Whiskered Tori and Chaotic Behavior in Nonlinear Waves

Proceedings of the International Congress of Mathematicians ◽

10.1007/978-3-0348-9078-6_145 ◽

1995 ◽

pp. 1484-1493

Author(s):

David W. McLaughlin

Keyword(s):

Nonlinear Waves ◽

Chaotic Behavior ◽

Whiskered Tori

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Whiskered Tori for Integrable Pde’s: Chaotic Behavior in Near Integrable Pde’s

Surveys in Applied Mathematics ◽

10.1007/978-1-4899-0436-2_2 ◽

1995 ◽

pp. 83-203 ◽

Cited By ~ 38

Author(s):

David W. McLaughlin ◽

Edward A. Overman

Keyword(s):

Chaotic Behavior ◽

Whiskered Tori

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CHAOS IN A NEAR-INTEGRABLE HAMILTONIAN LATTICE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127402005431 ◽

2002 ◽

Vol 12 (08) ◽

pp. 1743-1754 ◽

Cited By ~ 4

Author(s):

VASSILIOS M. ROTHOS ◽

CHRIS ANTONOPOULOS ◽

LAMBROS DROSSOS

Keyword(s):

Lyapunov Exponents ◽

Numerical Experiments ◽

Chaotic Dynamics ◽

Large Scale ◽

Hamiltonian Structure ◽

Chaotic Behavior ◽

Original System ◽

Homoclinic Orbits ◽

Hamiltonian Lattice ◽

Whiskered Tori

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 ε = 0, whose homoclinic orbits are shown to persist for ε ≠ 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 in the ε = 0 case. We conjecture that this large scale chaos is due to the occurrence of saddle-center type fixed points in a perturbed 1 d.o.f Hamiltonian to which the original system can be reduced for all N. As ε > 0 increases, the transient character of this chaotic behavior becomes apparent as the positive Lyapunov exponents steadily increase and the orbits escape to infinity.

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