Instability in Hamiltonian systems

A. Pumarino, Universidad de Oviedo, Oviedo, Spain
C. Valls, Instituto Superior Técnico, Lisboa, Portugal

E. J. Qualitative Theory of Diff. Equ., Monograph Series, No. 1. (2005), pp. 1-204.

Abstract: From the time of the pioneer Poincar\'e's essay up to the present days, chaos in conservative dynamics has been identified with the presence of heteroclinic motions in the phase space. The existence of this unlimited dynamical richness leads, in an unmistakable way, to the instability of the studied system. V. I. Arnold even discovered that, surprisingly, these situations often arise in a persistent way when an integrable Hamiltonian system is perturbed.
The global strategy designed by Arnold was based on the control of the so-called splitting of separatrices, which takes place when a parametric family of perturbations of the initial integrable system is considered. The method used by Arnold furnished orbits drifting along invariant objects and therefore giving rise to the presence of (nowadays called) Arnold diffusion. From the quantitative point of view, those events were observed for an open, but small, set of parameter values.
Besides proving the existence of Arnold diffusion for a new family of three degrees of freedom Hamiltonian systems, another goal of this book is not only to show how Arnold-like results can be extended to substantially larger sets of parameters, but also how to obtain effective estimates on the splitting of separatrices size when the {\em frequency} of the perturbation belongs to open real sets.

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