Numerical Chaos, Symplectic Integrators, and Exponentially Small Splitting Distances

Two discrete versions of Duffing's equation are derived as reductions of discretizations of the nonlinear Schrödinger (NLS) equation. Inheriting the properties of the NSL discretizations, one is shown to be integrable and, using a Mel'nikov type analysis, the other discretization is shown...

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Bibliographic Details
Published inJournal of computational physics Vol. 105; no. 1; pp. 122 - 132
Main Authors Herbst, B.M., Ablowitz, Mark J.
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 01.03.1993
Elsevier
Subjects
Exact sciences and technology
Mathematical methods in physics
Numerical approximation and analysis
Physics
Chaos
Non linear equation
Melnikov method
Numerical computation
Integrability
Homoclinic orbit
Hamiltonian system
Duffing equation
Symplectic integrator
Schrödinger equation
Finite difference method
KAM theory
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Summary:Two discrete versions of Duffing's equation are derived as reductions of discretizations of the nonlinear Schrödinger (NLS) equation. Inheriting the properties of the NSL discretizations, one is shown to be integrable and, using a Mel'nikov type analysis, the other discretization is shown to be nonintegrable. Arguments are given why it is possible for the nonintegrable scheme to restore the homoclinic orbit at an exponentially fast rate which is faster than the order of the approximation, as h → 0. This suggests why there can be fundamental differences between symplectic and nonsymplectic discretizations of certain continuous Hamiltonian systems.
ISSN:0021-9991
1090-2716
DOI:10.1006/jcph.1993.1058