Chaos in the case of two fixed black holes

Abstract

We study the structure of chaos in the relativistic problem of particles moving in the field of two fixed black holes M1 and M2 by considering the asymptotic curves of some simple unstable periodic orbits on a surface of section. These curves consist of infinite arcs reaching the two black holes. Most orbits starting along these curves (asymptotic orbits) escape, i.e., they fall into the black holes M1 (type I orbits) and M2 (type II orbits). The number of the remaining orbits after n iterations (intersections with the surface of section) decreases exponentially with n. The asymptotic curves intersect at infinite homoclinic and heteroclinic points. We study in detail the forms of the asymptotic orbits with emphasis on the homoclinic and heteroclinic orbits. The homoclinic and heteroclinic intersections are confined in certain intervals along the asymptotic curves. Every homoclinic and heteroclinic orbit is the limit of infinite more homoclinic and heteroclinic orbits. Between an asymptotic orbit falling on the black hole M1 and another orbit falling on the black hole M2 there are infinite homoclinic and heteroclinic orbits and infinite transitions between type I and type II orbits. Therefore the orbits of type I and II form fractal sets. The nonasymptotic curves falling on the black holes M1 and M2 also form fractal sets. The black holes act as attractors and the areas on the surface of section are not conserved despite the fact that the system is conservative.