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.     

Stochastic biperiodic oscillations in the electroreceptors of paddlefish

We report that the electroreceptors in paddlefish possess the novel property of being biperiodic, that is, being composed of two intrinsic self-sustained noisy oscillators, one residing in the hair cells, and another in the terminals of primary afferent neurons. The two oscillators are coupled unidirectionally. Thus the receptor system as a whole undergoes stochastic biperiodic oscillations. We characterize the spontaneous activity of this system of coupled biological oscillators, and also discuss the impact of the biperiodic organization on the transduction of external sensory stimuli. In particular, we show that the existence of hair cell oscillations leads to additional variability of afferent spike trains.     

Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators

Synaptically coupled neurons show in-phase or antiphase synchrony depending on the chemical and dynamical nature of the synapse. Deterministic theory helps predict the phase differences between two phase-locked oscillators when the coupling is weak. In the presence of noise, however, deterministic theory faces difficulty when the coexistence of multiple stable oscillatory solutions occurs. We analyze the solution structure of two coupled neuronal oscillators for parameter values between a subcritical Hopf bifurcation point and a saddle node point of the periodic branch that bifurcates from the Hopf point, where a rich variety of coexisting solutions including asymmetric localized oscillations occurs. We construct these solutions via a multiscale analysis and explore the general bifurcation scenario using the lambda-omega model. We show for both excitatory and inhibitory synapses that noise causes important changes in the phase and amplitude dynamics of such coupled neuronal oscillators when multiple oscillatory solutions coexist. Mixed-mode oscillations occur when distinct bistable solutions are randomly visited. The phase difference between the coupled oscillators in the localized solution, coexisting with in-phase or antiphase solutions, is clearly represented in the stochastic phase dynamics.     

耦合相振子系统同步的序参量理论

节律行为,即系统行为呈现随时间的周期变化,在我们的周围随处可见.不同节律之间可以通过相互影响、相互作用产生自组织,其中同步是最典型、最直接的有序行为,它也是非线性波、斑图、集群行为等的物理内在机制.不同的节律可以用具有不同频率的振子(极限环)来刻画,它们之间的同步可以用耦合极限环系统的动力学来加以研究.微观动力学表明,随着耦合强度增强,振子同步伴随着动力学状态空间降维到一个低维子空间,该空间由序参量来描述.序参量的涌现及其所描述的宏观动力学行为可借助于协同学与流形理论等降维思想来进行.本文从统计物理学的角度讨论了耦合振子系统序参量涌现的几种降维方案,并对它们进行了对比分析.序参量理论可有效应用于耦合振子系统的同步自组织与相变现象的分析,通过进一步研究序参量的动力学及其分岔行为,可以对复杂系统的涌现动力学有更为深刻的理解.     

Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators

Experimental observations of time-delay-induced amplitude death in two coupled nonlinear electronic circuits that are individually capable of exhibiting limit-cycle oscillations are described. The existence of multiply connected death islands in the parameter space of coupling strength and time delay for coupled identical oscillators is established. The existence of such regions was predicted earlier on theoretical grounds [Phys. Rev. Lett. 80, 5109 (1998); Physica (Amsterdam) 129D, 15 (1999)]. The experiments also reveal the occurrence of multiple frequency states, frequency suppression of oscillations with increased time delay, and the onset of both in-phase and antiphase collective oscillations.     

Collective phase sensitivity

The collective phase response to a macroscopic external perturbation of a population of interacting nonlinear elements exhibiting collective oscillations is formulated for the case of globally coupled oscillators. The macroscopic phase sensitivity is derived from the microscopic phase sensitivity of the constituent oscillators by a two-step phase reduction. We apply this result to quantify the stability of the macroscopic common-noise-induced synchronization of two uncoupled populations of oscillators undergoing coherent collective oscillations.     

Forced Synchronization of Electroconvective Roll Oscillations in Nematic Liquid Crystals

The forced phase locking in a system of the oscillating electroconvective rolls that form in a nematic liquid crystal layer in a dc electric field is studied. As a result of the action of an additive ac electric field with a small amplitude, the system of oscillators is found to be divided into clusters, where oscillations are fully phase locked. The electroconvective rolls in neighboring clusters oscillate in antiphase and the clusters are separated by Ising walls. The phase locking is shown to be maximal at the forcing frequency that is close to the double frequency of the natural oscillations of rolls. A model is proposed to describe spatially distributed phase oscillators, and it takes into account the symmetries of a system of electroconvective rolls and external forcing. The results of numerical simulations agree well with the experimental data.     

Spatial disorder and pattern formation in lattices of coupled bistable elements

The spatio-temporal dynamics of discrete lattices of coupled bistable elements is considered. It is shown that both regular and chaotic spatial field distributions can be realized depending on parameter values and initial conditions. For illustration, we provide results for two lattice systems: the FitzHugh-Nagumo model and a network of coupled bistable oscillators. For the latter we also prove the existence of phase clusters, with phase locking of elements in each cluster.     

Phase synchronization between collective rhythms of globally coupled oscillator groups: noisy identical case

We theoretically investigate the collective phase synchronization between interacting groups of globally coupled noisy identical phase oscillators exhibiting macroscopic rhythms. Using the phase reduction method, we derive coupled collective phase equations describing the macroscopic rhythms of the groups from microscopic Langevin phase equations of the individual oscillators via nonlinear Fokker-Planck equations. For sinusoidal microscopic coupling, we determine the type of the collective phase coupling function, i.e., whether the groups exhibit in-phase or antiphase synchronization. We show that the macroscopic rhythms can exhibit effective antiphase synchronization even if the microscopic phase coupling between the groups is in-phase, and vice versa. Moreover, near the onset of collective oscillations, we analytically obtain the collective phase coupling function using center-manifold and phase reductions of the nonlinear Fokker-Planck equations.     

