Phase synchronization in time-delay systems

Phase coherence is not related to topology

Phase coherence is an important measure in nonlinear science. Whereas there is no generally accepted definition for phase and therefore for phase coherence, many works associate this feature with topological aspects of the systems, such as having a well-defined rotating center. Given the relevance of this concept for synchronization problems, one aim of this paper is to argue by means of a couple of counterexamples that phase coherence is not related to the topology of the attractor. A second aim is to introduce a phase-coherence measure based on recurrence plots, for which probabilities of recurrences for two different trajectories are similar for a phase-coherent system and dissimilar for non-phase-coherent systems. The measure does not require a phase variable defined a priori.

Generalized variable projective synchronization of time delayed systems

We study generalized variable projective synchronization between two unified time delayed systems with constant and modulated time delays. A novel Krasovskii-Lyapunov functional is constructed and a generalized sufficient condition for synchronization is derived analytically using the Lyapunov stability theory and adaptive techniques. The proposed scheme is valid for a system of n-numbers of first order delay differential equations. Finally, a new neural oscillator is considered as a numerical example to show the effectiveness of the proposed scheme.

The curvature index and synchronization of dynamical systems

We develop a quantity, named the curvature index, for dynamical systems. This index is defined as the limit of the average curvature of the trajectory during evolution, which measures the bending of the curve on an attractor. The curvature index has the ability to differentiate the topological change of an attractor, as its alterations exhibit the structural changes of a dynamical system. Thus, the curvature index may indicate thresholds of some synchronization regimes. The Rössler system and a time-delay system are simulated to demonstrate the effectiveness of the index, respectively.

Noise-enhanced phase synchronization in time-delayed systems

We investigate the phenomenon of noise-enhanced phase synchronization (PS) in coupled time-delay systems, which usually exhibit non-phase-coherent attractors with complex topological properties. As a delay system is essentially an infinite dimensional in nature with multiple characteristic time scales, it is interesting and crucial to understand the interplay of noise and the time scales in achieving PS. In unidirectionally coupled systems, the response system adjust all its time scales to that of the drive, whereas both subsystems adjust their rhythms to a single (main time scale of the uncoupled system) time scale in bidirectionally coupled systems. We find similar effects for both a common and an independent additive Gaussian noise.

Synchronization in a chaotic neural network with time delay depending on the spatial distance between neurons

The synchronization is investigated in a two-dimensional Hindmarsh-Rose neuronal network by introducing a global coupling scheme with time delay, where the length of time delay is proportional to the spatial distance between neurons. We find that the time delay always disturbs synchronization of the neuronal network. When both the coupling strength and length of time delay per unit distance (i.e., enlargement factor) are large enough, the time delay induces the abnormal membrane potential oscillations in neurons. Specifically, the abnormal membrane potential oscillations for the symmetrically placed neurons form an antiphase, so that the large coupling strength and enlargement factor lead to the desynchronization of the neuronal network. The complete and intermittently complete synchronization of the neuronal network are observed for the right choice of parameters. The physical mechanism underlying these phenomena is analyzed.

Synchronization transitions in coupled time-delay electronic circuits with a threshold nonlinearity

Experimental observations of typical kinds of synchronization transitions are reported in unidirectionally coupled time-delay electronic circuits with a threshold nonlinearity and two time delays, namely feedback delay τ(1) and coupling delay τ(2). We have observed transitions from anticipatory to lag via complete synchronization and their inverse counterparts with excitatory and inhibitory couplings, respectively, as a function of the coupling delay τ(2). The anticipating and lag times depend on the difference between the feedback and the coupling delays. A single stability condition for all the different types of synchronization is found to be valid as the stability condition is independent of both the delays. Further, the existence of different kinds of synchronizations observed experimentally is corroborated by numerical simulations and from the changes in the Lyapunov exponents of the coupled time-delay systems.

