General approach to the localization of unstable periodic orbits in chaotic dynamical systems

Abstract

We present a method to detect the unstable periodic orbits of a multidimensional chaotic dynamical system. Our approach allows us to locate in an efficient way unstable cycles of, in principle, arbitrary length with a high accuracy. Based on a universal set of linear transformations the originally unstable periodic orbits are transformed into stable ones, and can consequently be detected and analyzed easily. This method is applicable to dynamical systems of any dimension, and requires no preknowledge with respect to the solutions of the original chaotic system. As an example of application of our method, we investigate the Ikeda attractor in some detail.