Detecting Unstable Periodic Orbits of Chaotic Dynamical Systems

A method to detect the unstable periodic orbits of a chaotic dynamical system is developed. For a given dynamical system our approach allows us to locate the unstable periodic cycles of, in principle, arbitrary length with a high accuracy. Preknowledge of the dynamical system is not required. To demonstrate its reliability as well as efficiency we apply it to several two-dimensional chaotic maps. In the case of short chaotic time series we develop a dynamical algorithm which is based on a mean-field approach via the Voronoi diagram belonging to the time series. This algorithm enables us to detect low period cycles using a very small set of data points. The influence of noise is investigated in some detail.