Analytical expressions for stable and unstable manifolds in higher degree of freedom Hamiltonian systems

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Perturbations of completely integrable Hamiltonian systems with three or more degrees of freedom are studied. In particular, the unperturbed systems are assumed to be separable into a product of simple oscillator-type systems and a system containing homo- or heteroclinic connections consisting of stable and unstable manifolds of saddle points. Under a perturbation, the manifolds persist but separate and may no longer intersect. In this paper we show how, with proper choices for initial conditions, one may solve the variational equations to obtain analytical expressions for orbits on the perturbed manifolds in the form of expansions in the small parameter characterizing the perturbation. The derivation also shows how the distance between the manifolds can be uniquely defined, and thus provides an alternative to the traditional higher dimensional Melnikov method. It is finally argued that the approximate knowledge of the shape and position of the perturbed manifolds could be utilized for the study of large-scale phase-space motions, such as those associated with Arnold diffusion. The theory is further illuminated in two example problems.

Original languageEnglish (US)
Pages (from-to)1997-2013
Number of pages17
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Issue number11
StatePublished - Nov 1996
Externally publishedYes

ASJC Scopus subject areas

  • Modeling and Simulation
  • Applied Mathematics


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