Splitting of separatrices for rapid degenerate perturbations of the classical pendulum
In this work we study the splitting distance of a rapidly perturbed pendulum H(x, y, t) = 1 2 y 2 + (cos(x) - 1) + µ(cos(x) - 1)g t e with g(t ) = P |k|>1 g [k] e ikt a 2p-periodic function and µ, e 1. Systems of this kind undergo exponentially small splitting and, when µ 1, it is known that the...
| Autores: | , , |
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| Tipo de recurso: | informe técnico |
| Fecha de publicación: | 2023 |
| 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:2117/387479 |
| Acceso en línea: | https://hdl.handle.net/2117/387479 https://dx.doi.org/https://doi.org/10.48550/arXiv.2302.07705 |
| Access Level: | acceso abierto |
| Palabra clave: | Differentiable dynamical systems Splitting of separatrices Exponentially small phenomena Hamiltonian systems Sistemes dinàmics diferenciables Classificació AMS::37 Dynamical systems and ergodic theory::37D Dynamical systems with hyperbolic behavior 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 H(x, y, t) = 1 2 y 2 + (cos(x) - 1) + µ(cos(x) - 1)g t e with g(t ) = P |k|>1 g [k] e ikt a 2p-periodic function and µ, e 1. Systems of this kind undergo exponentially small splitting and, when µ 1, it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided g [±1] 6= 0. Our study focuses on the case g [±1] = 0 and it is motivated by two main reasons. On the one hand the general understanding of the splitting, as current results fail for a perturbation as simple as g(t ) = cos(5t ) + cos(4t ) + cos(3t ). On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency p/q in Arnold’s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with g(t ) = sin p · t e + cos q · t e , where, for most p, q ¿ Z the perturbation satisfies g [±1] 6= 0. 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 µ n , where n is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it. |
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