Splitting potential and Poincaré-Melnikov method for whiskered tori in Hamiltonian Systems
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 Lagr...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 1999 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/768 |
| Acceso en línea: | https://hdl.handle.net/2117/768 |
| Access Level: | acceso abierto |
| Palabra clave: | Hamiltonian dynamical systems Lagrangian functions Global analysis (Mathematics) Dynamical systems Hamiltonian system Melnikov potential Morse function Poincaré-Melnikov method Hamilton, Sistemes de Lagrange, Funcions de Varietats (Matemàtica) Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Classificació AMS::58 Global analysis, analysis on manifolds::58E Variational problems in infinite-dimensional spaces Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics |
| Sumario: | 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 Poincare-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. |
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