Synchronization of chaos in coupled systems

Sinusoidal and nonsinusoidal patterns in amplitude envelope synchronization

In this work, amplitude envelope synchronization (AES), as a general phenomenon characterized with highly correlated amplitude envelope but uncorrelated phases and frequencies in coupled nonidentical nonlinear systems, is investigated theoretically and numerically. Two different types of AES patterns, including sinusoidal and nonsinusoidal, are widely observable in coupled periodic and/or chaotic oscillators. They both come from modulation of phase mismatch on amplitude but show different patterns due to different behaviors of phase mismatch. With increase of frequency mismatch, the system tends to crossover from nonsinusoidal to sinusoidal AES. With the aid of synchronization manifold and transverse stability analyses of the AES state, the physical mechanism and scale relations for the AES are well revealed. We expect that all these results could uncover the generality of AES in coupled nonlinear oscillators and help to understand the rich dynamics of phase and amplitude coupling in multidisciplinary fields.

Origin of amplitude synchronization in coupled nonidentical oscillators

The origin of amplitude synchronization (AS), or amplitude envelope synchronization, as a peculiar form of strong correlation between amplitudes of oscillators is studied by using a model of coupled Landau-Stuart periodic oscillators. We find that the AS extensively occurs within the traditional phase drift region, and the amplitude correlation does not change with variation of the coupling strength but is dampened with increase of the frequency mismatch. The AS appears only at weak couplings and before the occurrence of phase synchronization (PS), and the oscillator amplitude is modulated by its phase. This study could build a solid foundation for AS, which has not drawn much attention in the nonlinear dynamics field before, providing a clear physical picture for synchronization including not only PS, but also AS, and arousing general interest in many interdisciplinary fields, such as neuronal systems, laser dynamics, nanomechanical resonators, and power systems, etc., where phase and amplitude are always mutually influenced and both are important.

Insensitivity of synchronization to network structure in chaotic pendulum systems with time-delay coupling

It has been generally believed that both time delay and network structure could play a crucial role in determining collective dynamical behaviors in complex systems. In this work, we study the influence of coupling strength, time delay, and network topology on synchronization behavior in delay-coupled networks of chaotic pendulums. Interestingly, we find that the threshold value of the coupling strength for complete synchronization in such networks strongly depends on the time delay in the coupling, but appears to be insensitive to the network structure. This lack of sensitivity was numerically tested in several typical regular networks, such as different locally and globally coupled ones as well as in several complex networks, such as small-world and scale-free networks. Furthermore, we find that the emergence of a synchronous periodic state induced by time delay is of key importance for the complete synchronization.

Time delay induced different synchronization patterns in repulsively coupled chaotic oscillators

Time delayed coupling plays a crucial role in determining the system's dynamics. We here report that the time delay induces transition from the asynchronous state to the complete synchronization (CS) state in the repulsively coupled chaotic oscillators. In particular, by changing the coupling strength or time delay, various types of synchronous patterns, including CS, antiphase CS, antiphase synchronization (ANS), and phase synchronization, can be generated. In the transition regions between different synchronous patterns, bistable synchronous oscillators can be observed. Furthermore, we show that the time-delay-induced phase flip bifurcation is of key importance for the emergence of CS. All these findings may light on our understanding of neuronal synchronization and information processing in the brain.

Quasiperiodic, periodic, and slowing-down states of coupled heteroclinic cycles

We investigate two coupled oscillators, each of which shows an attracting heteroclinic cycle in the absence of coupling. The two heteroclinic cycles are nonidentical. Weak coupling can lead to the elimination of the slowing-down state that asymptotically approaches the heteroclinic cycle for a single cycle, giving rise to either quasiperiodic motion with separate frequencies from the two cycles or periodic motion in which the two cycles are synchronized. The synchronization transition, which occurs via a Hopf bifurcation, is not induced by the commensurability of the two cycle frequencies but rather by the disappearance of the weaker frequency oscillation. For even larger coupling the motion changes via a resonant heteroclinic bifurcation to a slowing-down state corresponding to a single attracting heteroclinic orbit. Coexistence of multiple attractors can be found for some parameter regions. These results are of interest in ecological, sociological, neuronal, and other dynamical systems, which have the structure of coupled heteroclinic cycles.

