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3.8] What is the control of chaos? (nonlinear science)




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This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by others.

3.8] What is the control of chaos? (nonlinear science)

Control of chaos has come to mean the two things:
stabilization of unstable periodic orbits,
use of recurrence to allow stabilization to be applied locally.
Thus term "control of chaos" is somewhat of a misnomer--but the name has
stuck. The ideas for controlling chaos originated in the work of Hubler
followed by the Maryland Group.

Hubler, A. W. (1989). "Adaptive Control of Chaotic Systems." Helv. Phys.
Acta 62: 343-346.
Ott, E., C. Grebogi, et al. (1990). "Controlling Chaos." Physical Review
Letters 64(11): 1196-1199. http://www-
chaos.umd.edu/publications/abstracts.html#prl64.1196

The idea that chaotic systems can in fact be controlled may be
counterintuitive--after all they are unpredictable in the long term.
Nevertheless, numerous theorists have independently developed methods which
can be applied to chaotic systems, and many experimentalists have demonstrated
that physical chaotic systems respond well to both simple and sophisticated
control strategies. Applications have been proposed in such diverse areas of
research as communications, electronics, physiology, epidemiology, fluid
mechanics and chemistry.

The great bulk of this work has been restricted to low-dimensional systems;
more recently, a few researchers have proposed control techniques for
application to high- or infinite-dimensional systems. The literature on the
subject of the control of chaos is quite voluminous; nevertheless several
reviews of the literature are available, including:

Shinbrot, T. Ott, E., Grebogi, C. & Yorke, J.A., "Using Small Perturbations
to Control Chaos," Nature, 363 (1993) 411-7.
Shinbrot, T., "Chaos: Unpredictable yet Controllable?" Nonlin. Sciences
Today, 3:2 (1993) 1-8.
Shinbrot, T., "Progress in the Control of Chaos," Advance in Physics (in
press).
Ditto, WL & Pecora, LM "Mastering Chaos," Scientific American (Aug. 1993),
78-84.
Chen, G. & Dong, X, "From Chaos to Order -- Perspectives and Methodologies
in Controlling Chaotic Nonlinear Dynamical Systems," Int. J. Bif. & Chaos 3
(1993) 1363-1409.

It is generically quite difficult to control high dimensional systems; an
alternative approach is to use control to reduce the dimension before applying
one of the above techniques. This approach is in its infancy; see:

Auerbach, D., Ott, E., Grebogi, C., and Yorke, J.A. "Controlling Chaos in
High Dimensional Systems," Phys. Rev. Lett. 69 (1992) 3479-82
http://www-chaos.umd.edu/publications/abstracts.html#prl69.3479

 

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