Time-series analysis

The fractals of interest in this paper are sets of states attainable by a system whose dynamical behaviour is governed by nonlinear evolution equations. Experimentally such sets, normally associated with chaotic dynamics, can occur in chemical reaction systems, fluid flows, electronic oscillators, driven neurons: the list has been growing rapidly over the past 10-15 years. Experimental studies of these systems generally produce time series of measurements that have no obvious fractal properties. Classical signal-processing methods provide little help. However, recently, techniques have been developed that are capable of extracting geometrical information from time-series data. This paper will review some of these methods and their application to the study of the geometry of invariant sets underlying the dynamics and bifurcations of some experimental and model systems.