Periodic Orbits and Escapes in Dynamical Systems

TL;DR

This paper investigates the properties of periodic orbits and escape phenomena in two dynamical systems, revealing how energy levels influence orbit stability, chaos, and escape behaviors, with implications for understanding complex gravitational systems.

Contribution

It provides a detailed analysis of periodic orbits, bifurcations, and escape mechanisms in coupled oscillators and perturbed Kerr metrics, highlighting new insights into their stability and phase space structure.

Findings

Periodic orbits reach infinite period at escape energy

Chaotic and escaping orbit proportions increase sharply beyond escape energy

Bifurcations occur at transitions between stability and instability

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.