6 Targeting in Chaotic Dynamical Systems

6.1 Introduction "Targeting" refers to a process wherein one uses a suitable sequence of small controlling perturbations to steer an initial condition on an attractor to a neighborhood of a prespecified point (target) on the attractor. Here it is shown that targeting can be done in high dimensional cases, as well for one-and two-dimensional maps. The method is demonstrated with a mechanical system described by a four dimensional mapping whose attractor has two positive Lyapunov exponents and a Lyapunov dimension of 2.8. The target is reached by making very small successive changes in either one or two control parameters. Targeting often makes it possible to reach a neighborhood of a prespecified target point several orders of magnitude more quickly than would be the case without targeting. Targeting can be used to rapidly switch a chaotic process between different periodic orbits [1]. The distinguishing feature of chaotic processes is their sensitive dependence on initial conditions and highly irregular behavior that is difficult to predict except in the short term. The so-called "strange attractors" associated with chaotic processes often have a complex, fractal structure. The existence of chaotic behavior in a wide variety of mathematical, physical, and biological contexts is well known. There are many introductory texts on chaos; see for example [2]-[7]. See also the reprint collections by Cvitanovic [8] and Hao [9] for some important early papers on observations of chaotic dynamics in a variety of laboratory experiments. Reference [10], subsequent proceedings, and the references therein contain a wealth of additional applications. Ott et al. [11] introduced the idea that control of chaos could in some cases be attained by feedback stabilization of one of the infinite number of unstable periodic orbits that naturally occur in a chaotic attractor. Their method and variations thereof have been used in many experimental situations; see the other contributions in this volume and Ref. [12] for additional references. Although the ergodic nature of the dynamics on the attractor ensures that typical initial conditions eventually reach the neighborhood of almost any prespecified target point on the attractor, the time needed to reach the target might be very long. Targeting algorithms often reduce the waiting time by several orders of magnitude [13]. In this paper, we will briefly outline one approach to the targeting problem and describe some recent extensions to this work.