THE STRUCTURE OF CHAOS IN A POTENTIAL WITHOUT ESCAPES

We study the structure of chaos in a simple Hamiltonian system that do es no have an escape energy. This system has 5 main periodic orbits th at are represented on the surface of section (y, y) by the points (1) O(0,0), (2) C1, C2(+/-y(c), 0), (3) B1, B2(O, +/- 1) and (4) the bound ary y2 + y2 = 1. The periodic orbits (1) and (4) have infinite transit ions from stability (S) to instability (U) and vice-versa; the transit ion values of epsilon are given by simple approximate formulae. At eve ry transition S --> U a set of 4 asymptotic curves is formed at O. For larger epsilon the size and the oscillations of these curves grow unt il they destroy the closed invariant curves that surround O, and they intersect the asymptotic curves of the orbits C1, C2 at infinite heter oclinic points. At every transition U --> S these asymptotic curves ar e duplicated and they start at two unstable invariant points bifurcati ng from O. At the transition itself the asymptotic curves from O are t angent to each other. The areas of the lobes from O increase with epsi lon; these lobes increase even after O becomes stable again. The asymp totic curves of the unstable periodic orbits follow certain rules. Whe never there are heteroclinic points the asymptotic curves of one unsta ble orbit approach the asymptotic curves of another unstable orbit in a definite way. Finally we study the tangencies and the spirals formed by the asymptotic curves of the orbits B1, B2. We find indications th at the number of spiral rotations tends to infinity as epsilon --> inf inity. Therefore new tangencies between the asymptotic curves appear f or arbitrarily large epsilon. As a consequence there are infinite new families of stable periodic orbits that appear for arbitrarily large e psilon.