GEOMETRICAL AND DYNAMIC PROPERTIES OF HOMOCLINIC TANGLES IN A SIMPLE HAMILTONIAN SYSTEM

We study, qualitatively and quantitatively, the forms of the asymptotic curves from an unstable periodic orbit in a simple Hamiltonian for various values of the energy. The asymptotic curves define two resonance areas and form infinite elongated ''lobes.'' We give the exact (not schematic) forms of such lobes over long times, and formulate certain rules followed by them. The lengths of the lobes of order n are of the order of lambda(n), where lambda is the largest eigenvalue of the periodic orbit. The lobes surround the resonance areas, spiraling outwards, before going into the large stochastic region outside the resonances. This explains the ''stickiness'' property of the resonance areas over long times. As the energy increases, the number of rotations of the lobes decreases and the onset of chaos is faster. The lengths and the areas of the lobes increase considerably. The number of intersections of the lobes increases, and we find how new tangencies between the various lobes are formed. If the energy goes beyond the escape energy, certain lobes terminate at ''limiting asymptotic curves'' corresponding to asymptotic curves of the Lyapunov orbits at the various escape channels.