Symplectic integration of deviation vectors and chaos determination. Application to the H\'enon-Heiles model and to 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 both the orbit and the deviation vectors using a symplectic scheme, hereby called {\em global symplectic integrator}. In particular, the proposed method allows us to recover the correct orbits character with very large integration time steps, small energy losses and short CPU times. To illustrate the numerical performances of the global symplectic integrator we will apply it to two well-known and widely studied problems: the H\'enon-Heiles model and the restricted three-body problem.