2001, Volume 4, Number 2, pp.194-205

We study the distributions of periodic orbits in two simple dynamical systems of two degrees of freedom. We distinguish between regular periodic orbits (bifurcating from the periodic families of the unperturbed problem) and irregular periodic orbits (independent of the above). Most periodic orbits are unstable for large perturbations. The average instability is larger in regions of larger density of periodic orbits. This fact can be explained by an ergodic argument. Many regular periodic orbits bifurcate from the central family of periodic orbits. Irregular families are generated in pairs (one stable - one unstable). On a Poincaré surface of section the points representing the periodic orbits form two types of characteristic lines, namely lines of regular orbits forming Farey trees, and lines of irregular orbits close to the asymptotic curves of the main unstable periodic orbits. All irregular orbits probably appear inside lobes of the homoclinic tangle. But some regular orbits are also trapped in the homoclinic tangle, although they do not belong to lobes. Inside the tangle there are some stable orbits.
Key words: periodic orbits, bifurcation, homoclinic points

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