Eliminating delay-induced oscillation death by gradient coupling

In this work, we investigate gradient coupling effect on oscillation death in a ring of N delay-coupled oscillators. We find that the gradient coupling monotonically reduces the domain of death island in the parameter space of the diffusive coupling and time delay, and thus the death island can be completely eliminated once the gradient coupling strength exceeds a certain threshold, whose value is found to be a constant if N is sufficiently large. For two special cases, a ring with zero gradient coupling and a one-way ring for identical diffusive and gradient couplings, all previous results in the literature are recovered. In particular, for the one-way ring, a size effect of N is discovered, which indicates that under this situation the death can always be eliminated if N is above a critical N(max). All the described results are proved to hold generally in coupled oscillator systems.

Generic behavior of master-stability functions in coupled nonlinear dynamical systems

Master-stability functions (MSFs) are fundamental to the study of synchronization in complex dynamical systems. For example, for a coupled oscillator network, a necessary condition for synchronization to occur is that the MSF at the corresponding normalized coupling parameters be negative. To understand the typical behaviors of the MSF for various chaotic oscillators is key to predicting the collective dynamics of a network of these oscillators. We address this issue by examining, systematically, MSFs for known chaotic oscillators. Our computations and analysis indicate that it is generic for MSFs being negative in a finite interval of a normalized coupling parameter. A general scheme is proposed to classify the typical behaviors of MSFs into four categories. These results are verified by direct simulations of synchronous dynamics on networks of actual coupled oscillators.

Complete periodic synchronization in coupled systems

Recently, complete chaotic synchronization in coupled systems has been well studied. In this paper, we study complete synchronization in coupled periodic oscillators with diffusive and gradient couplings. Eight typical types of critical curve for the transverse Lyapunov exponent of standard mode, which give rise to different synchronization-desynchronization patterns, are classified. All possible desynchronous behaviors including steady state, periodic state, quasiperiodic state, low-dimensional chaotic state, and two types of high-dimensional chaotic state are identified, and two classical synchronization-desynchronizaiton bifurcations--the shortest wavelength bifurcation and Hopf bifurcation from synchronous periodic state--are classified.

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.

Chaos synchronization in coupled systems by applying pinning control

Chaos synchronization in coupled chaotic oscillator systems with diffusive and gradient couplings forced by only one local feedback injection signal (boundary pinning control) is studied. By using eigenvalue analysis, we obtain controllable regions directly in control parameter space for different types of coupling links (including diagonal coupling and nondiagonal couplings). The effects of both diffusive and gradient couplings on chaos synchronization become clear. Some relevant factors on control efficiency such as coupled system size, transient process, and feedback signal intensity are also studied.

Critical lattice size limit for synchronized chaotic state in one- and two-dimensional diffusively coupled map lattices

We consider diffusively coupled map lattices with P neighbors (where P is arbitrary) and study the stability of the synchronized state. We show that there exists a critical lattice size beyond which the synchronized state is unstable. This generalizes earlier results for nearest neighbor coupling. We confirm the analytical results by performing numerical simulations on coupled map lattices with logistic map at each node. The above analysis is also extended to two-dimensional P -neighbor diffusively coupled map lattices.

Experimental study of the transitions between synchronous chaos and a periodic rotating wave

In this work we characterize experimentally the transition between periodic rotating waves and synchronized chaos in a ring of unidirectionally coupled Lorenz oscillators by means of electronic circuits. The study is complemented by numerical and theoretical analysis, and the intermediate states and their transitions are identified. The route linking periodic behavior with synchronous chaos involves quasiperiodic behavior and a type of high-dimensional chaos known as chaotic rotating wave. The high-dimensional chaotic behavior is characterized, and is shown to be composed actually by three different behaviors. The experimental study confirms the robustness of this route.

Phase synchronization in unidirectionally coupled chaotic ratchets

We study chaotic phase synchronization of unidirectionally coupled deterministic chaotic ratchets. The coupled ratchets were simulated in their chaotic states and perfect phase locking was observed as the coupling was gradually increased. We identified the region of phase synchronization for the ratchets and show that the transition to chaotic phase synchronization is via an interior crisis transition to strange attractor in the phase space.

