Global generalized synchronization in networks of different time-delay systems

Abstract

Abstract. -We show that global generalized synchronization (GS) exists in structurally different time-delay systems, even with different orders, with quite different fractal (Kaplan-Yorke) dimensions, which emerges via partial GS in symmetrically coupled regular networks. We find that there exists a smooth transformation in such systems, which maps them to a common GS manifold as corroborated by their maximal transverse Lyapunov exponent. In addition, an analytical stability condition using the Krasvoskii-Lyapunov theory is deduced. This phenomenon of GS in strongly distinct systems opens a new way for an effective control of pathological synchronous activity by means of extremely small perturbations to appropriate variables in the synchronization manifold.Synchronization is a ubiquitous nonlinear phenomenon serving as a platform for information processing in diverse natural and man made systems [1,2]. It has been investigated mainly in identical systems and in systems with parameter mismatch, with rare exceptions of distinctly (structurally) nonidentical systems [1,2]. However, in reality very often synchronization emerges in distinctly nonidentical systems such as respiratory arrhythmia between cardiovascular and respiratory systems [3], visual and motor systems [4], paced maternal breathing on fetal [5], different populations of species [6,7], in epidemics [8], in climatology [9], and many more. Considering the coherent coordination of living systems involving multiple organs such as brain, heart, lungs, limbs, etc., or machines consisting of distinct parts, cooperative evolution of distinct and often time-delayed systems is essential and challenging.Among various kinds of synchronization admitted by coupled nonlinear dynamical systems [1,2], the intricate phenomenon of generalized synchronization (GS) refers to (static) functional relationship between interacting systems [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. While the phenomenon of GS has been well understood in unidirectionally coupled systems [10][11][12][13], it remain still in its infancy in bidirectionally coupled systems and in particular there exists only very limited results on GS, even in systems with parameter mismatches [14][15][16][17][18][19][20][21][22][23], and particularly in structurally different (nonidentical) systems with different fractal dimensions [24]. Thus, in general, the notion of GS in mutually coupled systems needs much indepth investigation and in particular in distinctly different systems with different fractal dimensions involving time-delay. Indeed, recent investigations have revealed that GS is essentially more likely to occur in complex networks (even with identical nodes) [15] due to the large heterogeneity (degree distribution) of many natural networks [21].It is important to recall that the above mentioned studies [3][4][5][6][7][8][9] have demonstrated only phase synchronization (PS) among such distinctly different complex systems, while the natural choice of GS in them has been largely negle...

Emergence of a common generalized synchronization manifold in network motifs of structurally different time-delay systems

“…In line with the above discussion, we have reported briefly the existence of GS in symmetrically coupled networks of structurally different scalar one-dimensional time-delay systems using the auxiliary system approach [42]. In this paper, we will extend our investigations to non-scalar, higher dimensional heterogeneous time-delay systems to examine whether GS can still persist between strongly heterogeneous systems (with different orders) and to understand the underlying dynamical transitions.…”

Attraction of Li–Yorke chaos by retarded SICNNs

“…The presence of the synchronization manifold y ¼ ϕðxÞ in the drive-response systems is mostly investigated by numerical analyses [33,67,70]. The concept of generalized synchronization for coupled systems with delay was considered in [72]. In the present study, we suggest an easy theoretical approach to verify the presence of synchronization based on the exponential convergence of outputs.…”

“…The second way is the extension of chaos from one network to another. One can consider the synchronization of chaos [63][64][65][66][67][68][69][70][71][72][73][74] within the scope of the latter way. However, synchronization of chaos relies deeply on its description as well as on the verification of asymptotic closeness between the outputs.…”

“…This type of asymptotic relation is strong and to weaken it, one should consider the theory of generalized synchronization [33,[67][68][69][70][71][72]. In this theory, the previous relation is replaced by lim t-1 J yðtÞÀ ϕðxðtÞÞJ ¼ 0, where ϕ is a transformation.…”

“…This also provides a contribution to the chaos theory. Moreover, we analyze the relation between generalized synchronization [33,[67][68][69][70][71][72] and our approach about the chaotification of neural systems in a detailed form, and such discussions have never been reported before for SICNNs in the literature. In Section 4, we take into account retarded SICNNs with external inputs in the form of relay functions.…”