Dynamics of 4 D symplectic maps near a double resonance
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. ISSN 0167-2789 (In Press)Full text not available from this repository.
Official URL: http://dx.doi.org/10.1016/j.physd.2012.10.001
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:||Submitted Journal Article|
|Divisions:||Faculty of Science > Mathematics|
|Journal or Publication Title:||Physica D: Nonlinear Phenomena|
|Page Range:||pp. 92-110|
|Publication Status:||In Press|
|Access rights to Published version:||Restricted or Subscription Access|
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