Transition from phase to generalized synchronization in time-delay systems

Senthilkumar, D. V.; Lakshmanan, M.; Kurths, Jürgen

doi:10.1063/1.2911541

Cited by 47 publications

(49 citation statements)

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“…Based on our generalized formulation, we will specifically demonstrate the existence of global GS via partial GS in symmetrically coupled networks, which even consist of distinctly different time-delay systems (Mackey-Glass [28], piecewise linear [29], threshold nonlinearity [30] and Ikeda [31]). It is important to note that such a phenomenon also occurs among delay systems of different orders, namely, Ikeda and a second order Hopfield neural network [32] as well as Mackey-Glass and a third order plankton model [33] with multiple delays.…”

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confidence: 99%

“…If the two systems are in CS state the CC = 1, otherwise CC < 1. Further, the existence of PS in highly non-phase-coherent hyperchaotic attractors of time-delay systems are characterized by the value of the correlation of probability of recurrence (CPR ≈ 1) [29,34].…”

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confidence: 99%

“…of the PWL time-delay system [29], respectively. The parameters of the MG systems are chosen as α 1 = 0.5, β 1 = 1.0, and τ 1 = 8.5 and for the PWL systems we choose α 2 = 1.0, β 2 = 1.2 and τ 2 = 10.0, exhibiting hyperchaotic attractors with two (D KY = 2.957) and three (D KY = 4.414) positive Lyapunov exponents (LEs) [2], respectively.…”

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confidence: 99%

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Global generalized synchronization in networks of different time-delay systems

Senthilkumar¹,

Suresh²,

Lakshmanan³

et al. 2013

EPL

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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...

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Global generalized synchronization in networks of different time-delay systems

Senthilkumar¹,

Suresh²,

Lakshmanan³

et al. 2013

EPL

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“…However, nodes in different networks usually have different dynamics ͑parameter mismatch or structural discrepancy͒, while the two networks may still behave in a synchronous way. This kind of synchronization is called generalized synchronization, [24][25][26][27] which represents another degree of coherence. For instance, in the aforementioned predator and prey networks, predators and preys may finally reach a synchronous state even though they have entirely different behaviors ͑even individuals inside a network may behave in quite diverse ways͒.…”

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confidence: 99%

Generalized outer synchronization between complex dynamical networks

Wu¹,

Zheng²,

Zhou³

2009

Chaos: An Interdisciplinary Journal of Nonlinear Science

157

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In this paper, the problem of generalized outer synchronization between two completely different complex dynamical networks is investigated. With a nonlinear control scheme, a sufficient criterion for this generalized outer synchronization is derived based on Barbalat’s lemma. Two corollaries are also obtained, which contains the situations studied in two lately published papers as special cases. Numerical simulations further demonstrate the feasibility and effectiveness of the theoretical results.

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“…For this purpose we will consider the nonlinear functions f (x) as the piecewise linear function, which has been studied in detail recently [16,18,19],…”

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Stability of synchronization in coupled time-delay systems using Krasovskii-Lyapunov theory

2009

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Stability of synchronization in unidirectionally coupled time-delay systems is studied using the Krasovskii-Lyapunov theory. We have shown that the same general stability condition is valid for different cases, even for the general situation (but with a constraint) where all the coefficients of the error equation corresponding to the synchronization manifold are time-dependent. These analytical results are also confirmed by numerical simulation of paradigmatic examples. Synchronization of coupled chaotic dynamical systems is an active area of research in different branches of science and technology [1,2]. Different kinds of chaos synchronization have been reported both theoretically and experimentally since its discovery in coupled chaotic oscillators [3]. Recent studies on synchronization have been focused on coupled time-delay systems with or without time-delay coupling because of its intrinsic nature of generating high dimensional chaotic signals [4] and the ease of experimental realization of these systems [5]. These systems have potential applications in secure communication, cryptography, controlling, long term prediction, optimization of nonlinear system performance, modelling brain activity, pattern recognition, etc. (cf. [4, 5, 6, 7, 8, 9, 10]). Specifically, exploiting synchronization, communication with chaotic lasers was demonstrated in [7] and digital information at gigabyte rates was transferred successfully [8] exploiting time-delayed feedback to generate high-dimensional, high-capacity waveforms at high bandwidths. With this kind of applications of high-dimensional chaotic signals of time-delay systems, it becomes extremely important to establish conditions under which synchronized states are asymptotically stable in coupled time-delay systems. For this purpose, the Krasvoskii-Lyapunov theory has become an extremely useful tool [9,10,11,16,17].On the other hand, recently, it has been pointed out * Electronic address: skumar@cnld.bdu.ac.in † Electronic address: Juergen.Kurths@pik-potsdam.de ‡ Electronic address: lakshman@cnld.bdu.ac.in that the Krasovskii-Lyapunov theory is not suitable for a more general case where the error equation corresponding to the synchronization manifold is time-dependent; especially when all its coefficients are time-dependent [11]. In this contribution we will show that the KrasovskiiLyapunov theory is not restricted to rather special cases discussed in the literature so far, but that it can be exploited as a powerful tool in identifying the synchronization thresholds and the stability of synchronization in general coupled time-delay systems. In particular, we will show that the same general stability condition resulting from the Krasovskii-Lyapunov theory is indeed valid for the general case, where the coefficients are time dependent. This general situation is very important in many applications such as synchronization via dynamical relaying [12], for instance, synchronization of neural activity in brain with complex functional architecture has been shown to underlie ...

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Transition from phase to generalized synchronization in time-delay systems

Cited by 47 publications

References 58 publications

Global generalized synchronization in networks of different time-delay systems

Global generalized synchronization in networks of different time-delay systems

Generalized outer synchronization between complex dynamical networks

Stability of synchronization in coupled time-delay systems using Krasovskii-Lyapunov theory

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