Routes to complete synchronization via phase synchronization in coupled nonidentical chaotic oscillators

Synchronizing Chaos with Imperfections

Previous research on nonlinear oscillator networks has shown that chaos synchronization is attainable for identical oscillators but deteriorates in the presence of parameter mismatches. Here, we identify regimes for which the opposite occurs and show that oscillator heterogeneity can synchronize chaos for conditions under which identical oscillators cannot. This effect is not limited to small mismatches and is observed for random oscillator heterogeneity on both homogeneous and heterogeneous network structures. The results are demonstrated experimentally using networks of Chua's oscillators and are further supported by numerical simulations and theoretical analysis. In particular, we propose a general mechanism based on heterogeneity-induced mode mixing that provides insights into the observed phenomenon. Since individual differences are ubiquitous and often unavoidable in real systems, it follows that such imperfections can be an unexpected source of synchronization stability.

Discontinuity-induced intermittent synchronization transitions in coupled non-smooth systems

The synchronization transition in coupled non-smooth systems is studied for increasing coupling strength. The average order parameter is calculated to diagnose synchronization of coupled non-smooth systems. It is found that the coupled non-smooth system exhibits an intermittent synchronization transition from the cluster synchronization state to the complete synchronization state, depending on the coupling strength and initial conditions. Detailed numerical analyses reveal that the discontinuity always plays an important role in the synchronization transition of the coupled non-smooth system. In addition, it is found that increasing the coupling strength leads to the coexistence of periodic cluster states. Detailed research illustrates that the periodic clusters consist of two or more coexisting periodic attractors. Their periodic trajectory passes from one region to another region that is divided by discontinuous boundaries in the phase space. The mutual interactions of the local nonlinearity and the spatial coupling ultimately result in a stable periodic trajectory.

Recurrent synchronization of coupled oscillators with spontaneous phase reformation

Self-organizing and spontaneous breaking are seemingly opposite phenomena and hardly captured in a single model. We develop a second order Kuramoto model with phase-induced damping which shows phase locking together with spontaneous synchrony breaking and reformation. In a relatively large regime where the interacting force and the damping ratio are of the same order, the dynamics of the oscillators alternates in an irregular cycle of synchronization, formation-breaking, and reorganization. While the oscillators keep coming back to phase-locked states, their phase distribution repeatedly reforms. Also, the interevent time between bursty deviation from the synchronization states follows a power-law distribution, which implies that the synchronized states are maintained near a tipping point.

Transition to complete synchronization and global intermittent synchronization in an array of time-delay systems

We report the nature of transitions from the nonsynchronous to a complete synchronization (CS) state in arrays of time-delay systems, where the systems are coupled with instantaneous diffusive coupling. We demonstrate that the transition to CS occurs distinctly for different coupling configurations. In particular, for unidirectional coupling, locally (microscopically) synchronization transition occurs in a very narrow range of coupling strength but for a global one (macroscopically) it occurs sequentially in a broad range of coupling strength preceded by an intermittent synchronization. On the other hand, in the case of mutual coupling, a very large value of coupling strength is required for local synchronization and, consequently, all the local subsystems synchronize immediately for the same value of the coupling strength and, hence, globally, synchronization also occurs in a narrow range of the coupling strength. In the transition regime, we observe a type of synchronization transition where long intervals of high-quality synchronization which are interrupted at irregular times by intermittent chaotic bursts simultaneously in all the systems and which we designate as global intermittent synchronization. We also relate our synchronization transition results to the above specific types using unstable periodic orbit theory. The above studies are carried out in a well-known piecewise linear time-delay system.

Generating and enhancing lag synchronization of chaotic systems by white noise

In this paper, we study the crucial impact of white noise on lag synchronous regime in a pair of time-delay unidirectionally coupled systems. Our result demonstrates that merely via white-noise-based coupling lag synchronization could be achieved between the coupled systems (chaotic or not). And it is also demonstrated that a conventional lag synchronous regime can be enhanced by white noise. Sufficient conditions are further proved mathematically for noise-inducing and noise-enhancing lag synchronization, respectively. Additionally, the influence of parameter mismatch on the proposed lag synchronous regime is studied, by which we announce the robustness and validity of the new strategy. Two numerical examples are provided to illustrate the validity and some possible applications of the theoretical result.

