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2013-01-15

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Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/193871

Dynamics of 4 $D$ symplectic maps near a double resonance

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Recurs relacionat

https://doi.org/10.1016/j.physd.2012.10.001

Resum

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.

Matèries

Sistemes hamiltonians, Funcions de Lagrange, Sistemes dinàmics diferenciables, Teoria ergòdica

Matèries (anglès)

Hamiltonian systems, Lagrangian functions, Differentiable dynamical systems, Ergodic theory

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Articles publicats en revistes (Matemàtiques i Informàtica)

Citació

GELFREICH, Vassili, SIMÓ, Carles and VIEIRO YANES, Arturo. Dynamics of 4 $D$ symplectic maps near a double resonance. Physica D. 2013. Vol. 243, num. 1, pags. 92-110. ISSN 0167-2789. [consulted: 15 of August of 2026]. Available at: https://hdl.handle.net/2445/193871

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