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

V. Gelfreich,
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
v.gelfreich@warwick.ac.uk

C. Sim\'o and A. Vieiro
Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona,
Gran Via 585, 08007 Barcelona, Spain
carles@maia.ub.es, vieiro@maia.ub.es

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