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Unfolding Complexity in Nonlinear Dynamical Systems

  • R. Badii
Part of the NATO ASI Series book series (NSSB, volume 208)

Abstract

Nonlinear dynamical systems produce complex temporal or spatial patterns by stretching and folding regions of phase-space in an iterative way. The topological and metric properties of this process can be extracted from the chaotic signal by using symbolic dynamics. The underlying “grammatical rules” are systematically detected and arranged on a logic tree. Predictions on the set of possible outcomes of the system are made and compared with the observation. The discrepancy between the two, evaluated through a generalization of the information gain, characterizes the complexity of the source. As a result of this unfolding procedure, the dynamics is described as a sequence of deterministic paths (blocks of symbols) which appear at random in time, with given transition probabilities. Fast hierarchical evaluations of invariant measures, dimensions, entropies and Lyapunov exponents are obtained from the logic tree, considering lower and lower levels (i.e., increasingly long symbol-sequences). Power spectra are accurately reproduced with a limited number of short orbits. The analysis applies to any system: dissipative or conservative, hyperbolic or not, invertible or not.

Keywords

Periodic Orbit Lyapunov Exponent Information Gain Nonlinear Dynamical System Logic Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • R. Badii
    • 1
  1. 1.Fakultät für PhysikUniversität KonstanzConstanceWest Germany

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