We investigate synchronization between two unidirectionally linearly coupled chaotic non-identical time-delayed systems and show that parameter mismatches are of crucial importance to achieve synchronization. We establish that independent of the relation between the delay time in the coupled systems and the coupling delay time, only retarded synchronization with the coupling delay time is obtained. We show that with parameter mismatch or without it neither complete nor anticipating synchronization occurs. We derive existence and stability conditions for the retarded synchronization manifold. We demonstrate our approach using examples of the Ikeda and Mackey Glass models. Also for the first time we investigate chaos synchronization in time-delayed systems with variable delay time and find both existence and sufficient stability conditions for the retarded synchronization manifold with the coupling-delay lag time. Also for the first time we consider synchronization between two unidirectionally coupled chaotic multi-feedback Ikeda systems and derive existence and stability conditions for the different anticipating, lag, and complete synchronization regimes. 1 of interacting systems, y(t) = x(t) [1]. Generalized synchronization is defined as the presence of some functional relation between the states of response and drive, i.e. y(t) = F (x(t)) [3]. Phase synchronization means entrainment of phases of chaotic oscillators, nΦ x − mΦ y = const, (n and m are integers) whereas their amplitudes remain chaotic and uncorrelated [4]. Lag synchronization for the first time was introduced by Rosenblum et al. [5] under certain approximations in studying synchronization between bi-directionally coupled systems described by the ordinary differential equations (no intrinsic delay terms) with parameter mismatches:Anticipating synchronization [6-8] also appears as a coincidence of shifted-in-time states of two coupled systems, but in this case the driven system anticipates the driver, y(t) = x(t + τ ) or x = y τ ,τ > 0. An experimental observation of anticipating synchronization in external cavity laser diodes [9] has been reported recently, see also [10] for the theoretical interpretation of the experimental results. The concept of inverse anticipating synchronization x = −y τ is introduced in [11].Due to finite signal transmission times, switching speeds and memory effects time-delayed systems are ubiquitous in nature, technology and society [12]. Therefore the study of synchronization phenomena in such systems is of high practical importance. Time-delayed systems are also interesting because the dimension of their chaotic dynamics can be made arbitrarily large by increasing their delay time. From this point of view these systems are especially appealing for secure communication schemes [13].Role of parameter mismatches in synchronization phenomena is quite versatile. In certain cases parameter mismatches are detrimental to the synchronization quality: in the case of small parameter mismatches the synchronization error does not dec...
We investigate inverse retarded synchronization between two uni-directionally linearly coupled chaotic non-identical Ikeda models and show that parameter mismatches are of crucial importance to achieve synchronization. We establish that independent of the relation between the delay time in the coupled systems and the coupling delay time, only inverse retarded synchronization with the coupling delay time is obtained. We derive existence and stability conditions for the inverse retarded synchronization manifold. We show that with parameter mismatch or without it neither inverse complete nor inverse anticipating synchronization occurs. Numerical simulations fully support the analytical approach.
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