Resonant control of the Ro ¨ ssler system Valery Tereshko 1,* and Elena Shchekinova 2,† 1 Communications Research Laboratory, McMaster University, 1280 Main Street West, Hamilton, Ontario Canada L8S 4K1 and The Nonlinear Centre, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, United Kingdom 2 Center for Dynamical Systems and Nonlinear Studies, School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332 Received 3 December 1997 We develop a method of control, ‘‘resonant control,’’ when a weak resonant perturbation is tuned so as to drive the system into naturally occurring regimes, namely, periodic orbits, which happen to be unstable for some nominal parameter value. The results show that nonfeedback control by periodic perturbations can be goal oriented, and a final state can be predictably targeted. The method allows us to alter nonchaotic as well as chaotic dynamics using only small perturbations. S1063-651X9811507-6 PACS numbers: 05.45.+b The problem of dynamical system control consists of a goal-oriented alteration of its dynamics. For example, in many cases it is important not only to suppress a chaotic behavior but to convert it into a desired regular one. A feed- back method for stabilization of unstable periodic orbits em- bedded in the chaotic attractor was developed for that 1. Its advantage is the use of weak perturbations. Indeed, because of the ergodicity of chaotic systems, sooner or later the tra- jectory will fall into the vicinity of the desired unstable orbit. Then it is sufficient only to move the stable manifold of the corresponding unstable fixed point in the Poincare ´ section to the system state point to stabilize the former. In cases when the reciprocal of the maximal Lyapunov exponent is short compared to the time between perturbations, that can lead to occasional bursts of lost control, or when the dynamics is too fast for real-time computation of the control signal, continu- ous control strategies based on delayed self-controlling feed- back, or a combination of feedback with a periodic perturba- tion, can be applied 2. These methods and their modifications 3 have been experimentally verified for dif- ferent chaotic systems 4. However, in the case when the system state is not immediately accessible, the only way of control is the use of nonfeedback techniques. One of the approaches to nonfeedback control is the non- linear entrainment method 5. It requires a knowledge of the system equations to construct control forces which, however, can have a large amplitude and a complicated shape, and the basins of entrainment can have a very complicated structure. Typically, this method can require as many control forces as there are dimensions of the system. In contrast, there are many examples of converting chaos to a periodic motion by exposing a system to only one peri- odic force or parameter modulation 6–14. Provided that a small perturbation is applied, the controlled periodic orbit closely traces the corresponding unperturbed one. Although periodic perturbation methods are easy to real- ize in practice, their main deficiency is that often it is diffi- cult to anticipate the result of perturbation, which can give rise to an undesired behavior of system. In other words, there is no common concept for construction of appropriate per- turbations which direct the trajectory to the target. In this paper we propose a method of nonfeedback control when the perturbation is tuned so as to goal-orientedly drive the system into naturally occurring regimes, namely, periodic orbits, which happen to be unstable for some nominal param- eter value. The amplitude of this perturbation is very small, but its resonant effect is enough to alter the system dynamics drastically. We call this type of control resonant control. Let us formulate the conditions of resonant control for regular dynamics. The generalization to the chaotic case is straightforward. The main idea of resonant control is that the wave form of the perturbation must be tailored to suit the wave forms of both the controlled variable and the desired response. This means that i the period, phase, amplitude, and shape of perturbation have to be tuned so as to compen- sate for the difference between current and desired wave forms of the controlled variable to provide goal-oriented tar- geting; and ii the symmetry of the perturbation must corre- spond to the symmetry of the desired wave form so as not to destabilize the latter. Provided that the period of the desired response is a multiple of the period of the current wave form, condition i guarantees the perturbation to be resonant, i.e., its period has to be equal to or a multiple of the period of current cycle. The concept of resonant control can be considered as a generalization of the original concept of geometrical reso- nance 7, requiring that the control force preserve a natural response from the underlying conservative system, to the case when the weak perturbation drives the trajectory to fol- low an a priori chosen natural response from the unperturbed dissipative system. *Author to whom correspondence should be addressed. Present address: The Nonlinear Centre, Department of Applied Mathemat- ics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, U.K. Electronic address: V.Tereshko@damtp.cam.ac.uk † Electronic address: elenash@math.gatech.edu PHYSICAL REVIEW E JULY 1998 VOLUME 58, NUMBER 1 PRE 58 1063-651X/98/581/4234/$15.00 423 © 1998 The American Physical Society