Generalized synchronization in relay systems with instantaneous coupling

Abstract

We demonstrate the existence of generalized synchronization in systems that act as mediators between two dynamical units that, in turn, show complete synchronization with each other. These are the so-called relay systems. Specifically, we analyze the Lyapunov spectrum of the full system to elucidate when complete and generalized synchronization appear. We show that once a critical coupling strength is achieved, complete synchronization emerges between the systems to be synchronized, and at the same point, generalized synchronization with the relay system also arises. Next, we use two nonlinear measures based on the distance between phase-space neighbors to quantify the generalized synchronization in discretized time series. Finally, we experimentally show the robustness of the phenomenon and of the theoretical tools here proposed to characterize it. Synchronization is a common phenomenon in a diversity of natural and technological systems [1]. Synchrony, however, is not always achieved spontaneously, and reaching or maintaining a synchronous state often requires an external action. An elegant way to enhance synchronization is the use of relay units between the systems to be synchronized [see Fig. 1(a)]. Relay synchronization (RS) consists of achieving complete synchronization (CS) of two dynamical systems by indirect coupling through a relay unit, whose dynamics does not necessary join the synchronous state. RS is especially useful in bidirectionally coupled systems with a certain delay in the coupling line. In these cases, indeed, the coupling delay may induce instability of the synchronous state [2], which can be restored again thanks to a relay system. Lasers [3,4] and electronics circuits [5] have been the benchmark for experimental demonstration of the feasibility of RS, showing its robustness against noise or parameter mismatch. In semiconductor lasers, for instance, zero-lag synchronization between two delaycoupled oscillators can be achieved by relaying the dynamics via a third mediating element, which surprisingly lags behind the synchronized outer elements. With electronic circuits, RS has been used as a technique for transmitting and recovering encrypted messages, which can be sent bidirectionally and simultaneously [6]. Apart from its technological applications, RS has also been proposed as a possible mechanism at the basis of isochronous synchronization between distant areas of the brain [7]. Despite such evidence of RS, there are still open questions of a fundamental nature. The main issue is to characterize properly the relationship, established in RS, between the dynamics of the relay system and that of the synchronized systems. When a certain delay is introduced in the coupling lines, lag synchronization has been reported [3,4]. Nevertheless, relay units may have certain parameter mismatch [8] or even be completely different systems [5], thus having dynamics with unclear a priori relationship with the systems they are synchronizing.In this paper, we give evidence that RS in fact corresponds to t...

Diversity of coupled oscillators can enhance their synchronization

“…Heterogeneous units with both amplitude and phase dynamics profit from their amplitude dynamics [which is neglected in phase (Kuramoto) oscillators] to lower the synchronization threshold. Further, studies on synchronization in networks of oscillators indirectly coupled through a medium (the so-called relay and remote synchronization) highlighted the role of heterogeneity in inducing or enhancing synchronization [23][24][25][26][27][28][29]. Here, we also emphasize the enhancing effect of heterogeneity on synchronization.…”

“…Many factors affect the synchrony of coupled oscillators, e.g., the coupling strength [17], the number of coupled oscillators [18], the network topology [19,20], or the heterogeneity [21][22][23][24][25][26][27][28][29]. In [21], the authors reported an enhancement of synchrony in an array of Josephson junctions with disorder, which refers to the situation in which the elements of the array are not identical.…”

Multistability Analysis and Function Projective Synchronization in Relay Coupled Oscillators

“…Some demonstrations of chaotic masking of communication as well as selected secure communication lines were observed. Gutiérrez et al in [22] showed that a generalized synchronization (GS) in relay systems with instantaneous coupling could be obtained. The authors proved the existence of GS in unidirectional coupled units (drive system → response system) by checking the ability of the response system to react identically to different initial conditions of the same driver system, which can be quantified by evaluating the mutual false nearest neighbors or by measuring the conditional Lyapunov exponents.…”

Synchronization of two Rössler systems with switching coupling

“…The phenomenon is even more varied when more than two units interact according to a pattern of connectivity through which they share information about their current state: complete synchronization [6], phase synchronization [7], cluster synchronization [8], partial synchronization [9], chimera states [10,11], relay synchronization [12], and remote synchronization [13] are only examples of the behaviors observed in a network of coupled oscillators. For complete synchronization as investigated in this paper, a series of well-known results provides the conditions for the onset of the synchronous motion.…”