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Dynamics of 4 D symplectic maps near a double resonance
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Gelfreich, Vassili, Simó, C. and Vieiro, A. (2013) Dynamics of 4 D symplectic maps near a double resonance. Physica D: Nonlinear Phenomena, Vol.243 (No.1). pp. 92-110. doi:10.1016/j.physd.2012.10.001 ISSN 0167-2789.
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Official URL: http://dx.doi.org/10.1016/j.physd.2012.10.001
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. © 2012 Elsevier B.V. All rights reserved.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Physica D: Nonlinear Phenomena | ||||
Publisher: | Elsevier BV | ||||
ISSN: | 0167-2789 | ||||
Official Date: | 15 January 2013 | ||||
Dates: |
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Volume: | Vol.243 | ||||
Number: | No.1 | ||||
Page Range: | pp. 92-110 | ||||
DOI: | 10.1016/j.physd.2012.10.001 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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