Controlling chaos

14 Chaotic Behavior near the Ferroelectric Phase Transition 

Two typical properties of nonlinear dynamical systems are responsible for the realization of controlling chaos. Firstly, nonlinear systems show a sensitive dependence on initial conditions. This is represented in Table 14.1 by the nonlinear equation 

Small changes of the start values x0 produce a growing separation between the neighbouring trajectories. 

The new concepts of the nonlinear dynamics are a useful tool for the study of structural phase transitions and nonlinear properties. In particular, the comparison of calculated and exper- 

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Controlled chaos may also factor into the generation of rhythmic behavior in living systems. A recently proposed modeL describes the central circadian oscillator as a chaotic attractor. Limit cycle mechanisms have been previously offered to explain circadian clocks and related phenomena, but they are limited to a single stable periodic behavior. In contrast, a chaotic attractor can generate rich dynamic behavior. Attractive features of such a model include versatility of period selection as well as use of control elements of the type already well known for metabolic circuitry. 

The need to be able to control chaos has attracted considerable attention. Methods already available include a variety of minimal forms of interaction [150-155] and methods of strong control [156,157] that necessarily require a large modification of the system s dynamics, for at least a limited period of time. For example, in Refs. 158 and 159, the procedure of controlling chaos by means of minimal forms of interaction (saddle cycle stabilization) is realized for different laser systems. 

Secondly, the chaotic state of a resonant system is an infinite collection of unstable periodic motions (see Figure 14.9). The basic idea of the controlling chaos is to switch a chaotic system by a small perturbation among many different periodic orbits. Ott, Grebogy and Yorke [10] developed a scheme in which a chaotic system can be forced to follow one particular unstable periodic orbit. The problem is to calculate the perturbation that will shift the system towards the desired periodic orbit. This process is similar to balancing a marble on a saddle. To keep the marble from rolling off, one needs to move the saddle quickly from side to side. And... 

Kapitaniak, T., Controlling Chaos Theoretical and Practical Methods in Non-linear Dynamics, Academic Press, London, 1996. 

Bau H H and Wang Y-Z 1991 Chaos a heat transfer perspective Annual Reviews in Heat Transfer vol IV, ed C L Tien, pp 1-50 Ott E, Grebogi C and Yorke J A 1990 Controlling chaos Phys. Rev. Lett. 64 1196-9 Shah R K and London A L 1978 Laminar Flow Forced Convection in Ducts (New York Academic)... 

D. W. Sukow, M. E. Bleicli, D. J. Gauthier, and J. E. S. Socolar Controlling chaos in a fast diode resonator using time-delay autosynchronisation Experimental observations and theoretical analysis, Chaos 7, 560 (1997). 

Petrov, V., V. Gaspar, J. Masere K. Showalter. 1993. Controlling chaos in the Belousov-Zhabotinsky reaction. Nature 361 240-3. 

Shinbrot, T., C. Grebogi, E. Ott J.A. Yorke. 1993. Using small pertubations to control chaos. Nature 363 411-17. 

Hasty, J. Isaacs, R Dolnik, M. McMUlen, D. Collins, J. J. Designer gene networks towards fundamental cellular control. Chaos 2001, II, 207-220. 

We describe here an approach to controlling chaos in chemical systems pioneered by Showalter and collaborators (Peng et al., 1991). The algorithm does not require knowledge of the underlying differential equations, but rather works from an experimentally determined 1-D map—for example, the next amplitude map for the system. We consider the problem of stabilizing an unstable period-1 orbit, that is, an unstable fixed point in the 1- D map, of a chaotic system. We assume that we have measured the 1-D map and express that map as... 

Controlling chaos, even in the rather limited fashion described here, is a powerful and attractive notion. Showalter (1995) compares the process to the instinctive, apparently random motions of a clown as he makes precise corrections aimed at stabilizing his precarious perch on the seat of a unicycle. A small, but carefully chosen, variation in the parameters of a system can convert that system s aperiodic behavior not only into periodic behavior, but also into any of a large (in principle, infinite) number of possible choices. In view of the appearance of chaos in systems ranging from chemical reactions in the laboratory, to lasers, to flames, to water faucets, to hearts, to brains, our ability first to understand and then to control this ubiquitous phenomenon is likely to have major consequences. 

Experimentally, delayed feedback is a potentially powerful tool for the analysis of dynamical systems. It can be used either to stabilize otherwise unstable steady states (Zimmerman et al., 1984) or to cause a steady state to become imstable, leading to periodic or chaotic behavior. By extension, it should be possible to use delayed feedback to stabilize unstable periodic orbits, providing an alternative to other methods for controlling chaos (Grebogi et al., 1982 Peng et al., 1991). 

Petrov, V. Caspar, V. Masere, J, Showalter, K, 1993, Controlling Chaos in the Belousov-Zhabotinsky Reaction, Nature 361, 240-243,... 

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