Invariant tori for two-dimensional non-integrable Hamiltonians

Abstract

The invariant tori of a two-dimensional non-integrable Hamiltonian form families which can be parametrized by the energy and the rotation number. Within each family, the variation with energy is continuous, and the variation with rotation number is governed by the KAM theorem and is everywhere discontinuous. In order to calculate the tori numericaly, we approximate them by highly winding periodic orbits, and we give a criterion for the validity of this approximation. For one particular Hamiltonian, we study the family of tori which connects two distinct families of periodic trajectories. As a by-product of this calculation, we draw the curve bounding the "mostly regular" region, on a plot whose coordinates are the two partial periods of the tori. © 1988.