Continuation and stability deduction of resonant periodic orbits in three dimensional systems

TL;DR

This paper investigates the stability of resonant periodic orbits in three-dimensional dynamical systems, specifically the three-body problem, to understand the long-term behavior of extrasolar planetary systems.

Contribution

It introduces a method to compute and analyze the stability of periodic orbits in 3D systems, highlighting their role as backbones of stable regions in phase space.

Findings

Stable periodic orbits are surrounded by regular motion.

Chaos occurs near unstable periodic orbits.

Stable families of periodic orbits form the backbone of stability domains.

Abstract

In dynamical systems of few degrees of freedom, periodic solutions consist the backbone of the phase space and the determination and computation of their stability is crucial for understanding the global dynamics. In this paper we study the classical three body problem in three dimensions and use its dynamics to assess the long-term evolution of extrasolar systems. We compute periodic orbits, which correspond to exact resonant motion, and determine their linear stability. By computing maps of dynamical stability we show that stable periodic orbits are surrounded in phase space with regular motion even in systems with more than two degrees of freedom, while chaos is apparent close to unstable ones. Therefore, families of stable periodic orbits, indeed, consist backbones of the stability domains in phase space.