Experimental confirmation of chaotic phase synchronization in coupled time-delayed electronic circuits

We report the experimental demonstration of chaotic phase synchronization (CPS) in unidirectionally coupled time-delay systems using electronic circuits. We have also implemented experimentally an efficient methodology for characterizing CPS, namely, the localized sets. Snapshots of the evolution of coupled systems and the sets as observed from the oscilloscope confirming CPS are shown experimentally. Numerical results from different approaches, namely, phase differences, localized sets, changes in the largest Lyapunov exponents, and the correlation of probability of recurrence (C(CPR)) corroborate the experimental observations.

Practical time-delay synchronization of a periodically modulated self-excited oscillators with uncertainties

This paper studies time-delay synchronization of a periodically modulated Duffing Van der Pol (DVP) oscillator subjected to uncertainties with emphasis on complete synchronization. A robust adaptive response system is designed to synchronize with the uncertain drive periodically modulated DVP oscillator. Adaptation laws on the upper bounds of uncertainties are proposed to guarantee the boundedness of both the synchronization error and the estimated feedback coupling gains. Numerical results are presented to check the effectiveness of the proposed synchronization scheme. The results suggest that the linear and nonlinear terms in the feedback coupling play a complementary role in increasing the synchronization regime in the parameter space of the synchronization manifold. The proposed method can be successfully applied to a large variety of physical systems.

Global phase synchronization in an array of time-delay systems

We report the identification of global phase synchronization (GPS) in a linear array of unidirectionally coupled Mackey-Glass time-delay systems exhibiting highly non-phase-coherent chaotic attractors with complex topological structure. In particular, we show that the dynamical organization of all the coupled time-delay systems in the array to form GPS is achieved by sequential synchronization as a function of the coupling strength. Further, the asynchronous ones in the array with respect to the main sequentially synchronized cluster organize themselves to form clusters before they achieve synchronization with the main cluster. We have confirmed these results by estimating instantaneous phases including phase difference, average phase, average frequency, frequency ratio, and their differences from suitably transformed phase coherent attractors after using a nonlinear transformation of the original non-phase-coherent attractors. The results are further corroborated using two other independent approaches based on recurrence analysis and the concept of localized sets from the original non-phase-coherent attractors directly without explicitly introducing the measure of phase.

Synchronization in coupled time-delayed systems with parameter mismatch and noise perturbation

In this paper, a design of coupling and effective sufficient condition for stable complete synchronization and antisynchronization of a class of coupled time-delayed systems with parameter mismatch and noise perturbation are established. Based on the LaSalle-type invariance principle for stochastic differential equations, sufficient conditions guaranteeing complete synchronization and antisynchronization with constant time delay are developed. Also delay-dependent sufficient conditions for the case of time-varying delay are derived by using the Lyapunov approach for stochastic differential equations. Numerical examples fully support the analytical results.

Increased phase synchronization and decreased cerebral autoregulation during fainting in the young

Vasovagal syncope may be due to a transient cerebral hypoperfusion that accompanies frequency entrainment between arterial pressure (AP) and cerebral blood flow velocity (CBFV). We hypothesized that cerebral autoregulation fails during fainting; a phase synchronization index (PhSI) between AP and CBFV was used as a nonlinear, nonstationary, time-dependent measurement of cerebral autoregulation. Twelve healthy control subjects and twelve subjects with a history of vasovagal syncope underwent 10-min tilt table testing with the continuous measurement of AP, CBFV, heart rate (HR), end-tidal CO2 (ETCO2), and respiratory frequency. Time intervals were defined to compare physiologically equivalent periods in fainters and control subjects. A PhSI value of 0 corresponds to an absence of phase synchronization and efficient cerebral autoregulation, whereas a PhSI value of 1 corresponds to complete phase synchronization and inefficient cerebral autoregulation. During supine baseline conditions, both control and syncope groups demonstrated similar oscillatory changes in phase, with mean PhSI values of 0.58+/-0.04 and 0.54+/-0.02, respectively. Throughout tilt, control subjects demonstrated similar PhSI values compared with supine conditions. Approximately 2 min before fainting, syncopal subjects demonstrated a sharp decrease in PhSI (0.23+/-0.06), representing efficient cerebral autoregulation. Immediately after this period, PhSI increased sharply, suggesting inefficient cerebral autoregulation, and remained elevated at the time of faint (0.92+/-0.02) and during the early recovery period (0.79+/-0.04) immediately after the return to the supine position. Our data demonstrate rapid, biphasic changes in cerebral autoregulation, which are temporally related to vasovagal syncope. Thus, a sudden period of highly efficient cerebral autoregulation precedes the virtual loss of autoregulation, which continued during and after the faint.

