Breathing chimera in a system of phase oscillators

Chimera states consisting of synchronous and asynchronous domains in a medium of nonlinearly coupled phase oscillators have been considered. Stationary inhomogeneous solutions of the Ott–Antonsen equation for a complex order parameter that correspond to fundamental chimeras have been constructed. The direct numerical simulation has shown that these structures under certain conditions are transformed to oscillatory (breathing) chimera regimes because of the development of instability.     

Order parameter analysis of synchronization transitions on star networks

The collective behaviors of populations of coupled oscillators have attracted significant attention in recent years. In this paper, an order parameter approach is proposed to study the low-dimensional dynamical mechanism of collective synchronizations, by adopting the star-topology of coupled oscillators as a prototype system. The order parameter equation of star-linked phase oscillators can be obtained in terms of the Watanabe–Strogatz transformation, Ott–Antonsen ansatz, and the ensemble order parameter approach. Different solutions of the order parameter equation correspond to the diverse collective states, and different bifurcations reveal various transitions among these collective states. The properties of various transitions in the star-network model are revealed by using tools of nonlinear dynamics such as time reversibility analysis and linear stability analysis.     

Periodic synchronization in a system of coupled phase oscillators with attractive and repulsive interactions

In this study we investigate the collective behavior of the generalized Kuramoto model with an external pinning force in which oscillators with positive and negative coupling strengths are conformists and contrarians, respectively. We focus on a situation in which the natural frequencies of the oscillators follow a uniform probability density. By numerically simulating the model, it is shown that the model supports multistable synchronized states such as a traveling wave state, π state and periodic synchronous state: an oscillating π state. The oscillating π state may be characterized by the phase distribution oscillating in a confined region and the phase difference between conformists and contrarians oscillating around π periodically. In addition, we present the parameter space of the oscillating π state and traveling wave state of the model.     

Chimera states in a two–population network of coupled pendulum–like elements

More than a decade ago, a surprising coexistence of synchronous and asynchronous behavior called the chimera state was discovered in networks of nonlocally coupled identical phase oscillators. In later years, chimeras were found to occur in a variety of theoretical and experimental studies of chemical and optical systems, as well as models of neuron dynamics. In this work, we study two coupled populations of pendulum-like elements represented by phase oscillators with a second derivative term multiplied by a mass parameter m and treat the first order derivative terms as dissipation with parameter ? > 0. We first present numerical evidence showing that chimeras do exist in this system for small mass values 0 < m ? 1. We then proceed to explain these states by reducing the coherent population to a single damped pendulum equation driven parametrically by oscillating averaged quantities related to the incoherent population.     

Convective and absolute instabilities in the subcritical Ginzburg-Landau equation

We study the nature of the instability of the homogeneous steady states of the subcritical real Ginzburg-Landau equation in the presence of group velocity. The shift of the absolute instability threshold of the trivial steady state, induced by the destabilizing cubic nonlinearities, is confirmed by the numerical analysis of the evolution of its perturbations. It is also shown that the dynamics of these perturbations is such that finite size effects may suppress the transition from convective to absolute instability. Finally, we analyze the instability of the subcritical middle branch of steady states, and show, analytically and numerically, that this branch may be convectively unstable for sufficiently high values of the group velocity. Received 17 December 1998     

Relaxation dynamics of the Kuramoto model with uniformly distributed natural frequencies

The Kuramoto model describes a system of globally coupled phase-only oscillators with distributed natural frequencies. The model in the steady state exhibits a phase transition as a function of the coupling strength, between a low-coupling incoherent phase in which the oscillators oscillate independently and a high-coupling synchronized phase. Here, we consider a uniform distribution for the natural frequencies, for which the phase transition is known to be of first order. We study how the system close to the phase transition in the supercritical regime relaxes in time to the steady state while starting from an initial incoherent state. In this case, numerical simulations of finite systems have demonstrated that the relaxation occurs as a step-like jump in the order parameter from the initial to the final steady state value, hinting at the existence of metastable states. We provide numerical evidence to suggest that the observed metastability is a finite-size effect, becoming an increasingly rare event with increasing system size.     

