Free motion around black holes with discs or rings: between integrability and chaos -- VI. The Melnikov method
arXiv:1911.09790 · doi:10.1103/PhysRevD.100.103013
Abstract
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.
arXiv comment32 pages (PRD preprint style), 7 figures
References in corpus (10)
- Black holes, hidden symmetries, and complete integrability
- A Periodic Table for Black Hole Orbits
- Observable Properties of Orbits in Exact Bumpy Spacetimes
- An observable signature of a background deviating from Kerr
- Free motion around black holes with discs or rings: between integrability and chaos - II
- Free motion around black holes with discs or rings: between integrability and chaos -- IV
- Chaos in black holes surrounded by gravitational waves
- Free motion around black holes with discs or rings: between integrability and chaos -- V
- Static axisymmetric rings in general relativity: How diverse they are
- Chaos in Periodically Perturbed Monopole + Quadrupole Like Potentials
Cited by in corpus (7)
- Probing Phase Structure of Black Holes with Lyapunov Exponents
- Extreme mass ratio inspirals into black holes surrounded by matter
- Non-linear effects in EMRI dynamics and their imprints on gravitational waves
- Chaotic Motion around a Black Hole under Minimal Length Effects
- Temporal and spatial chaos in the Kerr-AdS black hole in an extended phase space
- Toward relativistic inspirals into black holes surrounded by matter
- Chaotic particle motion around a homogeneous circular ring