Splitting of separatrices for rapid degenerate perturbations of the classical pendulum
In this work we study the splitting distance of a rapidly perturbed pendulum with a -periodic function and . Systems of this kind undergo exponentially small splitting, and, when , it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided . Ou...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2024 |
| 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:2117/426335 |
| Acceso en línea: | https://hdl.handle.net/2117/426335 https://dx.doi.org/10.1137/23M1550992 |
| Access Level: | acceso abierto |
| Palabra clave: | Hamiltonian systems Differentiable dynamical systems Splitting of separatrices Exponentially small phenomena Sistemes dinàmics diferenciables Hamilton, Sistemes de Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics |
| Sumario: | In this work we study the splitting distance of a rapidly perturbed pendulum with a -periodic function and . Systems of this kind undergo exponentially small splitting, and, when , it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided . Our study focuses on the case , and it is motivated by two main reasons. On the one hand, our study is motivated by the general understanding of the splitting, as current results fail for a perturbation as simple as . On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency in Arnold’s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with where, for most , the perturbation satisfies . As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton–Jacobi formalism. The leading exponentially small term appears at order , where is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it. |
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