REGULAR AND IRREGULAR PERIODIC-ORBITS

We distinguish between regular orbits, that bifurcate from the main fa milies of periodic orbits (those that exist also in the unperturbed ca se) and irregular periodic orbits, that are independent of the above. The genuine irregular families cannot be made to join the regular fami lies by changing some parameters. We present evidence that all irregul ar families appear inside lobes formed by the asymptotic curves of the unstable periodic orbits. We study in particular a dynamical system o f two degrees of freedom, that is symmetric with respect to the x-axis , and has also a triple resonance in its unperturbed form. The distrib ution of the periodic orbits (points on a Poincare surface of section) shows some conspicuous lines composed of points of different multipli cities. The regular periodic orbits along these lines belong to Farey trees. But there are also lines composed mainly of irregular orbits. T hese are images of the x-axis in the map defined on the Poincare surfa ce of section. Higher order iterations of this map, close to the unsta ble triple periodic orbit, produce lines that are close to the asympto tic curves of this unstable orbit. The homoclinic tangle, formed by th ese asymptotic curves, contains many regular orbits, that were generat ed by bifurcation from the central orbit but were trapped inside the t angle as the perturbation increased. We found some stable periodic orb its inside the homoclinic tangle, both regular and irregular. This pro ves that the homoclinic tangle is not completely chaotic, but contains gaps (islands of stability) filled with KAM curves.