Phase synchronization in time-delay systems

Abstract

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...

Anticipating synchronization of chaotic Lur’e systems

“…Cherrier et al 14 and Cruz 15 used a time-delayed Chua-type oscillator for studying secure communication schemes. Finally, Senthilkumar et al 16 discussed phase synchronization in time-delay systems.…”

Zero-lag synchronization in coupled time-delayed piecewise linear electronic circuits

“…Synchronizing such time-delay systems is very challenging and has potential applications in diverse areas involving physical, chemical, biological, neurological and electrical systems [13][14][15][16][17]. Different types of synchronization have been recently observed numerically along with experimental evidence in coupled time-delay systems [18][19][20][21][22].…”

Intermittency transition to generalized synchronization in coupled time-delay systems

“…In this paper, we investigate the characteristic properties of nature of onset of GS from asynchronous state in unidirectionally coupled piece-wise linear time-delay systems exhibiting highly non-phase coherent hyperchaotic attractors [26]. We find that the onset of GS is preceded by on-off intermittency mechanism from the desynchronized state.…”

“…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].…”