Dynamics of 4 $D$ symplectic maps near a double resonance
We study the dynamics of a family of $4 D$ symplectic mappings near a doubly resonant elliptic fixed point. We derive and discuss algebraic properties of the resonances required for the analysis of a Takens type normal form. In particular, we propose a classification of the double resonances adapted...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2013 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2445/193871 |
| Acceso en línea: | https://hdl.handle.net/2445/193871 |
| Access Level: | acceso abierto |
| Palabra clave: | Sistemes hamiltonians Funcions de Lagrange Sistemes dinàmics diferenciables Teoria ergòdica Hamiltonian systems Lagrangian functions Differentiable dynamical systems Ergodic theory |
| Sumario: | We study the dynamics of a family of $4 D$ symplectic mappings near a doubly resonant elliptic fixed point. We derive and discuss algebraic properties of the resonances required for the analysis of a Takens type normal form. In particular, we propose a classification of the double resonances adapted to this problem, including cases of both strong and weak resonances. Around a weak double resonance (a junction of two resonances of two different orders, both being larger than 4) the dynamics can be described in terms of a simple (in general non-integrable) Hamiltonian model. The non-integrability of the normal form is a consequence of the splitting of the invariant manifolds associated with a normally hyperbolic invariant cylinder. We use a $4 D$ generalisation of the standard map in order to illustrate the difference between a truncated normal form and a full $4 D$ symplectic map. We evaluate numerically the volume of a $4 D$ parallelotope defined by 4 vectors tangent to the stable and unstable manifolds respectively. In good agreement with the general theory this volume is exponentially small with respect to a small parameter and we derive an empirical asymptotic formula which suggests amazing similarity to its $2 D$ analog. Different numerical studies point out that double resonances play a key role to understand Arnold diffusion. This paper has to be seen, also, as a first step in this direction. |
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