Introduction: Control and synchronization of chaos.

The hallmark of deterministic chaos, an extreme sensitivity to initial conditions, suggests that chaotic systems might be difficult if not impossible to control, since any perturbations used for control would grow exponentially in time. Indeed, this quite reasonable view was widely held until only a few years ago. Surprisingly, the basis for controlling chaos is provided by just this property, which allows carefully chosen, tiny perturbations to be used for stabilizing virtually any of the unstable periodic orbits making up a strange attractor. Ergodicity is another property of chaotic systems that makes them particularly amenable to control, since most points of interest are eventually visited in the natural evolution of the system. Other characteristics of nonlinear systems—together with the myriad tools of dynamical systems theory—allow the ‘‘dynamicist of the ’90s’’ to control complex behavior to an extent no one would have believed possible only a decade ago. In the papers in this Focus Issue, recent advances in methods for controlling dynamical systems along with applications of these methods in a wide range of experimental settings are described. Advances in the closely related topic of synchronization of chaotic systems are also featured. Research on controlling chaotic systems has seen remarkable growth in a short time span, with the ‘‘early’’ studies in the field appearing less than ten years ago. In the late 1980s, Hübler and co-workers carried out a series of studies on manipulating chaotic systems to achieve a desired ‘‘goal dynamics,’’ with forcing terms appropriately incorporated into the corresponding governing equations. In 1990, Ott, Grebogi, and Yorke introduced a linear feedback method for stabilizing unstable periodic orbits in chaotic systems, which did not require a knowledge of the governing equations. The OGY method generated widespread interest, and various modifications and reductions of the scheme quickly followed as well as alternative approaches. ~See Fig. 1.! Methods for synchronizing chaotic systems developed virtually simultaneously with the developments in chaos control. In 1990, Pecora and Carroll demonstrated how chaotic systems could be synchronized, using an electronic circuit coupled unidirectionally to a subsystem made up of components of the parent system. This innovation provided a new perspective on chaotic dynamics and inspired many other studies on synchronizing chaotic systems. Cuomo and Oppenheim further expanded the area by demonstrating how synchronized chaotic systems could be used in a scheme for private communication. There are now hundreds of papers on chaos control and