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A method for visualization of invariant sets of dynamical systems based on the ergodic partition

Mezić, Igor and Wiggins, Stephen (1999) A method for visualization of invariant sets of dynamical systems based on the ergodic partition. Chaos, 9 (1). pp. 213-218. ISSN 1054-1500. doi:10.1063/1.166399.

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We provide an algorithm for visualization of invariant sets of dynamical systems with a smooth invariant measure. The algorithm is based on a constructive proof of the ergodic partition theorem for automorphisms of compact metric spaces. The ergodic partition of a compact metric space A, under the dynamics of a continuous automorphism T, is shown to be the product of measurable partitions of the space induced by the time averages of a set of functions on A. The numerical algorithm consists of computing the time averages of a chosen set of functions and partitioning the phase space into their level sets. The method is applied to the three-dimensional ABC map for which the dynamics was visualized by other methods in Feingold et al.

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Additional Information:Copyright © 1999 American Institute of Physics. Received 13 April 1998; accepted 29 October 1998. We would like to thank Jean-Christophe Nave of the University of California, Santa Barbara for coding the program to compute the partition and producing the figures. This research was partially supported by ONR Grant No. N00014-98-1-0056, AFOSR Grant No. F49620-98-1-0146, and National Science Foundation Grant No. DMS-9803555 to IM and ONR Grant No. N00014-97-1-0071 to SW.
Subject Keywords:nonlinear dynamical systems; chaos
Issue or Number:1
Record Number:CaltechAUTHORS:MEZchaos99a
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1939
Deposited By: Tony Diaz
Deposited On:24 Feb 2006
Last Modified:08 Nov 2021 19:43

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