Waves and patterns in ring lattices with delays

We investigate the existence and the stability of waves and phase locked states in rings of coupled oscillators with delayed interactions. Using center manifold reduction and the normal form method, we reduce the equation governing the dynamics of the whole network to an amplitude-phase model (i.e. a set of coupled ordinary differential equations describing the evolution of both the amplitudes and the phases of the oscillators). Then we prove the existence of traveling waves, in-phase and anti-phase locked oscillations, in both one-dimensional and two-dimensional lattices. The influence of the interaction strength and the number of oscillators is investigated, and the possible coexistence of waves and phase locked oscillations is shown.     

Experimental evidence of anomalous phase synchronization in two diffusively coupled Chua oscillators

We study the transition to phase synchronization in two diffusively coupled, nonidentical Chua oscillators. In the experiments, depending on the used parameterization, we observe several distinct routes to phase synchronization, including states of either in-phase, out-of-phase, or antiphase synchronization, which may be intersected by an intermediate desynchronization regime with large fluctuations of the frequency difference. Furthermore, we report the first experimental evidence of an anomalous transition to phase synchronization, which is characterized by an initial enlargement of the natural frequency difference with coupling strength. This results in a maximal frequency disorder at intermediate coupling levels, whereas usual phase synchronization via monotonic decrease in frequency difference sets in only for larger coupling values. All experimental results are supported by numerical simulations of two coupled Chua models.     

Inferring phase equations from multivariate time series

An approach is presented for extracting phase equations from multivariate time series data recorded from a network of weakly coupled limit cycle oscillators. Our aim is to estimate important properties of the phase equations including natural frequencies and interaction functions between the oscillators. Our approach requires the measurement of an experimental observable of the oscillators; in contrast with previous methods it does not require measurements in isolated single or two-oscillator setups. This noninvasive technique can be advantageous in biological systems, where extraction of few oscillators may be a difficult task. The method is most efficient when data are taken from the nonsynchronized regime. Applicability to experimental systems is demonstrated by using a network of electrochemical oscillators; the obtained phase model is utilized to predict the synchronization diagram of the system.     

New universality type in chaotic synchronization of dynamic systems

A quite universal mechanism of establishing chaotic synchronization regime in coupled dynamic systems is found. It is shown that the synchronous regime arises due to the phase coupling between the Fourier-spectrum components of the interacting chaotic oscillators.     

Chaotic synchronization of unidirectionally coupled electron-wave media with interacting counterpropagating waves

Chaotic synchronization of two electron-wave media with interacting counterpropagating waves and cubic phase nonlinearity (transverse-field backward-wave oscillators) is studied. Analysis is based on considering a continuous set of the phases of a chaotic signal. The parameters of chaotic synchronization in a system of unidirectionally coupled backward-wave oscillators are found, and the complex dynamics of establishing the chaotic synchronization conditions in an active medium is investigated.     

Phase Synchronization of Coupled Rossler Oscillators: Amplitude Effect

Phase synchronization of two linearly coupled Rossler oscillators with parameter misfits is explored.It is found that depending on parameter mismatches,the synchronization of phases exhibits different manners.The synchronization regime can be divided into three regimes.For small mismatches,the amplitude-insensitive regime gives the phase-dominant synchronization; When the parameter misfit increases,the amplitudes and phases of oscillators are correlated,and the amplitudes will dominate the synchronous dynamics for very large mismatches.The lag time among phases exhibits a power law when phase synchronization is achieved.     

Phase-flip transition in nonlinear oscillators coupled by dynamic environment

We study the dynamics of nonlinear oscillators indirectly coupled through a dynamical environment or a common medium. We observed that this form of indirect coupling leads to synchronization and phase-flip transition in periodic as well as chaotic regime of oscillators. The phase-flip transition from in- to anti-phase synchronization or vise-versa is analyzed in the parameter plane with examples of Landau-Stuart and Ro?ssler oscillators. The dynamical transitions are characterized using various indices such as average phase difference, frequency, and Lyapunov exponents. Experimental evidence of the phase-flip transition is shown using an electronic version of the van der Pol oscillators.     

Long-term fluctuations in globally coupled phase oscillators with general coupling: finite size effects

We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models such as the Ising model, the scaling property of D is D～O(1) for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of D is different for the coherent and incoherent regimes: D～O(1/N(a)) with a certain constant a>0 in the coherent regime and D～O(1) in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.     

Stochastic resonance in a locally excited system of bistable oscillators

Stochastic resonance is studied in a one-dimensional array of overdamped bistable oscillators in the presence of a local subthreshold periodic perturbation. The system can be treated as an ensemble of pseudospins tending to align parallel which are driven dynamically by an external periodic magnetic field. The oscillators are subjected to a dynamic white noise as well as to a static topological disorder. The latter is quantified by the fraction of randomly added long-range connections among ensemble elements. In the low connectivity regime the system displays an optimal global stochastic resonance response if a small-world network is formed. In the mean-field regime we explain strong changes in the dynamic disorder strength provoking a maximal stochastic resonance response via the variation of fraction of long-range connections by taking into account the ferromagnetic-paramagnetic phase transition of the pseudospins. The system size analysis shows only quantitative power-law type changes on increasing number of pseudospins.     

