Periodic Orbits and Escapes in Dynamical Systems

Abstract

We study the periodic orbits and the escapes in two different dynamical systems, namely (1) a classical system of two coupled oscillators, and (2) the Manko-Novikov metric (1992) which is a perturbation of the Kerr metric (a general relativistic system). We find their simple periodic orbits, their characteristics and their stability. Then we find their ordered and chaotic domains. As the energy goes beyond the escape energy, most chaotic orbits escape. In the first case we consider escapes to infinity, while in the second case we emphasize escapes to the central "bumpy" black hole. When the energy reaches its escape value a particular family of periodic orbits reaches an infinite period and then the family disappears (the orbit escapes). As this family approaches termination it undergoes an infinity of equal period and double period bifurcations at transitions from stability to instability and vice versa. The bifurcating families continue to exist beyond the escape energy. We study the forms of the phase space for various energies, and the statistics of the chaotic and escaping orbits. The proportion of these orbits increases abruptly as the energy goes beyond the escape energy.

28 pages, 23 figures, accepted in "Celestial Mechanics and Dynamical Astronomy"

References in corpus (5)

• Observable Properties of Orbits in Exact Bumpy Spacetimes
• How to observe a non-Kerr spacetime
• On universality of algebraic decays in Hamiltonian systems
• Universality of algebraic laws in Hamiltonian systems
• Constraint on the quadrupole moment of super-massive black hole candidates from the estimate of the mean radiative efficiency of AGN

Cited by in corpus (9)

• Growth of resonances and chaos for a spinning test particle in the Schwarzschild background
• The Astrophysics of Resonant Orbits in the Kerr Metric
• A Symmetric Integrator for non-integrable Hamiltonian Relativistic Systems
• Testing the existence of regions of stable orbits at small radii around black hole candidates
• Geodesics and gravitational waves in chaotic extreme-mass-ratio inspirals: the curious case of Zipoy-Voorhees black-hole mimickers
• A Hamiltonian system of three degrees of freedom with eight channels of escape: The Great Escape
• Beyond-Newtonian dynamics of a planar circular restricted three-body problem with Kerr-like primaries
• Orbit classification in the Hill problem: I. The classical case
• Elucidating the escape dynamics of the four hill potential