25 May 1998 Physics Letters A 242 (1998) 1521162 Dynamical tracking of unstable periodic orbits A.N. Pisarchik 1 Stepanov Institute of Physics, National Academy of Sciences of Belarus, Skaryna Avenue 70, 220072 Minsk, Belarus 2 Received 15 December 1997; revised manuscript received 10 March 1998; accepted for publication 10 March 1998 Communicated by C.R. Doering Abstract Tracking unstable periodic orbits and its stabilization by large periodic modulation of a control parameter are studied numerically in the H´ enon map and laser equations. Some important scaling relations linking the tracking range to the modulation amplitude and frequency are deduced. The results obtained with both models are compared. Experimental realization of dynamical tracking is demonstrated in a loss-driven CO2 laser where cavity detuning or losses are periodically modulated. c  1998 Elsevier Science B.V. PACS: 05.45.+b; 42.65.Lt; 42.65.Sf Keywords: Nonlinear dynamics; Chaos control; CO 2 laser; Parameter modulation 1. Introduction An important task of the theory of dynamical systems is finding all stable and unstable solutions (branches) of algebraic nonlinear equations. Since the 1960s this problem has been attacked by contin- uation methods (branch tracing or path following), the procedure by which one can track a branch around a turning point [1]. At present there are very pow- erful methods for continuation in bifurcation theory for a wide variety of problems (for an overall re- view see Refs. [1,2]). Some years ago Schwartz and Triandaf [3] introduced the term of “tracking” as a continuation method designed for experiments in the context of chaos control. It considers the use of a time series of a measured quantity in an experiment or numerical simulations. Tracking an unstable periodic 1 Also at Departament de F´ ısica, Universitat Aut` onoma de Barcelona, E-08193 Bellaterra, Spain. E-mail: ifoph@cc.uab.es. 2 E-mail: lmk@dragon.bas-net.by. orbit (UPO) constrains a system’s trajectory within a periodic orbit as the system is moved, via large parameter shifts, through various bifurcations into a regime where the orbit is inherently unstable. The first experiments on tracking the UPOs and steady states were carried out using electronic [4] and laser systems [5]. Recently, the tracking technique has been also utilized to stabilize underlying UPOs in nonchaotic systems [6]. Tracking can be performed either by following a stable periodic orbit into a regime where the orbit is unstable [6], or by first controlling an UPO in the chaotic range and then following it through various bifurcations out of chaos [4,7,8]. In this paper we consider a kind of tracking which can be classified among the first type of tracking. However, in con- trast to classical tracking [318] that implies a very slow (adiabatic) change in the control parameter from a value at which the system behavior is stable towards unstable parameter range or vice versa, the 0375-9601/98/$19.00 c  1998 Elsevier Science B.V. All rights reserved. PII S0375-9601 ( 98 ) 00210-2 PLA 7936