Phase synchronization of chaotic oscillations in terms of periodic orbits Arkady Pikovsky, Michael Zaks, Michael Rosenblum, Grigory Osipov, and Ju ¨ rgen Kurths Department of Physics, University of Potsdam, Am Neuen Palais, PF 601553, D-14415, Potsdam, Germany Received 14 May 1997; accepted for publication 10 September 1997 We consider phase synchronization of chaotic continuous-time oscillator by periodic external force. Phase-locking regions are defined for unstable periodic cycles embedded in chaos, and synchronization is described in terms of these regions. A special flow construction is used to derive a simple discrete-time model of the phenomenon. It allows to describe quantitatively the intermittency at the transition to phase synchronization. © 1997 American Institute of Physics. S1054-15009702504-4 When a periodic self-sustained oscillator is governed by a periodic external force, the phenomenon of synchroniza- tion can be observed, i.e., the phase of the oscillator is locked to the phase of the driving force. For certain cha- otic autonomous dissipative systems the phase can be in- troduced as well. Such systems can also be synchronized by external periodic force. In this case the phase is locked, while the amplitude remains chaotic. We describe here the phase synchronization of chaotic oscillators through the phase-locking properties of the unstable pe- riodic orbits embedded in a chaotic attractor. For each such orbit the phase-locked region can be constructed, and when these regions overlap, full phase synchroniza- tion is observed. Transition to this state is shown to occur via a specific kind of intermittency, arising at the attractor – repeller collision in phase space. I. INTRODUCTION Synchronization is a basic nonlinear phenomenon in physics, discovered at the beginning of the modern age of science by Huygens. 1 In the classical sense, synchronization means adjustment or entrainment of frequencies of periodic oscillators due to a weak interaction cf. Refs. 2–4. This effect is well studied and finds a lot of practical applications in electrical and mechanical engineering. 5 Extensive investigations of chaotic oscillations have re- quired generalization of the notion of synchronization to this case. In this context, different phenomena have been found which are usually referred to as ‘‘synchronization.’’ Gener- ally, one speaks on synchronization if some nontrivial order is encountered in weakly interacting chaotic systems; e.g. the complete identical synchronization is observed if the states of interacting systems coincide while their dynamics remain chaotic; the attractor is then embedded into a symmetrical subspace of the phase space. 6–8 Another example is the gen- eralized synchronization, where also the dimension of the attractor decreases but the dynamics is restricted to some not necessarily symmetric subspace. 9–11 Recently, the effect of phase synchronization of chaotic systems has been described theoretically 12,13 and observed experimentally. 14 It appears in autonomous continuous-time oscillators, where one can introduce the notions of the am- plitude and the phase even for chaotic motions. Roughly speaking, the amplitude corresponds to a coordinate on a Poincare ´ surface of section, and the phase increases by 2 during the motion between the cross-sections. 15 The ampli- tude is chaotic, while the phase is characterized by zero Lyapunov exponent phase shifts are marginal, like time shifts. The phase synchronization of chaotic system can be defined as the occurrence of a certain relation between the phases of interacting systems or between the phase of a system and that of an external force, while the amplitudes can remain chaotic and are, in general, uncorrelated. This relation between the phases appears usually as frequency en- trainment. It can be easily observed also experimentally if one defines the mean frequency of chaotic oscillations as a number of maxima of the process per unit time more rigor- ously, one can introduce it as a number of iterations of the Poincare ´ mapping per unit time. If this frequency coincides or nearly coincides with the frequency of the external force, one can speak of frequency locking. Defined in this way, the phase synchronization appears to be a direct analog of phase locking of periodic oscillations. It describes the onset of long-range correlations in chaotic oscillations suppression of phase diffusion, and thus also corresponds to the appear- ance of certain order inside chaos. Different synchronization transitions can be character- ized with the help of the Lyapunov exponents. Because these are the transitions inside chaos, the largest Lyapunov expo- nent remains positive. The transition to complete synchroni- zation happens when a partial conditional Lyapunov expo- nent changes sign. The phase synchronization occurs when the zero Lyapunov exponent becomes negative. It is impor- tant, that these transitions occur in a chaotic environment, and therefore are not as ‘‘clean’’ as the order-chaos transi- tions. In fact, one has to consider these transitions statisti- cally, assuming some characteristic statistical properties of the underlying chaos. In this paper we exploit the analogy between synchroni- zation of periodic and chaotic oscillations to achieve deeper understanding of structural metamorphoses of strange attrac- tors at the phase synchronization transitions. Our approach is the investigation of phase-locking properties of unstable pe- riodic orbits embedded in strange attractor. 16 For each of this periodic orbits one can define phase-locking regions Arnold 680 Chaos 7 (4), 1997 1054-1500/97/7(4)/680/8/$10.00 © 1997 American Institute of Physics Downloaded 13 Apr 2001 to 141.89.178.57. Redistribution subject to AIP copyright, see http://ojps.aip.org/chaos/chocr.jsp