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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:
It is part of: Physica D, 2013, vol. 243, num. 1, p. 92-110
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ISSN: 0167-2789
Appears in Collections:Articles publicats en revistes (Matemàtiques i Informàtica)

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