyperbolic and elliptic
a ).
at = 0; ; cos 1( 4!a
xed points in the resonance annulus.
For 2 [0; 2] the perturbed solutions are asymptotic to the saddle points in the resonant area.
Since all manifolds, as well as the Mel'nikov function depend dierentiably on the parameters, there
must be some value of MH such that the trajectory corresponding to the zero of Mel'nikov function
asymptotes to a trajectory in W s(q") \ A! (where q" is a saddle point), i.e the boundary of the domain
of attraction of elliptic points. Thus, we have the existence of an orbit homoclinic to the perturbed
saddle point.
1
2
0
4
Hamiltonian Chaos and Numerical Simulations
When integrating numerically system (1) for 6= 0, we generally expect to nd chaotic solutions. In
this section, we present numerical simulations that display this chaotic behavior in the neighborhood
of the homoclinic orbits and for parameters which satisfy the Mel'nikov conditions, as explained in the
previous section. At these parameter values, we use as diagnostics the Lyapunov exponents to check
numerically for the presence of chaos.
We shall restrict ourselves to the case of system (1) with N = 4, displaying the phenomena of
interest. This may appear as a simplication, but recently a so{called 'homoclinic center{manifold'
theorem has been proved by Sandstede [1993, 1999]: Given a homoclinic orbit in an arbitrary (even
innite) dimensional system, there exists an invariant manifold along the homoclinic solution that is
at least class C 1 and which contains all recurrent dynamics in a neighborhood of the homoclinic orbit.
The dimension of this manifold depends on the linearization at the associated equilibrium point and the
nature of the homoclinic trajectory itself. Roughly speaking, it is the dimension of the smallest possible
phase space in which the particular homoclinic solution may generically arise.
For N = 4 we thus obtain from (1) a system of dierential equations of the form:
x_
=
y_
=
z_
=
h
i
y x
2
2
i 2 h2 + jxj y ! x + "f (x)
i z 2hy2 + x + jyj2 (x + z) 2!2 y + "f (y)
h
i
y z
2
2
i 2 h2 + jzj y ! z + "f (z)
(30)
with x = q1; y = q2; z = q3; x; y; z 2 C and
8
9
1
+
h2 jsj2 >
>
2
2
2
2
2
>
>
>
> ln(1 + h2 jsj )
f (s) = a1 (s + jsj ) + a2 (s + s )s + (a1 + 2a2 s) :
h2 ;
In describing the chaotic behavior of system (40) we nd it convenient to begin with simple solutions
(xed points and periodic orbits) in the neighborhood of the resonance annulus A! for ! = 4. As
mentioned above, there are saddle and elliptic type xed points in the invariant plane for "{small.
11
Figure 2: The chaotic dynamics of near-integrable Hamiltonian lattice (1) for N = 4; ! = 4:0; "a1 =
0:525; "a2 = 0:401 in projections: (a) (Re(x); Im(x)) coordinates and (b) (Re(x); Re(y); Im(z)) coordinates.
12
In our computations the integration of the above system of ordinary dierential equations is performed by the classical Runge{Kutta fourth order method, with initial conditions close to the resonance
annulus A! of the reduced system and external parameters satisfying the Mel'nikov conditions. For
small enough " > 0, the chaotic dynamics of system (1) is not yet apparent and the orbits are seen to
execute quasiperiodic behavior, as depicted in Fig. 2 for "a1 = 0:525; "a2 = 0:401; ! = 4:0; N = 4.
Observe, however, in Fig. 2(a) that the perfect regularity of the orbits displayed in Fig. 1 is not present.
This dierence is more vividly illustrated in the 3-D plots of Fig. 2(b).
Lyapunov exponents measure the degree of divergence of close trajectories in phase space in the
course of time. They are a direct tool to infer the presence of chaos in the case where at least one
Lyapunov exponent is larger than 0.
Given the dynamical system (1) in a n = 2(M + 1)-dimensional phase space (cf. Sec 1), we monitor
the long-term evolution of an innitesimal n-sphere of initial conditions, which becomes an n-ellipsoid
due to the locally deforming nature of the ow. The Lyapunov exponents can either be estimated by
making use of the equations, or else by using only a time series which has been generated by the system
under scrutiny. We follow the rst way, and apply the algorithms proposed in [Wolf et al., 1985].
