NDLI: Splitting Potential and the Poincaré-Melnikov Method for Whiskered Tori in Hamiltonian Systems

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Splitting Potential and the Poincaré-Melnikov Method for Whiskered Tori in Hamiltonian Systems

Content Provider	Springer Nature Link
Author	Delshams, A. Gutiérrez, P.
Copyright Year	2000
Abstract	We deal with a perturbation of a hyperbolic integrable Hamiltonian system with n+1 degrees of freedom. The integrable system is assumed to have n -dimensional hyperbolic invariant tori with coincident whiskers (separatrices). Following Eliasson, we use a geometric approach closely related to the Lagrangian properties of the whiskers, to show that the splitting distance between the perturbed stable and unstable whiskers is the gradient of a periodic scalar function of n phases, which we call splitting potential. This geometric approach works for both the singular (or weakly hyperbolic) case and the regular (or strongly hyperbolic) case, and provides the existence of at least n+1 homoclinic intersections between the perturbed whiskers. In the regular case, we also obtain a first-order approximation for the splitting potential, that we call Melnikov potential. Its gradient, the (vector) Melnikov function, provides a first-order approximation for the splitting distance. Then the nondegenerate critical points of the Melnikov potential give rise to transverse homoclinic intersections between the whiskers. Generically, when the Melnikov potential is a Morse function, there exist at least 2 n critical points. The first-order approximation relies on the n -dimensional Poincaré-Melnikov method, to which an important part of the paper is devoted. We develop the method in a general setting, giving the Melnikov potential and the Melnikov function in terms of absolutely convergent integrals, which take into account the phase drift along the separatrix and the first-order deformation of the perturbed hyperbolic tori. We provide formulas useful in several cases, and carry out explicit computations that show that the Melnikov potential is a Morse function, in different kinds of examples.
Starting Page	433
Ending Page	476
Page Count	44
File Format	PDF
ISSN	09388974
Journal	Journal of Nonlinear Science
Volume Number	10
Issue Number	4
e-ISSN	14321467
Language	English
Publisher	Springer-Verlag
Publisher Date	2000-08-01
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Analysis Theoretical, Mathematical and Computational Physics Mechanics ApplicationMathematics/Computational Methods of Engineering Economic Theory
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Engineering Modeling and Simulation

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