881 ISSN 1064-2269, Journal of Communications Technology and Electronics, 2007, Vol. 52, No. 8, pp. 881–890. © Pleiades Publishing, Inc., 2007. Original Russian Text © A.A. Koronovskii, O.I. Moskalenko, A.E. Hramov, 2007, published in Radiotekhnika i Elektronika, 2007, Vol. 52, No. 8, pp. 949–960. INTRODUCTION Being a main nonlinear effect important for appli- cations, chaotic synchronization has been studied intensively [1] in recent years. The development of the theory of dynamical chaos has enabled researchers to reveal a great number of various types of chaotic behavior of flow coupled dynamical systems [2–9]: phase synchronization, generalized synchronization, lag synchronization, complete synchronization, and time scale synchronization. Each of these types of synchronous chaotic dynamics exhibits specific fea- tures and can be diagnosed by means of specific meth- ods. The possible relationships between these types of synchronous behavior are actively discussed in the lit- erature. Various types of synchronization of chaotic oscillators can be interpreted as various manifesta- tions of common processes developing in coupled nonlinear systems (see, e.g., [8–12]). Analysis of the relationships between the different kinds of synchronous behavior of coupled chaotic oscillators, in particular, the relationship between gen- eralized synchronization and phase synchronization, is one of the most interesting problems. Initially, phase synchronization was believed to be a weaker type of chaotic behavior [13]. This concept means that, when unidirectionally coupled chaotic oscillators exhibit generalized synchronization, phase synchronization is always observed, while phase synchronization may occur in the absence of generalized synchronization. However, it was shown later [14] that, depending on the mismatch of the control parameters of coupled cha- otic oscillators, phase synchronization may occur at values of the oscillators’ coupling parameter that are smaller than those necessary for formation of general- ized synchronization. 1 This result means that, on the plane of control parameters, there are regions where 1 Mechanisms responsible for such behavior of coupled oscillators are described in [9, 15]. generalized synchronization is observed and phase syn- chronization is not realized. In particular, it has been found for the coupled Rössler systems considered in [14] that, at small mismatches of chaotic oscillators (including the zero mismatch, which corresponds to identical oscillators), the value of the coupling parame- ter at which generalized synchronization is realized is approximately twice the value corresponding to larger mismatches of control parameters (see also [16]). For the other known types of chaotic synchronization (phase synchronization, lag synchronization, complete synchronization, and time-scale synchronization), the threshold of the synchronous regime (as a function of the mismatch parameter) exhibits antipodal behavior: As the mismatch of the systems’ control parameters decreases, the value of the coupling parameter at which the corresponding synchronous regime is formed decreases. Thus, in this context, the regime of complete synchronization differs from the other types of chaotic synchronization. Moreover, this specific feature contra- dicts the seemingly evident statement that the lesser the mismatch between systems, the easier their synchroni- zation and the smaller the coupling parameter neces- sary for synchronization. The purpose of this study is to reveal mechanisms that provide for generalized synchronization of unidi- rectionally coupled chaotic oscillators. 1. GENERALIZED SYNCHRONIZATION AND THE MODIFIED-SYSTEM APPROACH Consider interacting unidirectionally coupled drive and response chaotic oscillators (t) and (t) described by the equalities (1) x d x r x ˙ d t () G x d t () ( ) , = x ˙ r t () H x r t () ( ) ε A x d t () x r t () – ( ) , + = The Threshold of Generalized Synchronization of Chaotic Oscillators A. A. Koronovskii, O. I. Moskalenko, and A. E. Hramov Received March 31, 2006 Abstract—The behavior of two unidirectionally coupled chaotic oscillators exhibited at the threshold of gen- eralized chaotic synchronization is considered. The modified-system approach is applied to explain physical mechanisms of formation of this regime in the cases of large and small mismatches of interacting systems. PACS numbers: 05.45.-a, 05.45.Xt DOI: 10.1134/S1064226907080086 DYNAMICAL CHAOS IN RADIOPHYSICS AND ELECTRONICS