Detecting the temporal structure of intermittent phase locking

This study explores a method to characterize the temporal structure of intermittent phase locking in oscillatory systems. When an oscillatory system is in a weakly synchronized regime away from a synchronization threshold, it spends most of the time in parts of its phase space away from the synchronization state. Therefore characteristics of dynamics near this state (such as its stability properties and Lyapunov exponents or distributions of the durations of synchronized episodes) do not describe the system's dynamics for most of the time. We consider an approach to characterize the system dynamics in this case by exploring the relationship between the phases on each cycle of oscillations. If some overall level of phase locking is present, one can quantify when and for how long phase locking is lost, and how the system returns back to the phase-locked state. We consider several examples to illustrate this approach: coupled skewed tent maps, the stability of which can be evaluated analytically; coupled Rössler and Lorenz oscillators, undergoing through different intermittency types on the way to phase synchronization; and a more complex example of coupled neurons. We show that the obtained measures can describe the differences in the dynamics and temporal structure of synchronization and desynchronization events for the systems with a similar overall level of phase locking and similar stability of the synchronized state.

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.

Synchronization of two different chaotic systems using novel adaptive fuzzy sliding mode control

In this paper, an adaptive fuzzy sliding mode control (AFSMC) scheme is proposed for the synchronization of two chaotic nonlinear systems in the presence of uncertainties and external disturbance. To design the reaching phase of the sliding mode control (SMC), a fuzzy controller is used. This will reduce the chattering and improve the robustness. An AFSMC is used (as an equivalent control part of the SMC) to approximate the unknown parts of the uncertain chaotic systems. Although the above schemes have been proposed in the past as separate stand-alone control schemes, in this paper, we integrate these methods to propose an effective control scheme having the benefits of each. The stability analysis for the proposed control scheme is provided and simulation examples are presented to verify the effectiveness of the method.

Delay time modulation induced oscillating synchronization and intermittent anticipatory/lag and complete synchronizations in time-delay nonlinear dynamical systems

Existence of a new type of oscillating synchronization that oscillates between three different types of synchronizations (anticipatory, complete, and lag synchronizations) is identified in unidirectionally coupled nonlinear time-delay systems having two different time-delays, that is feedback delay with a periodic delay time modulation and a constant coupling delay. Intermittent anticipatory, intermittent lag, and complete synchronizations are shown to exist in the same system with identical delay time modulations in both the delays. The transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay with suitable stability condition is discussed. The intermittent anticipatory and lag synchronizations are characterized by the minimum of the similarity functions and the intermittent behavior is characterized by a universal asymptotic -32 power law distribution. It is also shown that the delay time carved out of the trajectories of the time-delay system with periodic delay time modulation cannot be estimated using conventional methods, thereby reducing the possibility of decoding the message by phase space reconstruction.

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

The existence of anticipatory, complete, and lag synchronization in a single system having two different time delays, that is, feedback delay tau1 and coupling delay tau2, is identified. The transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay tau2 with a suitable stability condition is discussed. In particular, it is shown that the stability condition is independent of the delay times tau1 and tau2. Consequently, for a fixed set of parameters, all the three types of synchronizations can be realized. Further, the emergence of exact anticipatory, complete, or lag synchronization from the desynchronized state via approximate synchronization, when one of the system parameters (b2) is varied, is characterized by a minimum of the similarity function and the transition from on-off intermittency via periodic structure in the laminar phase distribution.

Complete synchronization and generalized synchronization of one-way coupled time-delay systems

The complete synchronization and generalized synchronization (GS) of one-way coupled time-delay systems are studied. We find that GS can be achieved by a single scalar signal, and its synchronization threshold for different delay times shows the parameter resonance effect, i.e., we can obtain stable synchronization at a smaller coupling if the delay time of the driven system is chosen such that it is in resonance with the driving system. Near chaos synchronization, the desynchronization dynamics displays periodic bursts with the period equal to the delay time of the driven system. These features can be easily applied to the recovery of time-delay systems.

Transition from phase synchronization to complete synchronization in mutually coupled nonidentical Nd:YAG lasers

Using mutually coupled nonidentical continuous-wave Nd:YAG lasers, we experimentally confirmed the recently proposed transition route from phase synchronization to complete synchronization. As evidence of this transition we obtained the probability distribution of the intermittent synchronization time near the threshold of the complete synchronization transition.

Phase synchronization in the perturbed Chua circuit

We show experimental and numerical results of phase synchronization between the chaotic Chua circuit and a small sinusoidal perturbation. Experimental real-time phase synchronized states can be detected with oscilloscope visualization of the attractor, using specific sampling rates. Arnold tongues demonstrate robust phase synchronized states for perturbation frequencies close to the characteristic frequency of the unperturbed Chua.