Abstract:

A method to detect periodic orbits in classical dynamical systems is presented. It is based on the manipulation of the stability properties of the periodic orbits via transformations of the original system. We provide a classification of a set of possible transformations and relate them to the corresponding change of the stability of the periodic orbits. Applications to the Henon and Ikeda map are performed, allowing in particular the investigation of the least unstable periodic orbits up to higher periods. Having found the complete sets of periodic orbits for a given period, one can discuss asymptotic spectra of Lyapunov exponents and can compare it to some other similarly calculated dynamical quantities like the distribution fine-time Lyapunov exponents. By discussing these distributions for some two-dimensional dynamical systems numerically and analytically certain evidence can be found for the existence of a universal behaviour of the tails of the spectra.