caa a22 4500 475805356 CHVBK 20180406123745.0 cr unu---uuuuu 170329e20000801xx s 000 0 eng 10.1007/s003329910016 doi (NATIONALLICENCE)springer-10.1007/s003329910016 Splitting Potential and the Poincaré-Melnikov Method for Whiskered Tori in Hamiltonian Systems [Elektronische Daten] [A. Delshams, P. Gutiérrez] Summary. : 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. Springer-Verlag New York Inc., 2000 Delshams A. Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain. E-mail: amadeu@ma1.upc.es, pereg@ma1.upc.es, ES aut Gutiérrez P. Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain. E-mail: amadeu@ma1.upc.es, pereg@ma1.upc.es, ES aut Journal of Nonlinear Science Springer-Verlag 10/4(2000-08-01), 433-476 0938-8974 10:4<433 2000 10 332 https://doi.org/10.1007/s003329910016 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s003329910016 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Delshams A. Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain. E-mail: amadeu@ma1.upc.es, pereg@ma1.upc.es, ES aut NATIONALLICENCE 700 1- Gutiérrez P. Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain. E-mail: amadeu@ma1.upc.es, pereg@ma1.upc.es, ES aut NATIONALLICENCE 773 0- Journal of Nonlinear Science Springer-Verlag 10/4(2000-08-01), 433-476 0938-8974 10:4<433 2000 10 332 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer