Using white noise to enhance synchronization of coupled chaotic systems

In the paper, complete synchronization of two chaotic oscillators via unidirectional coupling determined by white noise distribution is investigated. It is analytically proved that chaos synchronization could be achieved with probability one merely via white-noise-based coupling. The established theoretical result supports the observation of an interesting phenomenon that a certain kind of white noise could enhance chaos synchronization between two chaotic oscillators. Furthermore, numerical examples are provided to illustrate some possible applications of the theoretical result.     

Switching phenomenon induced by breakdown of chaotic phase synchronization

We investigate chaotic phase synchronization (CPS) in three-coupled chaotic oscillator systems. According to the coupling strength and mismatches in the frequencies of these oscillators, we can observe complete CPS where all three oscillators exhibit CPS, and partial CPS where only two oscillators exhibit CPS. When the coupling strength is weakened, we observe a phenomenon that complete CPS among the three oscillators is suddenly disrupted without going through partial CPS. In this case oscillators exhibit quasi-CPS where two oscillators appear to exhibit CPS transiently, and the combination of the two oscillators changes with time. We call this phenomenon CPS switching D. It is revealed that phase fluctuation plays an important role in CPS switching D. It is also shown that the amplitude with a specific structure strengthens the degree of CPS switching. In the present paper, we characterize this CPS switching and discuss its mechanism.     

Cooperative behaviors of coupled nonidentical oscillators with the same equilibrium points

The cooperative behaviors resulted from the interaction of coupled identical oscillators have been investigated intensively. However, the coupled oscillators in practice are nonidentical, and there exist mismatched parameters. It has been proved that under certain conditions, complete synchronization can take place in coupled nonidentical oscillators with the same equilibrium points, yet other cooperative behaviors are not addressed. In this paper, we further consider two coupled nonidentical oscillators with the same equilibrium points, where one oscillator is convergent while the other is chaotic,and explore their cooperative behaviors. We find that the coupling mode and the coupling strength can bring the coupled oscillators to different cooperation behaviors in unidirectional or undirected couplings. In the case of directed coupling,death islands appear in two-parameter spaces. The mechanism inducing these transitions is presented.     

Effect of noise on coupled chaotic systems

The effect of noise in inducing order on various chaotically evolving systems is reviewed, with special emphasis on systems consisting of coupled chaotic elements. In many situations it is observed that the uncoupled elements when driven by identical noise, show synchronization phenomena where chaotic trajectories exponentially converge towards a single noisy trajectory, independent of the initial conditions. In a random neural network, with infinite range coupling, chaos is suppressed due to noise and the system evolves towards a fixed point. Spatiotemporal stochastic resonance phenomenon has been observed in a square array of coupled threshold devices where a temporal characteristic of the system resonates at a given noise strength. In a chaotically evolving coupled map lattice with the logistic map as local dynamics and driven by identical noise at each site, we report that the number ofstructures (a structure is a group of neighbouring lattice sites for values of the variable follow which the certain predefined pattern) follows a power-law decay with the length of the structure. An interesting phenomenon, which we callstochastic coherence, is also reported in which the abundance and lifetimes of these structures show characteristic peaks at some intermediate noise strength.     

Noise enhanced phase synchronization and coherence resonance in sets of chaotic oscillators with weak global coupling

The effect of noise on phase synchronization in small sets and larger populations of weakly coupled chaotic oscillators is explored. Both independent and correlated noise are found to enhance phase synchronization of two coupled chaotic oscillators below the synchronization threshold; this is in contrast to the behavior of two coupled periodic oscillators. This constructive effect of noise results from the interplay between noise and the locking features of unstable periodic orbits. We show that in a population of nonidentical chaotic oscillators, correlated noise enhances synchronization in the weak coupling region. The interplay between noise and weak coupling induces a collective motion in which the coherence is maximal at an optimal noise intensity. Both the noise-enhanced phase synchronization and the coherence resonance numerically observed in coupled chaotic R?ssler oscillators are verified experimentally with an array of chaotic electrochemical oscillators.     

Chaos control in a discrete time system through asymmetric coupling

We study a pair of asymmetrically coupled identical chaotic quadratic maps. We investigate, via numerical simulations, chaos suppression associated with the variation of both parameters, the coupling parameter and the parameter which measures the asymmetry. This is a new technique recently introduced for chaos suppression in continuous systems and, as far we know, not yet tested for discrete systems. Parameter-space regions where the chaotic dynamics is driven towards regular dynamics are shown. Lyapunov exponents and phase-space plots are also used to characterize the phenomenon observed as the parameters are changed.     

Effect of common noise on phase synchronization in coupled chaotic oscillators

We report a general phenomenon concerning the effect of noise on phase synchronization in coupled chaotic oscillators: the average phase-synchronization time exhibits a nonmonotonic behavior with the noise amplitude. In particular, we find that the time exhibits a local minimum for relatively small noise amplitude but a local maximum for stronger noise. We provide numerical results, experimental evidence from coupled chaotic circuits, and a heuristic argument to establish the generality of this phenomenon.     

From low-dimensional synchronous chaos to high-dimensional desynchronous spatiotemporal chaos in coupled systems

The dynamic behavior of coupled chaotic oscillators is investigated. For small coupling, chaotic state undergoes a transition from a spatially disordered phase to an ordered phase with an orientation symmetry breaking. For large coupling, a transition from full synchronization to partial synchronization with translation symmetry breaking is observed. Two bifurcation branches, one in-phase branch starting from synchronous chaos and the other antiphase branch bifurcated from spatially random chaos, are identified by varying coupling strength epsilon. Hysteresis, bistability, and first-order transitions between these two branches are observed.     

Experimental investigation of high-quality synchronization of coupled oscillators

We describe two experiments in which we investigate the synchronization of coupled periodic oscillators. Each experimental system consists of two identical coupled electronic periodic oscillators that display bursts of desynchronization events similar to those observed previously in coupled chaotic systems. We measure the degree of synchronization as a function of coupling strength. In the first experiment, high-quality synchronization is achieved for all coupling strengths above a critical value. In the second experiment, no high-quality synchronization is observed. We compare our results to the predictions of the several proposed criteria for synchronization. We find that none of the criteria accurately predict the range of coupling strengths over which high-quality synchronization is observed. (c) 2000 American Institute of Physics.     

Amplitude and phase effects on the synchronization of delay-coupled oscillators

We consider the behavior of Stuart-Landau oscillators as generic limit-cycle oscillators when they are interacting with delay. We investigate the role of amplitude and phase instabilities in producing symmetry-breaking/restoring transitions. Using analytical and numerical methods we compare the dynamics of one oscillator with delayed feedback, two oscillators mutually coupled with delay, and two delay-coupled elements with self-feedback. Taking only the phase dynamics into account, no chaotic dynamics is observed, and the stability of the identical synchronization solution is the same in each of the three studied networks of delay-coupled elements. When allowing for a variable oscillation amplitude, the delay can induce amplitude instabilities. We provide analytical proof that, in case of two mutually coupled elements, the onset of an amplitude instability always results in antiphase oscillations, leading to a leader-laggard behavior in the chaotic regime. Adding self-feedback with the same strength and delay as the coupling stabilizes the system in the transverse direction and, thus, promotes the onset of identically synchronized behavior.     

Strong and Weak Chaos in Weakly Nonintegrable Many-Body Hamiltonian Systems

We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology.     

Hierarchical synchronization in complex networks with heterogeneous degrees

We study synchronization behavior in networks of coupled chaotic oscillators with heterogeneous connection degrees. Our focus is on regimes away from the complete synchronization state, when the coupling is not strong enough, when the oscillators are under the influence of noise or when the oscillators are nonidentical. We have found a hierarchical organization of the synchronization behavior with respect to the collective dynamics of the network. Oscillators with more connections (hubs) are synchronized more closely by the collective dynamics and constitute the dynamical core of the network. The numerical observation of this hierarchical synchronization is supported with an analysis based on a mean field approximation and the master stability function.     

Observation of chimera patterns in a network of symmetric chaotic finance systems

The economic and financial systems consist of many nonlinear factors that make them behave as the complex systems. Recently many chaotic finance systems have been proposed to study the complex dynamics of finance as a noticeable problem in economics. In fact, the intricate structure between financial institutions can be obtained by using a network of financial systems. Therefore, in this paper, we consider a ring network of coupled symmetric chaotic finance systems, and investigate its behavior by varying the coupling parameters. The results show that the coupling strength and range have significant effects on the behavior of the coupled systems, and various patterns such as the chimera and multi-chimera states are observed. Furthermore, changing the parameters' values, remarkably influences on the oscillators attractors. When several synchronous clusters are formed, the attractors of the synchronized oscillators are symmetric, but different from the single oscillator attractor.     

