The Global Symplectic Integrator: an efficient tool for stability studies of dynamical systems. Application to the Kozai resonance in the restricted three-body problem

Abstract

Following the discovery of extrasolar systems, the study of long-term evolution and stability of planetary systems is enjoying a renewed interest. While non-symplectic integrators are very time-consuming because of the very long time-scales and the small integration steps required to have a good energy preservation, symplectic integrators are well suited for the study of such orbits on long time-spans. However, stability studies of dynamical systems generally rely on non-symplectic integrations of deviation vectors. In this work we propose a numerical approach to distinguish between regular and chaotic orbits in Hamiltonian systems, hereby called Global Symplectic Integrator. It consists of the simultaneous integration of the orbit and the deviation vectors using a symplectic scheme of any order. In particular, due to its symplectic properties, the proposed method allows us to recover the correct orbit characteristics using very large integration time-steps, fluctuations of energy around a constant value and short CPU times. It proves to be more efficient than non-symplectic schemes to correctly identify the behaviour of a given orbit, especially on dynamics acting on long time-scales. To illustrate the numerical performances of the global symplectic integrator, we will apply it to the well-known toy problem of Henon–Heiles and the challenging problem of the Kozai resonance in the restricted three-body problem, whose secular effects have periods of the order of 104–105 yr.