Numerical methods for the control of chaos
Stubbendieck, Gregg T.
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The study of chaos is a relatively recent phenomenon. There is little doubt that the modem computer has played an important role in the growth of interest in chaos. High speed computers and sophisticated grzphical displays make it possible to explore aspects of chaos that have not previously been accessible. Chaos is a type of nonlinear behavior that appears to be random although it is driven by deterministic processes. A defining aspect of chaotic behavior is its sensitivity to perturbations. SmaU changes in a chaotic system lead to large differences in behavior later on. Although chaos seems to be random when first encotmtered, there is an underlying structure to chaos that can be exploited. A method was developed by E. Ott, C. Grebogi and J. A. Yorke (OGY) to exploit the underlying stmcture of chaos and its sensitivity to perturbations to induce chaotic d5'namical systems into periodic behavior. The method has been successfully tested in a number of experimental situations. We have developed a reconfigurable Online ControUer that integrates several of the elements related to the OGY control method with the OGY control method itself. The control method is thus expanded to include the related elements, previously treated as offline procedures, as part of its definition. The OGY method requires certain properties of a chaotic system to be controUed. The Online ControUer does not add to the required properties. The specialized data structures and algorithms developed in the course of this research to implement the related elements in an online situation are described. Alternate methods for calculating the stability information required by the OGY method that take advantage of the properties of the Online ControUer are described. The alternate methods address inadequacies in the methods suggested by Ott et al. A number of experiments Ulustrating various features of the Online Controller are described, and their results are discussed. The Online ControUer provides a stable platform upon which the remaining methods required to fully automate the control of chaos can be developed.