Nonidentical Oscillators Research Articles

Dynamical hysteresis and spatial synchronization in coupled non-identical chaotic oscillators

We identify a novel phenomenon in distinct (namely non-identical) coupled chaotic systems, which we term dynamical hysteresis. This behavior, which appears to be universal, is defined in terms of the system dynamics (quantified for example through the Lyapunov exponents), and arises from the presence of at least two coexisting stable attractors over a finite range of coupling, with a change of stability outside this range. Further characterization via mutual synchronization indices reveals that one attractor corresponds to spatially synchronized oscillators, while the other corresponds to desynchronized oscillators. Dynamical hysteresis may thus help to understand critical aspects of the dynamical behavior of complex biological systems, e.g. seizures in the epileptic brain can be viewed as transitions between different dynamical phases caused by time dependence in the brain’s internal coupling.

Stability of phase locking in a ring of unidirectionally coupled oscillators

We discuss the dynamic behaviour of a finite group of phase oscillators unidirectionally coupled in a ring. The dynamics are based on the Kuramoto model. In the case of identical oscillators, all phase locking solutions and their stability properties are obtained. For nonidentical oscillators it is proven that there exist phase locking solutions for sufficiently strong coupling. An algorithm to obtain all phase locking solutions is proposed. These solutions can be classified into classes, each with its own stability properties. The stability properties are obtained by means of a novel extension of Gershgorin's theorem. One class of stable solutions has the property that all phase differences between neighbouring cells are contained in . Contrary to intuition, a second class of stable solutions is established with exactly one of the phase differences contained in . The stability results are extended from sinusoidal interconnections to a class of odd functions. To conclude, a connection with the field of active antenna arrays is made, generalizing some results earlier obtained in this field.

Adaptation through minimization of the phase lag in coupled nonidentical systems.

We show that the internal control of adaptation can be obtained from the properties of the phase lag that results from phase synchronization of two nonidentical chaotic oscillators. The direction and magnitude of the phase lag depend upon the relative internal properties of the coupled units, and they can be used as indicators during the adjustment of dynamics, i.e., adaptation of the target unit to match that of the control. The properties of the phase lag are obtained using a method based on the estimation of properties of the distributions of relative event times of both (target and control) units. The phase lag dependent mechanism to control the adaptation process was applied to a system of nonidentical Rössler oscillators and a system of nonidentical Lorenz oscillators. We also elucidate its importance as a control mechanism of the changes of neuronal activity showing its application to neural adaptation.

Noise-aided synchronization of coupled chaotic electrochemical oscillators.

We report experimental and numerical results on noise-enhanced synchronization of two coupled chaotic oscillators. Enhanced synchronization is achieved through superimposing small-amplitude Gaussian noise on a common system parameter of the two chaotic oscillators. A resonancelike behavior is found: at an optimum level of noise, maximum synchronization is attained. The simulations show that the resonance behavior occurs with both identical and nonidentical oscillators. Noncommon (asymmetric and independent) noise does not enhance synchronization; common noise seems to enhance synchronization.

GENERATION OF HOMOCLINIC OSCILLATION IN THE PHASE SYNCHRONIZATION REGIME IN COUPLED CHUA'S OSCILLATORS

An experimental method for generating homoclinic oscillations using two nonidentical Chua's oscillators coupled in unidirectional mode is described here. A homoclinic oscillation is obtained at the response oscillator in the weaker coupling limit of phase synchronization. Different phase locking phenomena of homoclinic oscillations with external periodic pulse have been observed when the frequency of the pulse is close to the natural frequency of the homoclinic oscillation or its subharmonics.

Synchronization of two non-scalar-coupled limit-cycle oscillators

Being one of the fundamental phenomena in nonlinear science, synchronization of oscillations has permanently remained an object of intensive research. Development of many asymptotic methods and numerical simulations has allowed an understanding and explanation of various phenomena of self-synchronization. But even in the classical case of coupled van der Pol oscillators a full description of all possible dynamical regimes, their mutual transitions and characteristics is still lacking. We present here a study of the phenomenon of mutual synchronization for two non-scalar-coupled non-identical limit-cycle oscillators and analyze phase, frequency and amplitude characteristics of synchronization regimes. A series of bifurcation diagrams that we obtain exhibit various regions of qualitatively different behavior. Among them we find mono-, bi- and multistability regions, beating and “oscillation death” ones; also a region, where one of the oscillators dominates the other one is observed. The frequency characteristics that we obtain reveal three qualitatively different types of synchronization: (i) on the mean frequency (the in-phase synchronization), (ii) with a shift from the mean frequency caused by a conservative coupling term (the anti-phase synchronization), and (iii) on the frequency of one of the oscillators (when one oscillator dominates the other).

Rapid convergence of time-averaged frequency in phase synchronized systems.

Numerical and experimental evidences are presented to show that many phase synchronized systems of nonidentical chaotic oscillators, where the chaotic state is reached through a period-doubling cascade, show rapid convergence of the time-averaged frequency. The speed of convergence toward the natural frequency scales as the inverse of the measurement period. The results also suggest an explanation for why such chaotic oscillators can be phase synchronized.

Anomalous phase synchronization in populations of nonidentical oscillators.

We report the phenomenon of anomalous phase synchronization in interacting oscillator systems with randomly distributed parameters. We show that coupling is first able to enlarge the frequency disorder leading to maximal decoherence for intermediate levels of coupling strength before reaching synchronization. Anomalous synchronization arises when the natural frequency covaries with nonisochronicity and allows for synchronization control by adjustment of system parameters.

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 Rössler oscillators are verified experimentally with an array of chaotic electrochemical oscillators.

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 Rössler, 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.

Stability and Duration Time of Chaos Synchronization of a Class of Nonidentical Oscillators

In this paper, we consider the problem of stability and duration of the synchronization of two nonidentical chaotic oscillators. Despite modeling errors and parametric variation, we use the Lyapunov stability theory to derive a robust controller that comprises an uncertainty estimator and a linearizing control law. The expression of the synchronization time is explicitly computed. Numerical simulation results are shown to demonstrate the effectiveness and efficiency of the proposed method.

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

We study the transition route to complete synchronization through phase synchronization in generic coupled nonidentical chaotic oscillators. Through numerical studies, two routes are found, i.e., one, via lag synchronization, the other, via the intermittent chaotic burst state without lag synchronization. We claim that these two routes are universal. As evidence, we analyze several examples on the basis of the conventional theory of intermittency in the presence of noise.

Collective Dynamics of a Weakly Coupled Electrochemical Reaction on an Array

Experiments on the collective behavior and phase synchronization of weakly coupled populations of nonidentical chaotic electrochemical oscillators are described. The weak coupling has only a small effect on the local dynamics, i.e., through changes in frequency, but it has a strong influence on the collective or overall behavior. With added weak global or short-range coupling, a deviation from the law of large numbers is observed; large, irregular, and periodic mean-field oscillations occur, along with (partial) phase synchronization.

Collective dynamics of chaotic chemical oscillators and the law of large numbers.

Experiments on the nontrivial collective dynamics and phase synchronization of populations of nonidentical chaotic electrochemical oscillators are presented. Without added coupling no deviation from the law of large numbers is observed. Deviations do arise with weak global or short-range coupling; large, irregular, and periodic mean field oscillations occur along with (partial) phase synchronization.

Collective phase locked states in a chain of coupled chaotic oscillators.

We discuss the emergence of a collective phase locked state in an open chain of N unidirectionally weakly coupled nonidentical chaotic oscillators. Such a regime is characterized by a Lyapunov spectrum where N-1 exponents that were zero in the uncoupled regime assume negative values as the coupling strength increases. The dynamics of such collective state is studied, and a comparison is drawn with the case of phase synchronization of a pair of coupled chaotic oscillators. In particular, it is shown that a full phase synchronized state cannot be constructed without at least partial correlation in the chaotic amplitudes.

Transitions from partial to complete generalized synchronizations in bidirectionally coupled chaotic oscillators.

Generalized synchronization in an array of mutually (bidirectionally) coupled nonidentical chaotic oscillators is studied. Coupled Lorenz oscillators and coupled Lorenz-Rossler oscillators are adopted as our working models. With increasing the coupling strengths, the system experiences a cascade of transitions from the partial to the global generalized synchronizations, i.e., different oscillators are gradually entrained through a clustering process. This scenario of transitions reveals an intrinsic self-organized order in groups of interacting units, which generalizes the idea of generalized synchronizations in drive-response systems.

Synchronization of noise-induced bursts in noncoupled sensory neurons.

We report experimental observation of phase synchronization in an array of nonidentical noncoupled noisy neuronal oscillators, due to stimulation with external noise. The synchronization derives from a noise-induced qualitative change in the firing pattern of single neurons, which changes from a quasiperiodic to a bursting mode. We show that at a certain noise intensity the onsets of bursts in different neurons become synchronized, even though the number of spikes inside the bursts may vary for different neurons. We demonstrate this effect both experimentally for the electroreceptor afferents of paddlefish, and numerically for a canonical phase model, and characterize it in terms of stochastic synchronization.

PHASE CLUSTERS IN 2D ARRAYS OF NONIDENTICAL OSCILLATORS

We study the collective behavior of a 2D network of nonidentical oscillators in the chaotic regime and find that ordered phase can emerge in the collective behavior, due to mutual coupling. The resulting phase cluster is a frequency-locking. The configuration of pattern depends on time and can be divided into several phase clusters when coupling is over some threshold. The frequency-locked spatiotemporal patterns are determined by the mismatched parameter and the coupling strength, and its regularity can be measured by an average nearest-neighbor distance.

Frequency-Locking in Coupled Chaotic Systems**The project partly supported by the Hong Kong Research Grant Council and the Hong Kong Baptist University Faculty Research Grant, National Natural Science Foundation of China and the Special Funds for Major State Basic Research Projects of China and the Foundation for University Key Teacher by the Ministry of Education

A novel approach is presented for measuring the phase synchronization (frequency-locking) of coupled nonidentical oscillators, which can characterize frequency-locking for chaotic systems without well-defined phase by measuring the mean frequency. Numerical simulations confirm the existence of frequency-locking. The relations between the mean frequency and the coupling strength and the frequency mismatch are given. For the coupled hyperchaotic systems, the frequency-locking can be better characterized by more than one mean frequency curves.

EXPERIMENTAL OBSERVATION OF LAG SYNCHRONIZATION IN COUPLED CHAOTIC SYSTEMS

Chaotic lag synchronization is a subtle dynamical phenomenon in which the states of two mutually coupled, nonidentical chaotic oscillators follow each other with a fixed time delay. We investigate to what extent lag synchronization can be observed in laboratory experiments. Our measurements indicate that due to the influence of noise, lag synchronization appears to occur intermittently in time. Numerical confirmation and a heuristic explanation to the observed intermittent behavior are given.