Free motion around black holes with discs or rings: between integrability and chaos -- VI. The Melnikov method

Motivated by black holes surrounded by accretion structures, we consider in this series static and axially symmetric black holes "perturbed" gravitationally as being encircled by a thin disc or a ring. In previous papers, we employed several different methods to detect, classify and evaluate chaos which can occur, due to the presence of the additional source, in time-like geodesic motion. Here we apply the Melnikov-integral method which is able to recognize how stable and unstable manifolds behave along the perturbed homoclinic orbit. Since the method standardly works for systems with one degree of freedom, we first suggest its modification applicable to two degrees of freedom (which is the our case), starting from a suitable canonical transformation of the corresponding Hamiltonian. The Melnikov function reveals that, after the perturbation, the asymptotic manifolds tend to split and intersect, consistently with the chaos found by other methods in the previous papers.