Network reorganization driven by temporal interdependence of its elements

We employ an adaptive parameter control technique based on detection of phase/lag synchrony between the elements of the system to dynamically modify the structure of a network of nonidentical, coupled Rossler oscillators. Two processes are simulated: adaptation, under which the initially different properties of the units converge, and rewiring, in which clusters of interconnected elements are formed based on the temporal correlations. We show how those processes lead to different network structures and investigate their optimal characteristics from the point of view of resulting network properties.     

Observability of lag synchronization of coupled chaotic oscillators

Lag synchronization is a recently discovered theoretical phenomenon where the dynamical variables of two coupled, nonidentical chaotic oscillators are synchronized with a time delay relative to each other. We investigate experimentally and numerically to what extent lag synchronization can be observed in physical systems where noise is inevitable. Our measurements and numerical computation suggest that lag synchronization is typically destroyed when the noise level is comparable to the amount of average system mismatch. At small noise levels, lag synchronization occurs in an intermittent fashion.     

The effect of time-delay on anomalous phase synchronization

Anomalous phase synchronization in nonidentical interacting oscillators is manifest as the increase of frequency disorder prior to synchronization. We show that this effect can be enhanced when a time-delay is included in the coupling. In systems of limit-cycle and chaotic oscillators we find that the regions of phase disorder and phase synchronization can be interwoven in the parameter space such that as a function of coupling or time-delay the system shows transitions from phase ordering to disorder and back.     

A new scheme to generalized (lag, anticipated, and complete) synchronization in chaotic and hyperchaotic systems

In this paper, a generalized (lag, anticipated, and complete) synchronization of a class of continuous-time systems is defined. A systematic, powerful and concrete scheme is developed to investigate the generalized (lag, anticipated, and complete) synchronization between the drive system and response system based on the active control idea. The hyperchaotic R?ssler system, transformed R?ssler and Chen system as well as two coupled nonidentical R?ssler oscillators are chosen to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme. The scheme can be also extended to research generalized (lag, anticipated, and complete) synchronization between other dynamical systems.     

Synchronization is enhanced in weighted complex networks

The propensity for synchronization of complex networks with directed and weighted links is considered. We show that a weighting procedure based upon the global structure of network pathways enhances complete synchronization of identical dynamical units in scale-free networks. Furthermore, we numerically show that very similar conditions hold also for phase synchronization of nonidentical chaotic oscillators.     

Phase synchronization between collective rhythms of globally coupled oscillator groups: noiseless nonidentical case

We theoretically study the synchronization between collective oscillations exhibited by two weakly interacting groups of nonidentical phase oscillators with internal and external global sinusoidal couplings of the groups. Coupled amplitude equations describing the collective oscillations of the oscillator groups are obtained by using the Ott-Antonsen ansatz, and then coupled phase equations for the collective oscillations are derived by phase reduction of the amplitude equations. The collective phase coupling function, which determines the dynamics of macroscopic phase differences between the groups, is calculated analytically. We demonstrate that the groups can exhibit effective antiphase collective synchronization even if the microscopic external coupling between individual oscillator pairs belonging to different groups is in-phase, and similarly effective in-phase collective synchronization in spite of microscopic antiphase external coupling between the groups.     

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.     

Entangled chimeras in nonlocally coupled bicomponent phase oscillators: From synchronous to asynchronous chimeras

Chimera states, a symmetry-breaking spatiotemporal pattern in nonlocally coupled identical dynamical units, have been identified in various systems and generalized to coupled nonidentical oscillators. It has been shown that strong heterogeneity in the frequencies of nonidentical oscillators might be harmful to chimera states. In this work, we consider a ring of nonlocally coupled bicomponent phase oscillators in which two types of oscillators are randomly distributed along the ring: some oscillators with natural frequency ω₁ and others with ω₂ . In this model, the heterogeneity in frequency is measured by frequency mismatch |ω₁−ω₂| between the oscillators in these two subpopulations. We report that the nonlocally coupled bicomponent phase oscillators allow for chimera states no matter how large the frequency mismatch is. The bicomponent oscillators are composed of two chimera states, one supported by oscillators with natural frequency ω₁ and the other by oscillators with natural frequency ω₂. The two chimera states in two subpopulations are synchronized at weak frequency mismatch, in which the coherent oscillators in them share similar mean phase velocity, and are desynchronized at large frequency mismatch, in which the coherent oscillators in different subpopulations have distinct mean phase velocities. The synchronization–desynchronization transition between chimera states in these two subpopulations is observed with the increase in the frequency mismatch. The observed phenomena are theoretically analyzed by passing to the continuum limit and using the Ott-Antonsen approach.     

Intermittent lag synchronization in a driven system of coupled oscillators

We study intermittent lag synchronization in a system of two identical mutually coupled Duffing oscillators with parametric modulation in one of them. This phenomenon in a periodically forced system can be seen as intermittent jump from phase to lag synchronization, during which the chaotic trajectory visits a periodic orbit closely. We demonstrate different types of intermittent lag synchronizations, that occur in the vicinity of saddle-node bifurcations where the system changes its dynamical state, and characterize the simplest case of period-one intermittent lag synchronization.     

