Transition from anticipatory to lag synchronization via complete synchronization in time-delay systems

Senthilkumar, D. V.; Lakshmanan, M.

doi:10.1103/physreve.71.016211

Cited by 52 publications

(49 citation statements)

References 45 publications

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“…We first consider the following unidirectionally coupled drive x 1 (t) and response x 2 (t) systems, which we have recently studied in detail in [22,23,24],…”

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confidence: 99%

“…Phase is calculated using the Poincaré method [4,5] after a new transformation of attractors of the time-delay systems, which looks then like a smeared limit cycle. The existence of PS and generalized synchronization (GS) in coupled time-delay systems is characterized by recently proposed methods based on recurrence quantification analysis and also in terms of Lyapunov exponents of the coupled time-delay systems.We first consider the following unidirectionally coupled drive x 1 (t) and response x 2 (t) systems, which we have recently studied in detail in [22,23,24], x 1 (t) = −ax 1 (t) + b 1 f (x 1 (t − τ )), (1a) x 2 (t) = −ax 2 (t) + b 2 f (x 2 (t − τ )) + b 3 f (x 1 (t − τ )), (1b) where b 1 , b 2 and b 3 are constants, a > 0, τ is the delay time and f (x) is the piece-wise linear equation of the form …”

mentioning

confidence: 99%

“…We first consider the following unidirectionally coupled drive x 1 (t) and response x 2 (t) systems, which we have recently studied in detail in [22,23,24], x 1 (t) = −ax 1 (t) + b 1 f (x 1 (t − τ )), (1a) x 2 (t) = −ax 2 (t) + b 2 f (x 2 (t − τ )) + b 3 f (x 1 (t − τ )), (1b) where b 1 , b 2 and b 3 are constants, a > 0, τ is the delay time and f (x) is the piece-wise linear equation of the form…”

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confidence: 99%

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Phase synchronization in time-delay systems

2006

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Though the notion of phase synchronization has been well studied in chaotic dynamical systems without delay, it has not been realized yet in chaotic time-delay systems exhibiting non-phase coherent hyperchaotic attractors. In this article we report the first identification of phase synchronization in coupled time-delay systems exhibiting hyperchaotic attractor. We show that there is a transition from non-synchronized behavior to phase and then to generalized synchronization as a function of coupling strength. These transitions are characterized by recurrence quantification analysis, by phase differences based on a new transformation of the attractors and also by the changes in the Lyapunov exponents. We have found these transitions in coupled piece-wise linear and in Mackey-Glass time-delay systems. Synchronization is a natural phenomenon that one encounters in daily life. Since the identification of chaotic synchronization [1, 2, 3], several papers have appeared in identifying and demonstrating basic kinds of synchronization both theoretically and experimentally (cf. [4,5]). Among them, chaotic phase synchronization (CPS) refers to the coincidence of characteristic time scales of the coupled systems, while their amplitudes of oscillations remain chaotic and often uncorrelated. Phase synchronization (PS) plays a crucial role in understanding the behavior of a large class of weakly interacting dynamical systems in diverse natural systems. Examples include circadian rhythm, cardio-respiratory systems, neural oscillators, population dynamics, electrical circuits, etc [4,5,6].The notion of CPS has been investigated so far in oscillators driven by external periodic force [7,8], chaotic oscillators with different natural frequencies and/or with parameter mismatches [9,10,11,12], arrays of coupled chaotic oscillators [13,14] and also in essentially different chaotic systems [15,16]. On the other hand PS in nonlinear time-delay systems, which form an important class of dynamical systems, have not yet been identified and addressed. A main problem here is to define even the notion of phase in time-delay systems due to the intrinsic multiple characteristic time scales in these systems. Studying PS in such chaotic time-delay systems is of considerable importance in many fields, as in understanding the behavior of nerve cells (neuroscience), where memory effects play a prominent role, in pathological and physiological studies, in ecology, in lasers etc * Electronic address: lakshman@cnld.bdu.ac.in † Electronic address: jkurths@gmx.de [4,5,6,17,18,19,20,21]. In this paper, we report the first identification of phase synchronization (PS) in nonidentical time-delay systems in the hyperchaotic regime with non-phase coherent attractors with unidirectional nonlinear coupling. We will show the entrainment of phases of a coupled piecewise linear time-delay system for weak coupling from the non-synchronized state. Phase is calculated using the Poincaré method [4,5] after a new transformation of attractors of the time-delay systems, w...

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“…We first consider the following unidirectionally coupled drive x 1 (t) and response x 2 (t) systems, which we have recently studied in detail in [22,23,24],…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Phase synchronization in time-delay systems

2006

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“…Time-delay systems form an important class of dynamical systems and recently they are receiving central importance in investigating the phenomenon of chaotic synchronizations in view of their infinite dimensional nature and feasibility of experimental realization [24,25,26,27]. While the concept of GS has been well established in low dimensional systems, it has not yet been studied in detail in coupled time-delay systems and only very few recent studies have been dealt with GS in time-delay systems [24,25].…”

Section: Introductionmentioning

confidence: 99%

“…Following Krasovskii-Lyapunov functional approach, we define a positive definite Lyapunov functional of the form [27,32,33] (details of stability analysis are given in appendix A)…”

Section: A Stability Conditionmentioning

confidence: 99%

Intermittency transition to generalized synchronization in coupled time-delay systems

Senthilkumar

Lakshmanan

2007

Phys. Rev. E

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In this paper, we report the nature of transition to generalized synchronization (GS) in a system of two coupled scalar piecewise linear time-delay systems using the auxiliary system approach. We demonstrate that the transition to GS occurs via on-off intermittency route and also it exhibits characteristically distinct behaviors for different coupling configurations. In particular, the intermittency transition occurs in a rather broad range of coupling strength for error feedback coupling configuration and in a narrow range of coupling strength for direct feedback coupling configuration. It is also shown that the intermittent dynamics displays periodic bursts of period equal to the delay time of the response system in the former case, while they occur in random time intervals of finite duration in the latter case. The robustness of these transitions with system parameters and delay times has also been studied for both linear and nonlinear coupling configurations. The results are corroborated analytically by suitable stability conditions for asymptotically stable synchronized states and numerically by the probability of synchronization and by the transition of subLyapunov exponents of the coupled time-delay systems. We have also indicated the reason behind these distinct transitions by referring to unstable periodic orbit theory of intermittency synchronization in low-dimensional systems.

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Transition from Phase to Generalized Synchronization

Lakshmanan

Senthilkumar

2010

Dynamics of Nonlinear Time-Delay Systems

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Transition from anticipatory to lag synchronization via complete synchronization in time-delay systems

Cited by 52 publications

References 45 publications

Phase synchronization in time-delay systems

Phase synchronization in time-delay systems

Intermittency transition to generalized synchronization in coupled time-delay systems

Transition from Phase to Generalized Synchronization

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