Desynchronization and clustering with pulse stimulations of coupled electrochemical relaxation oscillators

Effects of pulse stimulations on the dynamics of relaxation oscillator populations were experimentally studied in a globally coupled electrochemical system. Similar to smooth oscillations, weakly and moderately relaxational oscillations possess a vulnerable phase, ?_S; pulses applied at ?_S resulted in desynchronization followed by a return to the synchronized state. In contrast to smooth oscillators, weakly and moderately relaxational oscillators exhibited transient and itinerant cluster dynamics, respectively. With strongly relaxational oscillators the pulse applied at a vulnerable phase effected transitions to other cluster configurations without effective desynchronization. Repeated pulse administration resulted in a cluster state that is stable against the perturbation; the cluster configuration is specific to the pulse administered at the vulnerable phase. The pulse-induced transient clusters are interpreted with a phase model that includes first and second harmonics in the interaction function and exhibits saddle type cluster states with strongly stable intra-cluster and weakly unstable inter-cluster modes.     

Experiments on arrays of globally coupled chaotic electrochemical oscillators: Synchronization and clustering

Experiments on chaotically oscillating arrays of 64 nickel electrodes in sulfuric acid were carried out. External resistors in parallel and series are added to vary the extent of global coupling among the oscillators without changing the other properties of the system. The array is heterogeneous due to small variations in the properties of the electrodes and there is also a small amount of noise. The addition of global coupling transforms a system of independent elements to a state of complete synchronization. At intermediate coupling strengths stable clusters, or condensates of elements, form. All the elements in a cluster follow the same chaotic trajectory but each cluster has its own dynamics; the system is thus temporally chaotic but spatially ordered. Many cluster configurations occur under the same conditions and transitions among them can be produced. For values of the coupling parameter on either side of the stable cluster region a non-stationary behavior occurs in which clustered and synchronized states alternately form and break up. Some statistical properties of the cluster states are determined. (c) 2000 American Institute of Physics.     

Synchronization phenomena for a pair of locally coupled chaotic electrochemical oscillators: a survey

Chaotic synchronization of two locally coupled electrochemical oscillators is studied numerically. Both bidirectional and unidirectional couplings are considered. For both these coupling scenarios, varying the characteristics of the coupling terms (functional form and/or strength) reveals a wide variety of synchronization phenomena. Standard diagnostic tests are performed to verify and classify the different types of synchronizations observed.     

Two coupled impact oscillators

A forced impact oscillator is coupled to a second freely moving oscillator with no amplitude constraint. The influence of the coupling strength and the external frequency is studied in detail. For intermediate coupling strength many of the long-period or chaotic motions are suppressed and replaced by short-period motions at least for not too high frequencies. Phase space trajectories are discussed for some characteristic examples.     

Oscillation death in coupled oscillators

We study dynamical behaviors in coupled nonlinear oscillators and find that under certain conditions, a whole coupled oscillator system can cease oscillation and transfer to a globally nonuniform stationary state [i.e., the so-called oscillation death (OD) state], and this phenomenon can be generally observed. This OD state depends on coupling strengths and is clearly different from previously studied amplitude death (AD) state, which refers to the phenomenon where the whole system is trapped into homogeneously steady state of a fixed point, which already exists but is unstable in the absence of coupling. For larger systems, very rich pattern structures of global death states are observed. These Turing-like patterns may share some essential features with the classical Turing pattern.      

Chimera states for coupled oscillators

Arrays of identical oscillators can display a remarkable spatiotemporal pattern in which phase-locked oscillators coexist with drifting ones. Discovered two years ago, such "chimera states" are believed to be impossible for locally or globally coupled systems; they are peculiar to the intermediate case of nonlocal coupling. Here we present an exact solution for this state, for a ring of phase oscillators coupled by a cosine kernel. We show that the stable chimera state bifurcates from a spatially modulated drift state, and dies in a saddle-node bifurcation with an unstable chimera state.     

Coalescence in coupled DulTing oscillators

The forced Duffing oscillator has a pair of symmetrical attractors in a proper parameter regime. When a lot of Duffing oscillators are coupled linearly, the system tends to form dusters in which the neighboring oscillators fall onto the same attractor. When the coupling strength is strong, all of the oscillators fall onto one attractor. In this work, we investigate coalescence in the coupled forced Duffing oscillators. Some phenomena are found and explanations are presented.     

Transference of squeezing in coupled oscillators

We examine the possibility of obtaining the transference of the squeezing effect between two coupled oscillators, one of them described by a quadratic Hamiltonian in terms of the ladder operators, the other one being a linear harmonic oscillator, plus an interaction term. We obtain an exact solution for the time evolution of our coupled system which allows us to find the variances for one and two-mode oscillations. It is shown that the squeezing generated in one of the oscillators may or may not spread to the other oscillator, depending on the choice of the involved parameters. Other interesting features exhibited for the one- and two-mode oscillations are also discussed.     

Frequency analysis with coupled nonlinear oscillators

We present a method to obtain the frequency spectrum of a signal with a nonlinear dynamical system. The dynamical system is composed of a pool of adaptive frequency oscillators with negative mean-field coupling. For the frequency analysis, the synchronization and adaptation properties of the component oscillators are exploited. The frequency spectrum of the signal is reflected in the statistics of the intrinsic frequencies of the oscillators. The frequency analysis is completely embedded in the dynamics of the system. Thus, no pre-processing or additional parameters, such as time windows, are needed. Representative results of the numerical integration of the system are presented. It is shown, that the oscillators tune to the correct frequencies for both discrete and continuous spectra. Due to its dynamic nature the system is also capable to track non-stationary spectra. Further, we show that the system can be modeled in a probabilistic manner by means of a nonlinear Fokker-Planck equation. The probabilistic treatment is in good agreement with the numerical results, and provides a useful tool to understand the underlying mechanisms leading to convergence.     