Multistability of twisted states in non-locally coupled Kuramoto-type models

A ring of N identical phase oscillators with interactions between L-nearest neighbors is considered, where L ranges from 1 (local coupling) to N/2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sign which has a profound influence on its behavior. Without the limitation of the generality, the frequency of the free-running oscillators can be set to zero. The resulting system is of gradient type, and therefore, all its solutions converge to an equilibrium point. All so-called q-twisted states, where the phase difference between neighboring oscillators on the ring is 2πq/N, are equilibrium points, where q is an integer. Their stability in the limit N → ∞ is discussed along the line of Wiley et al. [Chaos 16, 015103 (2006)] In addition, we prove that when a twisted state is asymptotically stable for the infinite system, it is also asymptotically stable for sufficiently large N. Note that for smaller N, the same q-twisted states may become unstable and other q-twisted states may become stable. Finally, the existence of additional equilibrium states, called here multi-twisted states, is shown by numerical simulation. The phase difference between neighboring oscillators is approximately 2πq/N in one sector of the ring, -2πq/N in another sector, and it has intermediate values between the two sectors. Our numerical investigation suggests that the number of different stable multi-twisted states grows exponentially as N → ∞. It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discrete-time translational dynamical system where the space-variable (position on the ring) plays the role of time. The q-twisted states are then fixed points, and the multi-twisted states are periodic solutions of period N that are close to a heteroclinic cycle. Due to the apparently exponentially fast growing number of such stable periodic solutions, the system shows spatial chaos as N → ∞.     

耦合振子系统的多稳态同步分析

本文讨论了一维闭合环上Kuramoto相振子在非对称耦合作用下同步区域出现的多定态现象. 研究发现在振子数N≤3情形下系统不会出现多态现象, 而N≥4多振子系统则呈现规律的多同步定态. 我们进一步对耦合振子系统中出现的多定态规律及定态稳定性进行了理论分析, 得到了定态渐近稳定解. 数值模拟多体系统发现同步区特征和理论描述相一致. 研究结果显示在绝热条件下随着耦合强度的减小, 系统从不同分支的同步态出发最终会回到同一非同步态. 这说明, 耦合振子系统在非同步区由于运动的遍历性而只具有单一的非同步态, 在发生同步时由于遍历性破缺会产生多个同步定态的共存现象.     

Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities

A model dynamical system with a great many degrees of freedom is proposed for which the critical condition for the onset of collective oscillations, the evolution of a suitably defined order parameter, and its fluctuations around steady states can be studied analytically. This is a rotator model appropriate for a large population of limit cycle oscillators. It is assumed that the natural frequencies of the oscillators are distributed and that each oscillator interacts with all the others uniformly. An exact self-consistent equation for the stationary amplitude of the collective oscillation is derived and is extended to a dynamical form. This dynamical extension is carried out near the transition point where the characteristic time scales of the order parameter and of the individual oscillators become well separated from each other. The macroscopic evolution equation thus obtained generally involves a fluctuating term whose irregular temporal variation comes from a deterministic torus motion of a subpopulation. The analysis of this equation reveals order parameter behavior qualitatively different from that in thermodynamic phase transitions, especially in that the critical fluctuations in the present system are extremely small.Dedicated to Ilya Prigogine on the occasion of his 70th birthday.     

On the response of an oscillatory medium to defect generation

We investigate the response of a system far from equilibrium close to an oscillatory instability to the induction of phase singularities. We base our investigation on a numerical treatment of the complex Ginzburg–Landau equation (CGLE) in two spatial dimensions, which is considered as an order-parameter equation for lasers and other nonlinear optical systems. Defects are randomly generated by a spatially modulated linear growth rate. In the amplitude-turbulent regime, no qualitative change of behaviour can be detected. Phase-turbulent patterns emerging due to the Benjamin–Feir instability are destroyed by the externally injected defects. One observes either states consisting of spiral structures of various sizes which resemble the vortex glass states of the unperturbed system or a travelling wave pattern containing moving topological defects. In parameter space, both states are separated by a well-defined phase boundary which is close to the line separating convectively from absolutely stable travelling waves. PACS 47.54.+r; 89.75.Kd; 42.65.Sf; 47.32.Cc; 47.27.Cn; 05.45.-a     

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.     

