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Statistical and Nonlinear Physics

Targeting Control of Chaotic Systems

Profile image of Erik BolltErik Bollt

2003

Abstract

Targeting control of chaos is concerned with taking advantage of sen- sitive dependence to initial conditions to coax a dynamical system to following a desirable trajectory. In other words, it is taking advantage of the butterfly effect so that the rich spectrum of possible trajectories embedded within a chaotic attractor can be selected with extremely small energy input. We review historical and pop- ular approaches which fall under this general area in an attempt to reveal these techniques in a useful manner for applied scientists.

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We have achieved control of chaos in a physical system using the method of Ott, Grebogi, and Yorke [Phys. Rev. Lett. 64, 1196 (1990)]. The method requires only small time-dependent perturbations of a single-system parameter and does not require that one have model equations for the dynamics. We demonstrate the power of the method by controlling a chaotic system around unstable periodic orbits of order 1 and 2, switching between them at will.

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The objective of the present work is to propose a method for nonfeedback anticontrol of chaos with perturbations of minimum power for a preset control goal. The noted Lorenz system is employed as the test model for chaotification with the target state specified by a prescribed positive value of the largest Lyapunov exponent ͑LLE͒, Ͼ 0. Periodic and quasiperiodic perturbations are used as control signals, and the signals parameters are optimized using a genetic algorithm under restriction of minimum power. Performance of the optimized signals in triggering chaos at an ordered state, fixed point or periodic, as well as further enhancing chaoticity at a chaotic state is explored. The present numerical experiments reveal the following interesting physics about chaotification. In general, the power for chaotification increases with the preset value of and quasiperiodic signals can achieve the control goal with a lower power than periodic ones. Given the same increment of LLE from that of the uncontrolled state ͑ 1,0 ͒, i.e., ⌬ = − 1,0 , the further enhancement of chaoticity in a chaotic state needs a higher control power than the triggering of chaos from an ordered state. The minimum power required for chaotification of an ordered state increases relatively slowly for lower but increases drastically as the preset target LLE reaches a certain critical value. Most strikingly, the numerical experiments demonstrate that this critical value of corresponds to LLE of the nearest chaotic state in the neighborhood of the uncontrolled state. Robustness of applying the present method in the presence of external noise is also demonstrated.

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Victor Yacila Preciado

Physics Letters A, 2004

We developed the control technique for nonlinear oscillators when simple feedback alters the oscillation energy. The control does not require any computation nor knowledge of the system equations. Depending on the control perturbation parameters, two scenarios take place: (i) changing (linearly or nonlinearly) the oscillator damping, and (ii) suppressing (enhancing) the driving force. In the case of large deviations between phases of the chaotic oscillator and the driving force, only first scenario holds. Generalization of the technique to the case of oscillator networks is considered.

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The control of dynamical systems—recovering order from chaos—
Louis J. Dubé

AIP Conference Proceedings, 2000

Following a brief historical introduction of the notions of chaos in dynamical systems, we will present recent developments that attempt to profit from the rich structure and complexity of the chaotic dynamics. In particular, we will demonstrate the ability to control chaos in realistic complex environments. Several applications will serve to illustrate the theory and to highlight its advantages and weaknesses. The presentation will end with a survey of possible generalizations and extensions of the basic formalism as well as a discussion of applications outside the field of the physical sciences. Future research avenues in this rapidly growing field will also be addressed.

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