Dynamics of 4D4D symplectic maps near a double resonance

We study the dynamics of a family of 4D4D 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 4D4D generalisation of the standard map in order to illustrate the difference between a truncated normal form and a full 4D4D symplectic map. We evaluate numerically the volume of a 4D4D 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 2D2D 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.

► Arithmetic properties of double resonances. ► Classification of double resonances adapted to the normal form theory. ► Non-integrability of the normal form. ► Dynamics beyond the normal form theory.