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

We study, qualitatively and quantitatively, the forms of the asymptoti c curves from an unstable periodic orbit in a simple Hamiltonian for v arious values of the energy. The asymptotic curves define two resonanc e 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 per iodic orbit. The lobes surround the resonance areas, spiraling outward s, before going into the large stochastic region outside the resonance s. This explains the ''stickiness'' property of the resonance areas ov er 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 va rious lobes are formed. If the energy goes beyond the escape energy, c ertain lobes terminate at ''limiting asymptotic curves'' corresponding to asymptotic curves of the Lyapunov orbits at the various escape cha nnels.