Detection of Ordered and Chaotic Motion Using the Dynamical Spectra

Abstract

Two simple and efficient numerical methods to explore the phase space structure are presented, based on the properties of the “dynamical spectra”. 1) We calculate a “spectral distance” D of the dynamical spectra for two different initial deviation vectors. D → 0 in the case of chaotic orbits, while D → const ≠ 0 in the case of ordered orbits. This method is by orders of magnitude faster than the method of the Lyapunov Characteristic Number (LCN). 2) We define a sensitive indicator called ROTOR (ROtational TOri Recongnizer) for 2D maps. The ROTOR remains zero in time on a rotational torus, while it tends to infinity at a rate ∝ N = number of iterations, in any case other than a rotational torus. We use this method to locate the last KAM torus of an island of stability, as well as the most important cantori causing stickiness near it.