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Globally enumerating unstable periodic orbits for observed data using symbolic dynamics.
Michael Buhl1, Matthew B Kennel
1Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402, USA. mbuhl@ucsd.edu
Chaos (Woodbury, N.Y.)
|October 2, 2007
Summary
Finding periodic orbits in chaotic systems is challenging. This study introduces a global method using symbolic dynamics and Markov chains to identify these orbits from time series data, aiding in dynamical system analysis.
Area of Science:
- * Dynamical Systems and Chaos Theory
- * Information Theory and Statistical Modeling
Background:
- * Unstable periodic orbits are crucial for understanding chaotic system dynamics.
- * Identifying these orbits from observed time series data is a significant challenge.
- * Existing methods often struggle with global and comprehensive identification.
Purpose of the Study:
- * To develop a global method for finding periodic orbits in chaotic systems.
- * To leverage symbolic dynamics and Markov chain approximations for orbit identification.
- * To enable the estimation of key dynamical quantities from identified orbits.
Main Methods:
- * Utilized recent advancements in symbolic dynamics for time series partitioning.
- * Approximated symbolic dynamics using a Markov chain estimated via information-theoretical concepts.
- * Employed a deterministic algorithm to enumerate cycles in the probabilistic graph representation of the Markov chain.
Main Results:
- * Successfully generated a global, comprehensive list of symbolic names for periodic orbits.
- * Developed a method to localize identified periodic orbits back into the original state space.
- * Demonstrated the utility of found orbits for estimating Lyapunov exponent and topological entropy.
Conclusions:
- * The presented global method effectively identifies periodic orbits in chaotic systems.
- * Symbolic dynamics and Markov chain modeling provide a robust framework for this task.
- * This approach enhances the analysis of chaotic attractors and their properties.
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