Phase synchronization of chaotic rotators

We demonstrate the existence of phase synchronization of two chaotic rotators. Contrary to phase synchronization of chaotic oscillators, here the Lyapunov exponents corresponding to both phases remain positive even in the synchronous regime. Such frequency locked dynamics with different ratios of frequencies are studied for driven continuous-time rotators and for discrete circle maps. We show that this transition to phase synchronization occurs via a crisis transition to a band-structured attractor.     

Detecting phase synchronization between coupled non-phase-coherent oscillators

We compare two methods for detecting phase synchronization in coupled non-phase-coherent oscillators. One method is based on the locking of self-sustained oscillators with an irregular signal. The other uses trajectory recurrences in phase space. We identify the pros and cons of both methods and propose guidelines to detect phase synchronization in data series.     

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.     

Rotating class of parametric resonance processes in coupled oscillators

We single out and elucidate basic features of strongly coupled parametric oscillators as opposed by symmetry to the type reducible to uncoupled Mathieu equations. The strongly coupled type is essential to the oscillations and waves in rotating structures and chiral systems. It exhibits only the main parametric resonance, no higher-order ones. As well, the relaxation–fluctuation behavior is unusual, characterized by a negative-energy oscillation mode that is, nonetheless, stable so the Gibbs statistics lose its force. The models presented are exactly solvable.     

Standing waves, clustering, and phase waves in 1D simulations of kinetic relaxation oscillations in NO+NH3 on Pt(1 0 0) coupled by diffusion

The Lombardo–Imbihl–Fink (LFI) ODE model of the NO+NH₃ reaction on a Pt(1 0 0) surface shows stable relaxation oscillations with very sharp transitions for temperatures T between 404 and 433 K. Here we study numerically the effect of linear diffusive coupling of these oscillators in one spatial dimension. Depending on the parameters and initial conditions we find a rich variety of spatio-temporal patterns which we group into four main regimes: bulk oscillations (BOs), standing waves (SW), phase clusters (PC), and phase waves (PW). Two key ingredients for SW and PC are identified, namely the relaxation type of the ODE oscillations and a nonlocal (and nonglobal) coupling due to relatively fast diffusion of the kinetically slaved variables NH₃ and H. In particular, the latter replaces the global coupling through the gas phase used to obtain SW and PC in models of related surface reactions. The PW exist only under the assumption of (relatively) slow diffusion of NH₃ and H.     

Experimental observation of multifrequency patterns in arrays of coupled nonlinear oscillators

Frequency-related oscillations in coupled oscillator systems, in which one or more oscillators oscillate at different frequencies than the other oscillators, have been studied using group theoretical methods by Armbruster and Chossat [Phys. Lett. A 254, 269 (1999)] and more recently by Golubitsky and Stewart [in Geometry, Mechanics, and Dynamics, edited by P. Newton, P. Holmes, and A. Weinstein (Springer, New York, 2002), p. 243]. We demonstrate, experimentally, via electronic circuits, the existence of frequency-related oscillations in a network of two arrays of N oscillators, per array, coupled to one another. Under certain conditions, one of the arrays can be induced to oscillate at N times the frequency of the other array. This type of behavior is different from the one observed in a driven system because it is dictated mainly by the symmetry of the coupled system.     

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.     

Spiral wave chimeras in populations of oscillators coupled to a slowly varying diffusive environment

Chimera states are firstly discovered in nonlocally coupled oscillator systems. Such a nonlocal coupling arises typically as oscillators are coupled via an external environment whose characteristic time scale τ is so small (i.e., τ → 0) that it could be eliminated adiabatically. Nevertheless, whether the chimera states still exist in the opposite situation (i.e., τ ≫ 1) is unknown. Here, by coupling large populations of Stuart−Landau oscillators to a diffusive environment, we demonstrate that spiral wave chimeras do exist in this oscillator-environment coupling system even when τ is very large. Various transitions such as from spiral wave chimeras to spiral waves or unstable spiral wave chimeras as functions of the system parameters are explored. A physical picture for explaining the formation of spiral wave chimeras is also provided. The existence of spiral wave chimeras is further confirmed in ensembles of FitzHugh−Nagumo oscillators with the similar oscillator-environment coupling mechanism. Our results provide an affirmative answer to the observation of spiral wave chimeras in populations of oscillators mediated via a slowly changing environment and give important hints to generate chimera patterns in both laboratory and realistic chemical or biological systems.     

Synchronization of Kuramoto oscillators in random complex networks

We study the synchronization of coupled phase oscillators in random complex networks. The topology of the networks is assumed to be vary over time. Here we mainly study the onset of global phase synchronization when the topology switches rapidly over time. We find that the results are, to some extent, different from those in deterministic situations. In particular, the synchronizability of coupled oscillators can be enhanced in ER networks and scale-free networks under fast switching, while in stochastic small-world networks such enhancement is not significant.     

Bloch oscillations for circularly polarized light

All previous investigations on the Bloch oscillations of waves focus on scalar waves. Here we demonstrate, for the first time, the existence of Bloch oscillations of vector fields for circularly polarized light (CPL) propagating through a designed liquid crystal structure. To obtain the Wannier-Stark ladder of the CPL, we have designed a cholesteric liquid crystal structure with spatially varying pitch. The Bloch oscillations of the CPL have been observed in such a structure by exact numerical simulations. We have also shown that such a phenomenon can be easily detected in time-resolved reflection experiments.     

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.