To estimate the Lyapunov exponents with the help of the equations, the evolution of an innitesimal
6-sphere is monitored in tangent space under the action of the linearized equations. This sphere turns
into an ellipsoid, and we denote by pi the length of its i-th principal axis. Thus, the i-th Lyapunov
exponent is dened in terms of the ellipsoidal principal axis pi:
1 pi(t)
i := tlim
(31)
!1 i (t) := tlim
!1 t log pi (0)
"a1
"a2
0
0
0.525
0.401
0.8
0.8
0.95
0.98
1
0.98
1
2
3
4
5
6
0.0072
0.0048
0.0033
-0.0032
-0.0051
-0.0072
0.0074
0.0049
0.0038
-0.0035
-0.0052
-0.0075
0.0079
0.005
0.0053
-0.0049
-0.0055
-0.0077
0.009
0.0069
0.0059
-0.0057
-0.0074
-0.0088
0.1276
0.0149
0.0054
-0.0054
-0.0148
-0.1278
d
5.987
5.989
6.000
5.997
5.999
Sum -0.0002 -0.0001 0.0001 -0.0001 -0.0001
Table 1: Lyapunov exponents for N = 4 and ! = 4:0 and dierent sets of parameters
and initial condition (Re(x); Im(x); Re(y); Im(y); Re(z); Im(z)) =
"a1 ; "a2 for nal time ' 1000
(0:93; 1:558; 0:44; 0:6; 0:7; 0:8).
The signs of the Lyapunov exponents provide a qualitative picture of the dynamics since, if one of
them is positive, this constitutes evidence for the presence of chaotic behavior. These exponents are
13
closely related also to the dimension d of the associated chaotic dynamics through the equation
d = | +
where | is the largest integer for which
|
X
{=1
P|
{=1 {
{+1
j
j
(32)
{ 0
Let us now make some remarks concerning the accuracy of our computations: Observe that in the
rst column of Table 1, where " = 0 and our system of equations (40) is integrable, all Lyapunov
exponents should be zero. Furthermore, in all cases, they should appear in positive-negative parts (due
to the Hamiltonian structure of the equations) and hence should all add up to zero. Note also that two
of these exponents should be zero in the third and fourth row of Table 1 due to the constancy of the
Hamiltonian. Finally, our accuracy is valid to the extent that the dimension of the dynamics in phase
space is 6, as indicated by the value of d in table 1. Due to these remarks it is clear from Table 1 above
that our results are accurate to order 10 3 , i.e. we can generally be sure of the rst two digits. Clearly,
for " small enough, the chaotic behavior of the solutions is weak. The presence of large scale chaos is
evident, however, for "a1 = 1:0 and "a2 = 0:98 where the positive exponents attain appreciably large
values. Finally, when "a1 = "a2 = 1:0 we nd that the Lyapunov exponents steadily increase, until time
t ' 468, when we have escape of the orbits to innity. This is reminiscent of similar results found by
[M.N. Vrahatis et al, 1997] for the orbits of a 4-dimensional symplectic map of accelerator dynamics.
5
Conclusions
Our motivation for this work was to investigate the chaotic dynamics of the Ablowitz{Ladik lattice
with a Hamiltonian perturbation. In particular, we focused on the near-integrable Hamiltonian (1) and
studied some homoclinic solutions of its unperturbed phase space which persist under the perturbation.
Specically, we have found that one can address this question studying families of special solutions
of the lattice which are homoclinic in time to unstable plane wave solutions. We investigated these
solutions through a Mel'nikov technique that utilizes the Hamiltonian structure of the lattice. It is worth
mentioning, that the existence of conserved quantities, which Poisson commute with the functional S ,
is crucial to derive the Mel'nikov condition without using the geometric interpretation of the method.
Moreover, the
restriction of the perturbed problem to the center manifold is a 1{dof Hamiltonian system
in order O(p") and the linearized analysis gives the presence of saddle, center type xed points.
We then integrated numerically the perturbed system for N = 4 in the neighborhood of these homoclinic solutions and obtained the following: First, we veried the existence of chaos even for small values
of the external parameters, by showing via Mel'nikov conditions that the stable and unstable manifolds
of a circle of unstable equilibria intersect transversally. Secondly, using the dierential equations of (1)
we computed diagnostics such as Lyapunov exponents and veried that the system indeed has chaotic
behavior near a fundamental resonance of the unperturbed system. This chaos, however, is transient,
since for large enough perturbation our solutions eventually escape to innity.
We also noted the noncanonical structure of the symplectic form and the presence of dierent time
scales in our problem. These are main diÆculties which prevent us from applying the existing versions
of the KAM theory, [Gra, 1974]. KAM theory and diusion phenomena in similar near-integrable
Hamiltonian lattices are currently under study and results are expected to appear in a future publication.
14
6
Acknowledgments
The authors would like to thank Dr. Y. Li and Prof. T.Bountis for valuable discussion. V.M. Rothos
was funded in part by TMR Marie Curie fellowship No. ERBFMBICT983236 of Commission of the
European Communities, and an EPSRC Grant No. GR/R02702/01. Chris Antonopoulos acknowledge
the support of the Caratheodory Programme of the University of Patras, under grant no. 2464.
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