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

In this work we propose a new numerical approach to distinguish between regular and chaotic orbits in Hamiltonian systems, based on the simultaneous integration of the orbit and of the deviation vectors using a symplectic scheme, hereby called global symplectic integrator. In particular, the proposed method allows us to recover the correct orbits character with very large integration time steps, in some cases up to 1000 times larger than the one needed by a non-symplectic scheme. To illustrate the numerical performances of the global symplectic integrator we will apply it to two well-known and widely studied problems: the Hénon-Heiles model and the restricted three-body problem.