Distribution of Periodic Orbits in 2-D Hamiltonian Systems

Abstract

We study the distribution of periodic orbits in two Hamiltonian systems of two degrees of freedom, using Poincaré surfaces of section. We distinguish between regular periodic orbits (bifurcations of the periodic families of the unperturbed system) and irregular periodic orbits (not connected to the above). Regular orbits form characteristic lines joining periodic orbits of multiplicities following a Farey tree, while irregular periodic orbits form lines very close to the asymptotic curves of the main unstable families. We find that all irregular orbits are inside the lobes of the hornoclinic tangles. These tangles have gaps between various lobes containing regular periodic orbits. There are also gaps inside some lobes containing stable irregular periodic orbits.