Lagrangian descriptors and regular motion

Abstract

Lagrangian descriptors introduced a decade ago have revealed as a powerful tool to unveil the intricacies of the phase space of dynamical systems in a very easy way. They have been extensively used to study chaotic motion in a variety of different situations, but much less attention has been paid to applications to the regular regions of phase space. In this paper, we show the potential of this recent mathematical tool, when suitably manipulated, to compute and fully characterize invariant tori of generic systems. To illustrate the method, we present an application to the well known Hénon–Heiles Hamiltonian, a paradigmatic example in nonlinear science. In particular, we demonstrate that the Lagrangian descriptors associated with regular orbits oscillate around an asymptotic value when divided over the integration time, which enables the computation of the frequencies characterizing the invariant tori in which regular motion takes place.