Chaotic Motions in The Field of Two Fixed Black Holes

Abstract

We study the chaotic motions in the field of two fixed black holes M1, M2 by calculating (a) the asymptotic curves from the main unstable periodic orbits, (b) the asymptotic orbits, with particular emphasis on the homoclinic and heteroclinic orbits, and (c) the basins of attraction of the two black holes. The orbits falling on M1 and M2 form fractal sets. The asymptotic curves consist of many arcs, separated by gaps. Every gap contains orbits falling on a black hole. The sizes of various arcs decrease as the mass of M1 increases. The basins of attraction of the black holes M1, M2 consist of large compact regions and of thin filaments. The relative area of the basin M2 tends to 100% as M1→0, and it decreases as M1 increases. The total area of the basins is found analytically, while the relative area of the basin M2 is given by an empirical formula. Further empirical formulae give the exponential decrease of the number of asymptotic orbits that have not yet reached a black hole after n iterations.