Stability and instability in the anisotropic Kepler problem

Abstract

We study the degree of order and chaos in the anisotropic Kepler problem. The Lyapunov characteristic number (LCN) is constant for all orbits in the chaotic domain with a fixed mass ratio µ. It increases from zero as the mass ratio µ increases from µ = 1, reaches a maximum for µ = 4 and then decreases slowly to zero for a very large µ. We found the characteristics of many families of the periodic orbits of multiplicities up to 17. Several families have a stable part for µ close to µ = 1 (namely, µ < 1.748). We conjecture that there are stable orbits of high multiplicities for larger values of µ. The stable orbits are surrounded by islands of stability consisting of closed invariant curves. The rotation number along these invariant curves decreases from the periodic orbits outwards. Higher-order islands are enclosed in every main island around a low-order periodic orbit. In the case µ = 1.1, we find a large number of islands of multiplicities from 3 up to 17. Then we calculate the asymptotic curves from unstable periodic orbits. These curves go through the collision orbits but continue beyond these points. The intersections of the stable and unstable asymptotic curves are homoclinic and heteroclinic points. The corresponding orbits are the main characteristics of chaos.