Suppression of Chaos and Phase Locking in Two Coupled Nonidentical Neurons under Periodic Input

Dynamical behaviour of two coupled neurons with at least one of them being chaotic is presented. Bifurcation diagrams and Lyapunov exponents are calculated to diagnose the dynamical behaviour of the coupled neurons with the increasing coupling strength. It is found that when the coupling strength increases, a chaotic neuron can be controlled by the coupling between neurons. At the same time, phase locking is studied by the maxima of the differences of instantaneous phases and average frequencies between two coupled neurons, and the inherent connection of phase locking and the suppression of chaos is formulated. It is observed that the onset of phase locking is closely related to the suppression of chaos. Finally, a way for suppression of chaos in two coupled nonidentical neurons under periodic input is suggested.     

Dynamic synchronization of a time-evolving optical network of chaotic oscillators

We present and experimentally demonstrate a technique for achieving and maintaining a global state of identical synchrony of an arbitrary network of chaotic oscillators even when the coupling strengths are unknown and time-varying. At each node an adaptive synchronization algorithm dynamically estimates the current strength of the net coupling signal to that node. We experimentally demonstrate this scheme in a network of three bidirectionally coupled chaotic optoelectronic feedback loops and we present numerical simulations showing its application in larger networks. The stability of the synchronous state for arbitrary coupling topologies is analyzed via a master stability function approach.     

n:m phase synchronization with mutual coupling phase signals

We generalize the n:m phase synchronization between two chaotic oscillators by mutual coupling phase signals. To characterize this phenomenon, we use two coupled oscillators to demonstrate their phase synchronization with amplitudes practically noncorrelated. We take the 1:1 phase synchronization as an example to show the properties of mean frequencies, mean phase difference, and Lyapunov exponents at various values of coupling strength. The phase difference increases with 2pi phase slips below the transition. The scaling rules of the slip near and away from the transition are studied. Furthermore, we demonstrate the transition to a variety of n:m phase synchronizations and analyze the corresponding coupling dynamics. (c) 2002 American Institute of Physics.     

Consequential noise-induced synchronization of indirectly coupled self-sustained oscillators

We consider the dynamics of identical self-sustained oscillators coupled via a common linear system (beam), which is perturbed by noise. We demonstrate that increasing the noise intensity induces complete synchronization between the oscillators and, surprisingly, their in-phase synchronization with the beam. This new phenomenon of in-phase synchronization of both the oscillators and the oscillating beam arises when the noise intensity exceeds a threshold value, and can not appear in the deterministic case where the beam stably oscillates in anti-phase with the synchronized oscillators (as it is in the case of the Huygens clocks synchronization). Similar behavior persists for slightly non-identical oscillators.     

Aging transition and universal scaling in oscillator networks

Self-sustained oscillators may turn non-self-oscillatory as a result of some kind of deterioration, which we call aging for simplicity. We discuss the effect of aging on the behavior of globally and diffusively coupled oscillators which are either all periodic or chaotic. It is shown that at a certain level of aging, macroscopic oscillation stops in a way which depends on the coupling strength. A universal scaling function to describe it is analytically derived and numerically verified.     

Resurgence of oscillation in coupled oscillators under delayed cyclic interaction

This paper investigates the emergence of amplitude death and revival of oscillations from the suppression states in a system of coupled dynamical units interacting through delayed cyclic mode. In order to resurrect the oscillation from amplitude death state, we introduce asymmetry and feedback parameter in the cyclic coupling forms as a result of which the death region shrinks due to higher asymmetry and lower feedback parameter values for coupled oscillatory systems. Some analytical conditions are derived for amplitude death and revival of oscillations in two coupled limit cycle oscillators and corresponding numerical simulations confirm the obtained theoretical results. We also report that the death state and revival of oscillations from quenched state are possible in the network of identical coupled oscillators. The proposed mechanism has also been examined using chaotic Lorenz oscillator.     

Synchronizing multiple chaotic maps with a randomized scalar coupling

Synchronization between chaotic systems has recently been a topic of great interest among physicists and engineers. In addition to theoretical results, a number of applications in communications and control have also been proposed. We have previously shown that identical chaotic maps can, under certain conditions, be synchronized by a common noise-like input. This raises the question whether the chaotic output from a map can synchronize other maps to itself. In this paper, we present results on the synchronization of two identical maps where the output of one map is used to drive both maps. We then apply this method to synchronizing identical but internally inhomogeneous populations of chaotic oscillators using a randomly constructed scalar coupling signal, which tends to white noise as the number of oscillators increases. Such a system may have applications in secure communication.