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http://hdl.handle.net/2445/193871
Title: | Dynamics of 4 $D$ symplectic maps near a double resonance |
Author: | Gelfreich, Vassili Simó, Carles. Vieiro Yanes, Arturo |
Keywords: | Sistemes hamiltonians Funcions de Lagrange Sistemes dinàmics diferenciables Teoria ergòdica Hamiltonian systems Lagrangian functions Differentiable dynamical systems Ergodic theory |
Issue Date: | 15-Jan-2013 |
Publisher: | Elsevier B.V. |
Abstract: | 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. |
Note: | Versió postprint del document publicat a: https://doi.org/10.1016/j.physd.2012.10.001 |
It is part of: | Physica D, 2013, vol. 243, num. 1, p. 92-110 |
URI: | http://hdl.handle.net/2445/193871 |
Related resource: | https://doi.org/10.1016/j.physd.2012.10.001 |
ISSN: | 0167-2789 |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
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625611.pdf | 1.3 MB | Adobe PDF | View/Open |
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