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

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 the numerical simulation of paradigmatic examples.

Inverse synchronizations in coupled time-delay systems with inhibitory coupling

Transitions between inverse anticipatory, inverse complete, and inverse lag synchronizations are shown to occur as a function of the coupling delay in unidirectionally coupled time-delay systems with inhibitory coupling. We have also shown that the same general asymptotic stability condition obtained using the Krasovskii-Lyapunov functional theory can be valid for the cases where (i) both the coefficients of the Delta(t) (error variable) and Delta(tau)=Delta(t-tau) (error variable with delay) terms in the error equation corresponding to the synchronization manifold are time independent and (ii) the coefficient of the Delta term is time independent, while that of the Delta(tau) term is time dependent. The existence of different kinds of synchronization is corroborated using similarity function, probability of synchronization, and also from changes in the spectrum of Lyapunov exponents of the coupled time-delay systems.

Generalized projective synchronization in time-delayed systems: nonlinear observer approach

In this paper, we consider the projective-anticipating, projective, and projective-lag synchronization in a unified coupled time-delay system via nonlinear observer design. A new sufficient condition for generalized projective synchronization is derived analytically with the help of Krasovskii-Lyapunov theory for constant and variable time-delay systems. The analytical treatment can give stable synchronization (anticipatory and lag) for a large class of time-delayed systems in which the response system's trajectory is forced to have an amplitude proportional to the drive system. The constant of proportionality is determined by the control law, not by the initial conditions. The proposed technique has been applied to synchronize Ikeda and prototype models by numerical simulation.

Phase synchronization in tilted inertial ratchets as chaotic rotators

The phenomenon of phase synchronization for a particle in a periodic ratchet potential is studied. We consider the deterministic dynamics in the underdamped case where the inertia plays an important role since the dynamics can become chaotic. The ratchet potential is tilted due to a constant external force and is rocking by an external periodic forcing. This potential has to be tilted in order to obtain a rotator or self-sustained nonlinear oscillator in the absence of the external periodic forcing; this oscillator then acquires an intrinsic frequency that can be locked with the frequency of the external driving. We introduced an instantaneous linear phase, using a set of discrete time markers, and the associated average frequency, and show that this frequency can be synchronized with the frequency of the driving. We calculate Arnold tongues in a two-dimensional parameter space and discuss their implications for the chaotic transport in ratchets. We show that the local maxima in the current correspond to the borders of these Arnold tongues; in this way we established a link between optimal transport in ratchets and phase synchronization.

Synchronization of chaotic systems with delay using intermittent linear state feedback

This paper investigates the synchronization of coupled chaotic systems with time delay by using intermittent linear state feedback control. An exponential synchronization criterion is obtained by means of Lyapunov function and differential inequality method. Numerical simulations on the chaotic Ikeda and Lu systems are given to demonstrate the effectiveness of the theoretical results.