Quasiregular concentric waves in heterogeneous lattices of coupled oscillators

We study the pattern formation in a lattice of locally coupled phase oscillators with quenched disorder. In the synchronized regime quasiregular concentric waves can arise which are induced by the disorder of the system. Maximal regularity is found at the edge of the synchronization regime. The emergence of the concentric waves is related to the symmetry breaking of the interaction function. An explanation of the numerically observed phenomena is given in a one-dimensional chain of coupled phase oscillators. Scaling properties, describing the target patterns are obtained.     

Automatic control of phase synchronization in coupled complex oscillators

We present an automatic control method for phase locking of regular and chaotic nonidentical oscillations, when all subsystems interact via feedback. This method is based on the well known principle of feedback control which takes place in nature and is successfully used in engineering. In contrast to unidirectional and bidirectional coupling, the approach presented here supposes the existence of a special controller, which allows to change the parameters of the controlled systems. First we discuss general principles of automatic phase synchronization (PS) for arbitrary coupled systems with a controller whose input is given by a special quadratic form of coordinates of the individual systems and its output is a result of the application of a linear differential operator. We demonstrate the effectiveness of our approach for controlled PS on several examples: (i) two coupled regular oscillators, (ii) coupled regular and chaotic oscillators, (iii) two coupled chaotic Rössler oscillators, (iv) two coupled foodweb models, (v) coupled chaotic Rössler and Lorenz oscillators, (vi) ensembles of locally coupled regular oscillators, (vii) ensembles of locally coupled chaotic oscillators, and (viii) ensembles of globally coupled chaotic oscillators.     

Intermittent lag synchronization in a nonautonomous system of coupled oscillators

Synchronization properties of two identical mutually coupled Duffing oscillators with parametric modulation in one of them are studied. Intermittent lag synchronization is observed in the vicinity of saddle-node bifurcations where the system changes its dynamical state. This phenomenon is seen as intermittent jumps from phase to lag synchronization, during which the chaotic trajectory visits closely a periodic orbit. Different types of intermittent lag synchronization are demonstrated and the simplest case of period-one lag synchronization is analyzed.     

Chimera dynamics in nonlocally coupled moving phase oscillators

Chimera states, a symmetry-breaking spatiotemporal pattern in nonlocally coupled dynamical units, prevail in a variety of systems. However, the interaction structures among oscillators are static in most of studies on chimera state. In this work, we consider a population of agents. Each agent carries a phase oscillator. We assume that agents perform Brownian motions on a ring and interact with each other with a kernel function dependent on the distance between them. When agents are motionless, the model allows for several dynamical states including two different chimera states (the type-I and the type-II chimeras). The movement of agents changes the relative positions among them and produces perpetual noise to impact on the model dynamics. We find that the response of the coupled phase oscillators to the movement of agents depends on both the phase lag α, determining the stabilities of chimera states, and the agent mobility D. For low mobility, the synchronous state transits to the type-I chimera state for α close to π/2 and attracts other initial states otherwise. For intermediate mobility, the coupled oscillators randomly jump among different dynamical states and the jump dynamics depends on α. We investigate the statistical properties in these different dynamical regimes and present the scaling laws between the transient time and the mobility for low mobility and relations between the mean lifetimes of different dynamical states and the mobility for intermediate mobility.     

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.     

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.     

Using nonisochronicity to control synchronization in ensembles of nonidentical oscillators

We investigate the transition to synchronization in ensembles of coupled oscillators with quenched disorder. We find that small coupling is able to increase the frequency disorder and to induce a spread of oscillator frequencies. This new effect of anomalous desynchronization is studied with numerical and analytical means in a large class of systems including R?ssler, Lotka-Volterra, Landau-Stuart, and Van-der-Pol oscillators. We show that anomalous effects arise due to an interplay between nonisochronicity and natural frequency of each oscillator and can either increase or inhibit synchronization in the ensemble. This provides a novel possibility to control the synchronization transition in nonidentical systems by suitably distributing the disorder among system parameters. We conjecture that our results are of relevance for biological systems.     

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.     

Lag synchronization and scaling of chaotic attractor in coupled system

We report a design of delay coupling for lag synchronization in two unidirectionally coupled chaotic oscillators. A delay term is introduced in the definition of the coupling to target any desired lag between the driver and the response. The stability of the lag synchronization is ensured by using the Hurwitz matrix stability. We are able to scale up or down the size of a driver attractor at a response system in presence of a lag. This allows compensating the attenuation of the amplitude of a signal during transmission through a delay line. The delay coupling is illustrated with numerical examples of 3D systems, the Hindmarsh-Rose neuron model, the Ro?ssler system, a Sprott system, and a 4D system. We implemented the coupling in electronic circuit to realize any desired lag synchronization in chaotic oscillators and scaling of attractors.     

Phase synchronization in coupled nonidentical excitable systems and array-enhanced coherence resonance

We study the dynamics of a lattice of coupled nonidentical Fitz Hugh-Nagumo system subject to independent external noise. It is shown that these stochastic oscillators can lead to global synchronization behavior without an external signal. With the increase of the noise intensity, the system exhibits coherence resonance behavior. Coupling can enhance greatly the noise-induced coherence in the system.     

Spatiotemporal Regimes in the Kuramoto–Battogtokh System of Nonidentical Oscillators

Journal of Experimental and Theoretical Physics - We consider the spatiotemporal states of an ensemble of nonlocally coupled nonidentical phase oscillators, which correspond to different regimes of...