Targetting Chaos through Adaptive Control

We describe adaptive control algorithms whereby a chaotic dynamical system can be steered to a target state with desired characteristics. A specific implementation considered has the objective of directing the system to a state which is more chaotic or mixed than the uncontrolled one. This methodology is easy to implement in discrete or continuous dynamical systems. It is robust and efficient, and has the additional advantage that knowledge of the detailed behaviour of the system is not required. Adaptive control algorithms have hitherto been implemented for the purpose of maintaining periodic behaviour in nonlinear systems [1, 2, 3]. Recently there has been some interest in control algorithms whose aim is to target other types of dynamical behaviour. “Anti–control” algorithms, namely those wherein the objective is to maintain [4] or to enhance [5] the chaoticity of dynamical systems have been devised. These efforts have been motivated by practical situations where the enhancement or maintenance of chaos has desirable consequences. Examples of these can be found in contexts as diverse as mixing flows [6], electronic systems [7] and chemical reactions [8], where the enhancement of chaos can lead to improved performance, or in biological applications such as in neural systems, where the maintenance of chaos provides the key to the avoidance of pathological behaviour [9]. In this paper we describe an adaptive anti–control algorithm which is simple and easily implemented. The algorithm is set up to maintain a desired level of chaoticity, and to achieve a target value of the local Lyapunov [email protected], 2 [email protected], 3 [email protected]