Symbolic dynamics from chaotic time series.
Alain Destexhe, Grégoire Nicolis and Cathy Nicolis

In: Measures of Complexity and Chaos, Edited by Abraham, N.B., Albano, A.L., Passamente, A. and Rapp, P.E. NATO ASI Series, Plenum Press, Vol. B208, pp. 339-343, 1989.

PDF copy of the full paper

High-resolution PDF

Following the ideas of Ruelle and others [1,2], an embedding phase space can be reconstructed from experimental systems on the basis of time series data. The introduction of numerical methods for calculating dimensions, entropies, Lyapunov exponents and other related properties, has permitted extensive investigations of chaotic experimental systems these last years. However, severe restrictions about the applicability of these methods were noticed, especially for high dimensional systems [3-6]. In this paper, we use the symbolic description as an alternative approach to analyze chaotic dynamical systems, independently of any phase space reconstruction algorithms. The idea is to compress the information contained in continuous variables into a sequence of symbols which can be studied with standard statistical tools, in order to gain some quantitative knowledge about the system’s dynamics. In section 1, we show from a model system that a symbolic description can reveal useful information about the regular – or irregular – aspects inherent to the chaotic dynamics. In section 2, we study two biological systems for which patterns of activity are repeated at irregular intervals. Here again, the symbolic dynamics will provide interesting information on how these intervals are interrelated.