Homoclinic orbits of twist maps and billiards
The splitting of separatrices for hyperbolic fixed points of twist maps with $d$ degrees of freedom is studied through a real-valued function, called the Melnikov potential. Its non-degenerate critical points are associated to transverse homoclinic orbits and an asymptotic expression for the symplec...
| Authors: | , |
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| Format: | article |
| Publication Date: | 1997 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/1209 |
| Online Access: | https://hdl.handle.net/2117/1209 |
| Access Level: | Open access |
| Keyword: | Hamiltonian dynamical systems Lagrangian functions Differentiable dynamical systems Hamiltonian systems Homoclinic orbits Hamilton, Sistemes de Lagrange, Funcions de Sistemes dinàmics diferenciables Classificació AMS::37 Dynamical systems and ergodic theory::37E Low-dimensional dynamical systems Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics |
| Summary: | The splitting of separatrices for hyperbolic fixed points of twist maps with $d$ degrees of freedom is studied through a real-valued function, called the Melnikov potential. Its non-degenerate critical points are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, Morse theory can be applied to give lower bounds on the number of transverse homoclinic orbits. This theory is applied first to elliptic billiards, where non-integrability holds for any non-trivial entire symmetric perturbation. Next, symmetrically perturbed prolate billiards with $d>1$ degrees of freedom are considered. Several topics are studied about these billiards: existence of splitting, explicit computations of Melnikov potentials, existence of $8$ or $8d$ transverse homoclinic orbits, exponentially small splitting, etc |
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