Diffusion for weakly coupled quantum oscillators

We construct a simple model which exhibits some of the properties discussed by van Hove in his study of the Pauli master equation. The model consists of an infinite chain of quantum oscillators which are coupled so that the interaction Hamiltonian is quadratic. We suppose the chain is in equilibrium at an inverse temperature and study the return to equilibrium when a chosen oscillator is given an arbitrary perturbation. We show that in the limit as the interaction becomes weaker and of longer range, the evolution of the chosen oscillator becomes a diffusion equation. Moreover we give an explicit example where the evolution of the chosen oscillator has the Markov property and where the Pauli master equation is exactly satisfied.     

Coherence resonance in coupled chaotic oscillators

Existing works on coherence resonance, i.e., the phenomenon of noise-enhanced temporal regularity, focus on excitable dynamical systems such as those described by the FitzHugh-Nagumo equations. We extend the scope of coherence resonance to an important class of dynamical systems: coupled chaotic oscillators. In particular, we show that, when a system of coupled chaotic oscillators is under the influence of noise, the degree of temporal regularity of dynamical variables characterizing the difference among the oscillators can increase and reach a maximum value at some optimal noise level. We present numerical results illustrating the phenomenon and give a physical theory to explain it.     

Decoherence in strongly coupled quantum oscillators

In this paper, we present a comprehensive analysis of the coherence phenomenon of two coupled dissipative oscillators. The action of a classical driving field on one of the oscillators is also analyzed. Master equations are derived for both regimes of weakly and strongly interacting oscillators from which interesting results arise concerning the coherence properties of the joint and the reduced system states. The strong coupling regime is required to achieve a large frequency shift of the oscillator normal modes, making it possible to explore the whole profile of the spectral density of the reservoirs. We show how the decoherence process may be controlled by shifting the normal mode frequencies to regions of small spectral density of the reservoirs. Different spectral densities of the reservoirs are considered and their effects on the decoherence process are analyzed. For oscillators with different damping rates, we show that the worse-quality system is improved and vice versa, a result which could be useful for quantum state protection. State recurrence and swap dynamics are analyzed as well as their roles in delaying the decoherence process.     

The saturation bifurcation in coupled oscillators

We examine examples of weakly nonlinear systems whose steady states undergo a bifurcation with increasing forcing, such that a forced subsystem abruptly ceases to absorb additional energy, instead diverting it into an initially quiescent, unforced subsystem. We derive and numerically verify analytical predictions for the existence and behavior of such saturated states for a class of oscillator pairs. We also examine related phenomena, including zero-frequency response to periodic forcing, Hopf bifurcations to quasiperiodicity, and bifurcations to periodic behavior with multiple frequencies.     

Synchronization of coupled oscillators on small-world networks

We investigate the synchronous dynamics of Kuramoto oscillators and van der Pol oscillators on Watts-Strogatz type small-world networks. The order parameters to characterize macroscopic synchronization are calculated by numerical integration. We focus on the difference between frequency synchronization and phase synchronization. In both oscillator systems, the critical coupling strength of the phase order is larger than that of the frequency order for the small-world networks. The critical coupling strength for the phase and frequency synchronization diverges as the network structure approaches the regular one. For the Kuramoto oscillators, the behavior can be described by a power-law function and the exponents are obtained for the two synchronizations. The separation of the critical point between the phase and frequency synchronizations is found only for small-world networks in the theoretical models studied.     

Collective behavior of coupled nonuniform stochastic oscillators

Theoretical studies of synchronization are usually based on models of coupled phase oscillators which, when isolated, have constant angular frequency. Stochastic discrete versions of these uniform oscillators have also appeared in the literature, with equal transition rates among the states. Here we start from the model recently introduced by Wood et al. [K. Wood, C. Van den Broeck, R. Kawai, K. Lindenberg, Universality of synchrony: critical behavior in a discrete model of stochastic phase-coupled oscillators, Phys. Rev. Lett. 96 (2006) 145701], which has a collectively synchronized phase, and parametrically modify the phase-coupled oscillators to render them (stochastically) nonuniform. We show that, depending on the nonuniformity parameter 0≤α≤1, a mean field analysis predicts the occurrence of several phase transitions. In particular, the phase with collective oscillations is stable for the complete graph only for α≤α^′<1. At α=1 the oscillators become excitable elements and the system has an absorbing state. In the excitable regime, no collective oscillations were found in the model.     

耦合分数阶混沌振荡器的主-从同步

近年来对分数阶系统的动力学研究得到了较为广泛的关注。本文研究了基于主-从耦合同步法的同步技术并实现了两个耦合的分数阶振荡器的混沌同步。仿真结果表明：在适当的耦合强度的调节下，该方法可实现两个耦合分数阶混沌振荡器的准确同步，且分数阶混沌振荡器的同步率明显慢于整数阶混沌振荡器的同步率；而耦合分数阶混沌振荡器在实现同步的过程中，随着阶数的提高，同步误差曲线变得平滑，这表明，系统阶数的提高改善了耦合混沌振荡器实现同步的平稳性。     

Coarse graining the dynamics of coupled oscillators

We present an equation-free computational approach to the study of the coarse-grained dynamics of finite assemblies of nonidentical coupled oscillators at and near full synchronization. We use coarse-grained observables which account for the (rapidly developing) correlations between phase angles and natural frequencies. Exploiting short bursts of appropriately initialized detailed simulations, we circumvent the derivation of closures for the long-term dynamics of the assembly statistics.