Conference abstracts

Session B1 - Computational Dynamics

July 13, 17:00 ~ 17:30 - Room B1

Accurate measurement of Arnold diffusion using computer-algebraic optimal normal forms

Christos Efthymiopoulos

Research Center for Astronomy and Applied Mathematics, Academy of Athens, Greece   -   cefthim@academyofathens.gr

Abstract: the numerical measurement of Arnold diffusion, i.e., the slow drift of weakly chaotic orbits along resonances in nearly-integrable systems, has proven to be a particularly difficult task for computational dynamics. The talk will show how such measurement can be practically realized even for tiny values of the perturbation parameter, showing real examples in a model introduced in the literature some years ago by Froeschl\'{e} et al. (2000). Our technique combines three elements: i) a computer-algebraic determination of the resonant normal form which is optimal in the sense of minimizing the corresponding remainder. ii) Use of the normalizing transformation in order to remove all `deformation' effects. This allows to clearly see the diffusion in `clean' canonical variables, devoid of deformation noise. iii) A Melnikov-type determination of the most important remainder terms which drive the diffusion. In step (iii) we implement a stationary or quasi-stationary phase approach for estimating the contribution of each term to the chaotic jumps of the action variables which take place along every individual homoclinic loop. This allows, in turn, to quantify the time evolution of these jumps, in excellent agreement with numerical results. Finally, the same technique allows to visualize Arnold diffusion, i.e., to show how the diffusion proceeds along the resonance in a sequence of consecutive, in time, homoclinic loops.

Joint work with M. Guzzo (Dipt. di Matematica, Universit\`a degli Studi di Padova) and R.I. Paez (RCAAM, Academy of Athens).

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