Thursday, May 2, 2013

Dynamics of 4D symplectic maps near a doubly resonant point

Arturo Vieiro
(Universitat de Barcelona)

Abstract: I shall discuss about the results recently published in Gelfreich, V.; Simó, C.; Vieiro, A., Dynamics of 4D symplectic maps near a double resonance. Physica D: Nonlinear Phenomena, 243(1):92--110 (Holland): 2012. In this work we consider a family of 4D symplectic mappings near a doubly resonant elliptic fixed point. Around a weak double resonance (a junction of two resonances of different orders, both being larger than 4) the dynamics can be described in terms of a simple (in general non-integrable) Hamiltonian model. The main results are: 1) We show that the non-integrability of the normal form is expected because of the generic splitting of the invariant manifolds associated with a normally hyperbolic invariant cylinder. 2) Using an interpolating vector field of the truncated normal form we provide a strong numerical evidence of the fact that the normally hyperbolic invariant cylinder of the 4D standard-like map is not analytic and it has the regularity that can be expected in the generic case. 3) Finally, we use a 4D generalisation of the standard map to illustrate the difference between a truncated normal form (interpolating Hamiltonian) and the full standard-like map. In particular, we (numerically) evaluate 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.

Thursday, May 2, 2013
Time: 16h30
Room: Sala IAPMEI, Edificio Quelhas, ISEG