Desynchronization transitions in nonlinearly coupled phase oscillators

We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed. We demonstrate that for small ensembles it is typically mediated by stable cluster states, that disappear with creation of heteroclinic cycles, while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs.     

Modulation instabilities in a system of four coupled, nonlinear Schrödinger equations

The modulation instability of continuous waves for a system of four coupled nonlinear Schrödinger equations, two of which are in the unstable regime, is studied. In earlier studies, plane or continuous waves for a system of two coupled, nonlinear Schrödinger equations is shown to exhibit modulation instability (MI), even if both modes are in the normal dispersion regime, provided that the coefficient of cross phase modulation (XPM) is larger than that of self phase modulation (SPM). Requirements for MI in this system of four coupled, nonlinear Schrödinger equations can be relaxed. MI can occur even if the magnitude of XPM is less than that of SPM, and the magnitude of instability is generally larger than that of each mode alone. The implications for parametric process and wavelength exchange in optical physics with two pump waves are discussed.     

Resonance assisted synchronization of coupled oscillators: frequency locking without phase locking

Frequency locking without phase locking of two coupled nonlinear oscillators is experimentally demonstrated. This synchronization regime is found for two coupled laser modes, beyond the phase-locking range fixed by Adler's equation, because of a resonance mechanism. Specifically, we show that the amplitudes of the two modes exhibit strong fluctuations that produce average frequency synchronization, even if the instantaneous phases are unlocked. The experimental results are in good agreement with a theoretical model.     

Frequency clustering of coupled phase oscillators on small-world networks

We analyze the phenomenon of frequency clustering in a system of coupled phase oscillators. The oscillators, which in the absence of coupling have uniformly distributed natural frequencies, are coupled through a small-world network, built according to the Watts-Strogatz model. We study the time evolution and determine variations in the transient times depending on the disorder of the network and on the coupling strength. We investigate the effects of fluctuations in the average frequencies, and discuss the definition of the threshold for synchronization. We characterize the structure of clusters and the distribution of cluster sizes in the synchronization transition, and define suitable order parameters to describe the aggregation of the oscillators as the network disorder and the coupling strength change. The non-monotonic behavior observed in some order parameters is related to fluctuations in the mean frequencies.     

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.     

Zeno effect in quantum Newton's cradle

We describe a chain of quantum oscillators which behaves analogously to Newton's cradle. The energy swings between the ends of the chain with very low population in its interior. Moreover, the oscillators at the ends can entangle with each other with negligible entanglement with the intermediate oscillators that mediate the process. Up to a certain number of oscillators, the system evolves in a manner similar to two coupled oscillators. The conditions for such behavior and the characteristic periods are analyzed. When that number exceeds a threshold, the dynamical regime changes to virtually freezing. In the oscillatory regime, Zeno effect can be observed. The parallelism between the Zeno dynamics in quantum Newton's cradle and in two coupled oscillators is highlighted. Promising platforms to observe such phenomena in the laboratory are cavities in photonic-band-gap material and trapped ions.     

Role of unstable periodic orbits in phase and lag synchronization between coupled chaotic oscillators

An increase of the coupling strength in the system of two coupled R?ssler oscillators leads from a nonsynchronized state through phase synchronization to the regime of lag synchronization. The role of unstable periodic orbits in these transitions is investigated. Changes in the structure of attracting sets are discussed. We demonstrate that the onset of phase synchronization is related to phase-lockings on the surfaces of unstable tori, whereas transition from phase to lag synchronization is preceded by a decrease in the number of unstable periodic orbits.     

Effects of frustration on explosive synchronization

In this study, we consider the emergence of explosive synchronization in scale-free networks by considering the Kuramoto model of coupled phase oscillators. The natural frequencies of oscillators are assumed to be correlated with their degrees and frustration is included in the system. This assumption can enhance or delay the explosive transition to synchronization. Interestingly, a de-synchronization phenomenon occurs and the type of phase transition is also changed. Furthermore, we provide an analytical treatment based on a star graph, which resembles that obtained in scale-free networks. Finally, a self-consistent approach is implemented to study the de-synchronization regime. Our findings have important implications for controlling synchronization in complex networks because frustration is a controllable parameter in experiments and a discontinuous abrupt phase transition is always dangerous in engineering in the real world.     

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.     

Repulsive synchronization in an array of phase oscillators

We study the dynamics of a repulsively coupled array of phase oscillators. For an array of globally coupled identical oscillators, repulsive coupling results in a family of synchronized regimes characterized by zero mean field. If the number of oscillators is sufficiently large, phase locking among oscillators is destroyed, independently of the coupling strength, when the oscillators' natural frequencies are not the same. In locally coupled networks, however, phase locking occurs even for nonidentical oscillators when the coupling strength is sufficiently strong.