Synchronization and symmetry-breaking bifurcations in constructive networks of coupled chaotic oscillators

The spatiotemporal dynamics of networks based on a ring of coupled oscillators with regular shortcuts beyond the nearest-neighbor couplings is studied by using master stability equations and numerical simulations. The generic criterion for dynamic synchronization has been extended to arbitrary network topologies with zero row-sum. The symmetry-breaking oscillation patterns that resulted from the Hopf bifurcation from synchronous states are analyzed by the symmetry group theory.

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.

Transitions from spatiotemporal chaos to cluster and complete synchronization states in a shift-invariant set of coupled nonlinear oscillators

We study the spatiotemporal dynamics of a ring of diffusely coupled single-well Duffing oscillators. The transitions from spatiotemporal chaos to cluster and complete synchronization states are particularly investigated, as well as the Hopf bifurcations to instability. It is found that the underlying mechanism of these transitions relies on the motion of the representative points corresponding to the system's nondegenerated spatial transverse Fourier modes in the parametric Strutt diagram. A scaling law is used to demonstrate that the compact interval of the scalar coupling parameter values leading to cluster synchronization broadens in a square-power-like fashion as the number of oscillators is increased. The analytical approach is confirmed by numerical simulations.

General stability analysis of synchronized dynamics in coupled systems

We consider the stability of synchronized states (including equilibrium point, periodic orbit, or chaotic attractor) in arbitrarily coupled dynamical systems (maps or ordinary differential equations). We develop a general approach, based on the master stability function and Gershgörin disk theory, to yield constraints on the coupling strengths to ensure the stability of synchronized dynamics. Systems with specific coupling schemes are used as examples to illustrate our general method.

Generalized correlated states in a ring of coupled nonlinear oscillators with a local injection

In this paper, we study the spatiotemporal dynamics of a ring of diffusely coupled nonlinear oscillators. Floquet theory is used to investigate the various dynamical states of the ring, as well as the Hopf bifurcations between them. A local injection scheme is applied to synchronize the ring with an external master oscillator. The shift-invariance symmetry is thereby broken, leading to the emergence of generalized correlated states. The transition boundaries from these correlated states to spatiotemporal chaos and complete synchronization are also derived. Numerical simulations are performed to support the analytic approach.

Coherence resonance near the Hopf bifurcation in coupled chaotic oscillators

We uncover a coherence resonance near the Hopf bifurcation from chaos in coupled chaotic oscillators. At the bifurcation, a nearly periodic rotating wave becomes stable as the state of synchronous chaos is destabilized. We find that noise can induce the bifurcation and, more strikingly, it can enhance the temporal regularity of the wave pattern in the coupled system. This resonant phenomenon is expected to be robust and physically observable.

Transition from intermittency to periodicity in lag synchronization in coupled Rössler oscillators

The dynamical and statistical behavior of lag synchronization in two coupled self-sustained chaotic Rössler oscillators is reexamined. The lack of uniqueness in the conventional characterization of lag synchronization based on the similarity function has caused much skepticism about the existence of lag synchronization. We provide an evidence that the emergence of lag synchronization is associated with the transition from on-off intermittency to a periodic structure in the laminar phase distribution.

Phase locking in on-off intermittency

Dynamical behavior of on-off intermittency around chaos synchronization-desynchronization bifurcation parameter line is investigated in coupled identical chaotic oscillators. Along this parameter line, we find that on-off intermittency can transit from phase-unlocking status to phase-locking one in the phase space of variable differences, which can be regarded as a codimension-two bifurcation, i.e., combinative bifurcations of desynchronization and phase locking. In the phase-locking case, the motions of all oscillators are chaotic and they show on-off intermittency with respect to the synchronous manifold, however, spatial phase order of variable differences is clearly established.

Wave fronts and spatiotemporal chaos in an array of coupled Lorenz oscillators

The effects of coupling strength and single-cell dynamics (SCD) on spatiotemporal pattern formation are studied in an array of Lorenz oscillators. Different spatiotemporal structures (stationary patterns, propagating wave fronts, short wavelength bifurcation) arise for bistable SCD, and two well differentiated types of spatiotemporal chaos for chaotic SCD (in correspondence with the transition from stationary patterns to propagating fronts). Wave-front propagation in the bistable regime is studied in terms of global bifurcation theory, while a short wavelength pattern region emerges through a pitchfork bifurcation.