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

Polcar, L.; Semerák, O.

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 timelike 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 1 degree of freedom, we first suggest
its modification applicable to 2 degrees of freedom (which is 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, consistent with the chaos found by other methods in previous papers.
journal:Phys. Rev. D
grants:Sources of strong gravity and their astrophysical meaning, GAČR 17-13525S; 2017-2019; hlavní řešitel: Oldřich Semerák
physrevd.100.103013.pdf (1930.65 kB)

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