- Control of chaos
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This article is about a non-linear system. For the technique in the Sonic the Hedgehog games, see Chaos Control.
In chaos theory, control of chaos is based on the fact that any chaotic attractor contains an infinite number of unstable periodic orbits. Chaotic dynamics then consists of a motion where the system state moves in the neighborhood of one of these orbits for a while, then falls close to a different unstable periodic orbit where it remains for a limited time, and so forth. This results in a complicated and unpredictable wandering over longer periods of time.
Control of chaos is the stabilization, by means of small system perturbations, of one of these unstable periodic orbits. The result is to render an otherwise chaotic motion more stable and predictable, which is often an advantage. The perturbation must be tiny, to avoid significant modification of the system's natural dynamics.
Several techniques have been devised for chaos control, but most are developments of two basic approaches: the OGY (Ott, Grebogi and Yorke) method, and Pyragas continuous control. Both methods require a previous determination of the unstable periodic orbits of the chaotic system before the controlling algorithm can be designed.
In the OGY method, small, wisely chosen, swift kicks are applied to the system once per cycle, to maintain it near the desired unstable periodic orbit. In the Pyragas method, an appropriate continuous controlling signal is injected into the system, whose intensity is practically zero as the system evolves close to the desired periodic orbit but increases when it drifts away from the desired orbit.
Experimental control of chaos by one or both of these methods has been achieved in a variety of systems, including turbulent fluids, oscillating chemical reactions, magneto-mechanical oscillators, and cardiac tissues. Sarnobat et al. (2000) attempt the control of chaotic bubbling with the OGY method and using electrostatic potential as the primary control variable.
The number of publications devoted to control of chaos is huge, see e.g. Chaos control bibliography (1997-2000)
Forcing two systems into the same state is not the only way to achieve synchronization of chaos. Both control of chaos and synchronization constitute parts of Cybernetical Physics. Cybernetical physics is a research area on the border between Physics and Control Theory.
References
- Sarnobat, S.U., “Modification, Identification & Control of Chaotic Bubbling via Electrostatic Potential”, Masters Thesis, University of Tennessee, Knoxville, August 2000. link
- Cybernetical_physics
External links
Books
Eckehard Schöll and Heinz Georg Schuster (Eds). Handbook of Chaos Control Wiley-VCH; 2nd Revision, Enlarged edition (2007) Weinheim.
González-Miranda, J. M. (2004). Synchronization and Control of Chaos: An Introduction for Scientists and Engineers. London: Imperial College Press.
Fradkov A.L., Pogromsky A.Yu. (1998). Introduction to Control of Oscillations and Chaos. Singapore: World Scientific Publishers.
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