Generalized synchronization onset ( ∗ )

– The behavior of two unidirectionally coupled chaotic oscillators near the generalized synchronization onset has been considered. The character of the boundaries of the generalized synchronization regime has been explained by means of the modified system approach. Chaotic synchronization is one of the fundamental nonlinear phenomena actively studied recently [1]. Several different types of chaotic synchronization of coupled oscillators, i.e. generalized synchronization (GS) [2], phase synchronization (PS) [1], lag synchronization (LS) [3], complete synchronization (CS) [4] and time scale synchronization (TSS) [5] are well known. There are also attempts to find unifying framework for chaotic synchronization of coupled dynamical systems [6–9]. One of the interesting and intricate types of the synchronous behavior of unidirectionally coupled chaotic oscillators is the generalized synchronization. The presence of GS between the response x r (t) and drive x d (t) chaotic systems means that there is some functional relation x r (t) = F[x d (t)] between system states after the transient finished. This functional relation F[·] may be smooth or fractal. According to the properties of this relation, GS may be divided into the strong synchronization and weak synchronization, respectively [10]. There are several methods to detect the presence of GS between chaotic oscillators, such as the auxiliary system approach [11] or the method of nearest neighbors [2, 12]. It is also possible to calculate the conditional Lyapunov exponents (CLEs) [10, 13] to detect GS. The regimes of LS and CS are also the particular cases of GS. One of the interesting aspects of the generalized synchronization study is the analysis of the onset of this regime. In particular, the intermittent behavior is known to be revealed on the onset of GS [14,15] as well as in case of the lag [3,16,17] or phase [18–20] synchronization. At the same time, there are known examples of the unidirectionally coupled chaotic systems for which the location of the generalized synchronization onset on the " parameter mismatch — coupling strength " plane differs radically from the other synchronization types. Indeed, for two unidirectionally coupled Rössler oscillators with identical control parameters the value of the coupling strength corresponding to the onset of GS is twice as much as for the same