Coherent States of Position-Dependent Mass Oscillator

In this paper, we study Gazeau-Klauder and displacement-type coherent states of two-dimensional position-dependent mass oscillators, which is called Λ-dependent oscillators and Λ can be interpreted as the curvatures of the spherical and the hyperbolic spaces, on which oscillators are constrained. In addition, we consider the effect of Λ parameter on the physical properties of these coherent states, including minimized Heisenberg uncertainty relation and Mandel’s Q parameter. We also elaborate the relation between the curvature of the physical space and the curvature of the Λ-dependent coherent state manifold.     

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.     

First order transition to oscillation death through an environment

The emergence of dynamical abrupt transitions for the first time in an ensemble of identical limit-cycle and chaotic oscillators coupled via a common environment is reported. The transition from the oscillatory state to the death state and vice versa, in these networks of oscillators are found not only discontinuous as well as irreversible in the parameter space. This first order phase transition in these systems is termed as Explosive Death. The occurrence of such transition is studied in details by using an appropriate order parameter for both limit-cycle and chaotic oscillators, in particular, Stuart–Landau and Rössler oscillators. The backward transition point for this phenomenon is obtained analytically using linear stability analysis and is found to be consistent with the numerical results.     

Chimeras in random non-complete networks of phase oscillators

We consider the simplest network of coupled non-identical phase oscillators capable of displaying a "chimera" state (namely, two subnetworks with strong coupling within the subnetworks and weaker coupling between them) and systematically investigate the effects of gradually removing connections within the network, in a random but systematically specified way. We average over ensembles of networks with the same random connectivity but different intrinsic oscillator frequencies and derive ordinary differential equations (ODEs), whose fixed points describe a typical chimera state in a representative network of phase oscillators. Following these fixed points as parameters are varied we find that chimera states are quite sensitive to such random removals of connections, and that oscillations of chimera states can be either created or suppressed in apparent bifurcation points, depending on exactly how the connections are gradually removed.     

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.     

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.     

The Ginzburg-Landau approach to oscillatory media

Close to a supercritical Hopf bifurcation, oscillatory media may be described, by the complex Ginzburg-Landau equation. The most important spatiotemporal behaviors associated with this dynamics are reviewed here. It is shown, on a few concrete examples, how real chemical oscillators may be described by this equation, and how its coefficients may be obtained from the experimental data. Furthermore, the effect of natural forcings, induced by the experimental realization of chemical oscillators in batch reactors, may also be studied in the framework of complex Ginzburg-Landau equations and its associated phase dynamics. We show, in particular, how such forcings may locally transform oscillatory media into excitable ones and trigger the formation of complex spatiotemporal patterns.     

Secondary transverse instabilities in optical parametric oscillators

We investigate the effect of coupling between diffraction and walk-off on secondary instabilities in nondegenerate optical parametric oscillators. We show that traveling waves that propagate in the walk-off direction, which are generated at the onset of absolute instability, experience Eckhaus and zigzag phase instabilities. Each of these secondary instabilities splits into absolute and convective instabilities that modify the Eckhaus and zigzag instability boundaries. As a consequence, the stability domain of modulated traveling waves is enlarged and may coexist with uniform steady states. The predictions are consistent with the numerical solutions of the optical parametric oscillator model.     

A new approach to detecting weak signal in strong noise based on chaos system control

In this paper, a chaos system and proportional differential control are both used to detect the frequency of an unknown signal. In traditional methods the useful signal is obtained through the Duffing equation or other chaotic oscillators. But these methods are too complex because of using a lot of chaos oscillators. In this paper a new method is presented that uses the R?ssler equation and proportional differential control to detect a weak signal frequency. Substituting the detected signal frequency into the R?ssler equation leads the R?ssler phase state to be considerably changed. The chaos state can be controlled through the proportional differential method. Through its phase diagram and spectrum analysis, the unknown frequency is obtained. The simulation results verify that the presented method is feasible and that the detection accuracy is higher than those of other methods.