Universal occurrence of the phase-flip bifurcation in time-delay coupled systems

Recently, the phase-flip bifurcation has been described as a fundamental transition in time-delay coupled, phase-synchronized nonlinear dynamical systems. The bifurcation is characterized by a change of the synchronized dynamics from being in-phase to antiphase, or vice versa; the phase-difference between the oscillators undergoes a jump of pi as a function of the coupling strength or the time delay. This phase-flip is accompanied by discontinuous changes in the frequency of the synchronized oscillators, and in the largest negative Lyapunov exponent or its derivative. Here we illustrate the phenomenology of the bifurcation for several classes of nonlinear oscillators, in the regimes of both periodic and chaotic dynamics. We present extensive numerical simulations and compute the oscillation frequencies and the Lyapunov spectra as a function of the coupling strength. In particular, our simulations provide clear evidence of the phase-flip bifurcation in excitable laser and Fitzhugh-Nagumo neuronal models, and in diffusively coupled predator-prey models with either limit cycle or chaotic dynamics. Our analysis demonstrates marked jumps of the time-delayed and instantaneous fluxes between the two interacting oscillators across the bifurcation; this has strong implications for the performance of the system as well as for practical applications. We further construct an electronic circuit consisting of two coupled Chua oscillators and provide the first formal experimental demonstration of the bifurcation. In totality, our study demonstrates that the phase-flip phenomenon is of broad relevance and importance for a wide range of physical and natural systems.

Phase effects on synchronization by dynamical relaying in delay-coupled systems

Synchronization in an array of mutually coupled systems with a finite time delay in coupling is studied using the Josephson junction as a model system. The sum of the transverse Lyapunov exponents is evaluated as a function of the parameters by linearizing the equation about the synchronization manifold. The dependence of synchronization on damping parameter, coupling constant,and time delay is studied numerically. The change in the dynamics of the system due to time delay and phase difference between the applied fields is studied. The case where a small frequency detuning between the applied fields is also discussed.

Transition from phase to generalized synchronization in time-delay systems

The notion of phase synchronization in time-delay systems, exhibiting highly non-phase-coherent attractors, has not been realized yet even though it has been well studied in chaotic dynamical systems without delay. We report the identification of phase synchronization in coupled nonidentical piecewise linear and in coupled Mackey-Glass time-delay systems with highly non-phase-coherent regimes. We show that there is a transition from nonsynchronized behavior to phase and then to generalized synchronization as a function of coupling strength. We have introduced a transformation to capture the phase of the non-phase-coherent attractors, which works equally well for both the time-delay systems. The instantaneous phases of the above coupled systems calculated from the transformed attractors satisfy both the phase and mean frequency locking conditions. These transitions are also characterized in terms of recurrence-based indices, namely generalized autocorrelation function P(t), correlation of probability of recurrence, joint probability of recurrence, and similarity of probability of recurrence. We have quantified the different synchronization regimes in terms of these indices. The existence of phase synchronization is also characterized by typical transitions in the Lyapunov exponents of the coupled time-delay systems.

Intermittency transition to generalized synchronization in coupled time-delay systems

We report the nature of the 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 an on-off intermittency route and that it also exhibits characteristically distinct behaviors for different coupling configurations. In particular, the intermittency transition occurs in a rather broad range of coupling strength for the error feedback coupling configuration and in a narrow range of coupling strength for the direct feedback coupling configuration. It is also shown that the intermittent dynamics displays periodic bursts of periods 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 sub-Lyapunov exponents of the coupled time-delay systems. We have also indicated the reason behind these distinct transitions by referring to the unstable periodic orbit theory of intermittency synchronization in low-dimensional systems.

Synchronization of time-delayed systems

In this paper we study synchronization in linearly coupled time-delayed systems. We first consider coupled nonidentical Ikeda systems with a square wave coupling rate. Using the theory of the time-delayed equation, we derive less restrictive synchronization conditions than those resulting from the Krasovskii-Lyapunov theory [Yang Kuang, (Academic Press, New York, 1993)]. Then we consider a wide class of nonlinear nonidentical time-delayed systems. We also propose less restrictive synchronization conditions in an approximative sense, even if the coefficients in the linear time-delayed equation on the synchronization error are time dependent. Theoretical analysis and numerical simulations fully verify our main results.

Anticipating synchronization of chaotic Lur'e systems

In this paper we consider the anticipating synchronization of chaotic time-delayed Lur'e-type systems in a master-slave setting. We introduce three scenarios for anticipating synchronization, and give sufficient conditions for the existence of anticipating synchronizing slave systems in terms of linear matrix inequalities. The results obtained are illustrated on a time-delayed Rossler system and a time-delayed Chua oscillator.