Synchronization in a population of globally coupled chaotic oscillators

– We demonstrate synchronization transition in a large ensemble of non-identical chaotic oscillators, globally coupled via the mean field. We show that this coherent behaviour is due to synchronization of phases of these oscillators, while their amplitudes remain chaotic. Two types of transition, depending on the phase coherence properties of the individual systems, are described. A number of physical, chemical and biological systems can be viewed as large ensembles of weakly interacting non-identical oscillators [1]. One of the most popular models here is an ensemble of globally coupled non-linear oscillators. Such systems appear in the studies of Josephson junction arrays [2], oscillatory neuronal systems [3], multimode lasers [4], chargedensity waves [5], etc. Investigations of ensembles of non-linear continuous-time oscillators have revealed such interesting phenomena as clustering [6], existence of splay states [7], finite-size– induced transition [8], dephasing and bursting [9] and collective chaotic behaviour [6], [10]. A non-trivial transition to self-synchronization in a population of periodic oscillators with different natural frequencies coupled through a mean field has been described by Kuramoto [11]. In this system, as the coupling parameter increases, a sharp transition is observed for which the mean-field intensity serves as an order parameter. This transition owes to a mutual synchronization of the oscillators, so that their fields become coherent (i.e. their phases are locked), thus producing a macroscopic mean field. In its turn, this field acts on the individual oscillators, locking their phases, so that the synchronous state is self-sustained. Different aspects of this transition have been studied in [12], where also an analogy with a second-order phase transition has been exploited. In this letter we describe self-synchronization transitions in a population of chaotic systems. We explain this by the recently found phenomenon of phase synchronization of chaotic oscillators [13]. As a basic model we consider a population of non-identical Rössler oscillators  ẋi = −ωiyi − zi + εX, ẏi = ωixi + ayi, żi = 0.4 + zi(xi − 8.5), (1) (∗) A. von Humboldt Fellow. Permanent address: Mech. Eng. Res. Institute, Russian Academy of Sciences, Moscow, Russia. (∗∗) Homepage: www.agnld.uni-potsdam.de. c © Les Editions de Physique 166 EUROPHYSICS LETTERS 0.00 0.05 0.10 0.15 Coupling strength ε 10 -3 10 -2 10 -1 10 0 10 1 10 2 < (X -< X > ) 2 > a) b)