Synchronization of mutually coupled chaotic systems D. Y. Tang and N. R. Heckenberg Physics Department, The University of Queensland, Brisbane, Qld 4072, Australia Received 3 December 1996 We report on the experimental observation of both basic frequency locking synchronization and chaos synchronization between two mutually coupled chaotic subsystems. We show that these two kinds of synchro- nization are two stages of interaction between coupled chaotic systems. In particular the chaos synchronization could be understood as a state of phase locking between coupled chaotic oscillations. S1063-651X9703806-3 PACS numbers: 05.45.+b, 05.40.+j, 42.65.Sf, 42.60.Mi I. INTRODUCTION Synchronization between periodic oscillations of mutually coupled dynamical systems is a well-known phenomenon. Generally, when the oscillation frequencies of two coupled periodic systems are within a certain range called the locking range, the frequencies will automatically lock to a mutual value and consequently both systems oscillate with the same frequency. After frequency locking between their oscilla- tions, we say they are synchronized. Since the oscillation of a periodic system is regular, the effect of synchronization between them is clear and unique. The dynamics of a system can also be chaotic. Recently, there has been great interest in synchronization between cha- otic systems 1–11. In contrast to the oscillation of a peri- odic system, the oscillation of a chaotic system is dynami- cally intrinsically unstable: Its oscillation depends sensitively on the initial conditions and varies with time. Due to this special character of chaotic systems, there have devel- oped different versions of the definition of synchronization between chaotic oscillations. Mostly, synchronization of cha- otic oscillations is defined as the complete coincidence of the trajectories of the coupled individual chaotic systems sub- systems in the phase space 5. According to this definition, under the synchronization the dynamics of two coupled sys- tems subsystems become exactly the same, even though without coupling they are not dynamically identical. This kind of synchronization was called ‘‘chaos synchronization’’ and has been observed in coupled chaotic systems 2–10. Another definition takes account of the behavior of some chaotic attractors that in their power spectrum a basic fre- quency can be distinguished and defines synchronization of chaotic oscillations as meaning merely that their basic fre- quencies are locked together 1. We refer to this synchroni- zation as the ‘‘basic frequency locking synchronization.’’ A chaotic attractor whose power spectrum possesses this be- havior is called a ‘‘phase coherent attractor’’ 12–14.A major property of these chaotic attractors is that their chaotic behavior results mainly from chaotic amplitude modulation and the contribution from the chaotic phase modulation is very weak. Consequently, there exists a predominant fre- quency in the chaotic oscillation of these attractors. A char- acteristic of this synchronization is that the average oscilla- tion frequencies of the coupled chaotic systems are entrained, while the amplitudes of the oscillations remain chaotic and independent. An advantage of this definition is that, like the synchronization between periodic systems, the mechanism of synchronization is clear. Rosenblum, Pik- ovsky, and Kurths have reported an observation of ‘‘phase’’ synchronization between coupled chaotic systems 15. However, the synchronization they referred to seems to be exactly the basic frequency locking synchronization. Strictly speaking, despite the fact that the phase fluctuation of the oscillation of a phase coherent strange attractor is very small, under the basic frequency locking, the instantaneous phases of these coupled systems are not locked. In this paper we report on an experimental observation of both basic frequency locking synchronization and chaos syn- chronization between two mutually coupled chaotic sub- systems. We show that, like coupled periodic systems, coupled chaotic systems have a tendency to engage in mutual synchronization in the form of basic frequency locking or chaotic phase locking. Our experimental results demonstrate that the two observed synchronizations are in fact the two natural stages of interaction between coupled chaotic sys- tems. In particular the chaos synchronization between cha- otic systems could be physically understood as a result of phase locking between coupled chaotic oscillations. II. EXPERIMENT AND RESULTS Our experimental system is an optically pumped NH 3 bi- directional ring laser. Details of the configuration of the laser were reported in 16. This laser was chosen for the present experimental study because it lases in two modes simulta- neously and these two modes are mutually coupled. One mode field of the laser propagates in the same direction as the pump laser beam and is called the forward mode; the other mode field propagates against the direction of the pump laser beam and is called the backward mode. Due to the optical pumping of the laser that selectively excites NH 3 molecules with the same longitudinal velocity, the gain bandwidth of the laser is very narrow, limited by homoge- neous broadening. Both modes of the laser share the same population inversion, while, because of the Doppler effect resulting from the motion of the excited molecules, the ef- fective gain frequency of each mode is different. The fre- quency difference between them is determined by the pump frequency detuning relative to the NH 3 absorption line cen- ter. This relation between the two laser modes results in a PHYSICAL REVIEW E JUNE 1997 VOLUME 55, NUMBER 6 55 1063-651X/97/556/66186/$10.00 6618 © 1997 The American Physical Society