Generalized synchronization in mutually coupled oscillators and complex networks

Abstract

We introduce a novel concept of generalized synchronization, able to encompass the setting of collective synchronized behavior for mutually coupled systems and networking systems featuring complex topologies in their connections. The onset of the synchronous regime is confirmed by the dependence of the system's Lyapunov exponents on the coupling parameter. The presence of a generalized synchronization regime is verified by means of the nearest neighbor method.

Global generalized synchronization in networks of different time-delay systems

“…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].…”

“…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 neglected, except for the important study of Zheng etal [14] on low-dimensional systems without delay and without a substantial difference in their fractal dimension. Further, depending on the relation between PS and GS [26] in such systems, which remains unclear, our understanding on their evolutionary mechanism, dynamical and functional behavior may need to be reinvestigated.…”

“…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.…”

“…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].…”

Generalized synchronization of coupled chaotic systems

“…For the unidirectionally coupled systems the evolution of drive system does not depend on the evolution of response system, but for bidirectionally or mutually coupled systems the state variables of each system will depend on the state variables of the other system. So, for such a case the functional relation y = φ(x) is to be modified to the form [15,16],…”