Phase locking in networks of synaptically coupled McKean relaxation oscillators

We use geometric dynamical systems methods to derive phase equations for networks of weakly connected McKean relaxation oscillators. We derive an explicit formula for the connection function when the oscillators are coupled with chemical synapses modeled as the convolution of some input spike train with an appropriate synaptic kernel. The theory allows the systematic investigation of the way in which a slow recovery variable can interact with synaptic time scales to produce phase-locked solutions in networks of pulse coupled neural relaxation oscillators. The theory is exact in the singular limit that the fast and slow time scales of the neural oscillator become effectively independent. By focusing on a pair of mutually coupled McKean oscillators with alpha function synaptic kernels, we clarify the role that fast and slow synapses of excitatory and inhibitory type can play in producing stable phase-locked rhythms. In particular we show that for fast excitatory synapses there is coexistence of a stable synchronous, a stable anti-synchronous, and one stable asynchronous solution. For slower synapses the anti-synchronous solution can lose stability, whilst for even slower synapses it can regain stability. The case of inhibitory synapses is similar up to a reversal of the stability of solution branches. Using a return-map analysis the case of strong pulsatile coupling is also considered. In this case it is shown that the synchronous solution can co-exist with a continuum of asynchronous states.     

Bifurcation, amplitude death and oscillation patterns in a system of three coupled van der Pol oscillators with diffusively delayed velocity coupling

In this paper, we study a system of three coupled van der Pol oscillators that are coupled through the damping terms. Hopf bifurcations and amplitude death induced by the coupling time delay are first investigated by analyzing the related characteristic equation. Then the oscillation patterns of these bifurcating periodic oscillations are determined and we find that there are two kinds of critical values of the coupling time delay: one is related to the synchronous periodic oscillations, the other is related to eight branches of asynchronous periodic solutions bifurcating simultaneously from the zero solution. The stability of these bifurcating periodic solutions are also explicitly determined by calculating the normal forms on center manifolds, and the stable synchronous and stable phase-locked periodic solutions are found. Finally, some numerical simulations are employed to illustrate and extend our obtained theoretical results and numerical studies also describe the switches of stable synchronous and phase-locked periodic oscillations.     

Desynchronization by phase slip patterns in networks of pulse-coupled oscillators with delays

Coupling delays may cause drastic changes in the dynamics of oscillatory networks. In the present paper we investigate how coupling delays alter synchronization processes in networks of all-to-all coupled pulse oscillators. We derive an analytic criterion for the stability of synchrony and study the synchronization areas in the space of the delay and coupling strength. Specific attention is paid to the scenario of destabilization on the borders of the synchronization area. We show that in bifurcation points the system possesses homoclinic loops, which give rise to complex long- or quasi-periodic solutions. These newly born solutions are characterized by a synchronous group, from which an oscillator periodically escapes, laps one period, and rejoins. We call such a dynamical regime “phase slip patterns”.     

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.     

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.     

Discrete Breathers in Lattices of Coupled Oscillators

Discrete breathers are generic solutions for the dynamics of nonlinearly coupled oscillators. We show that discrete breathers can be observed in low-dimensional and high-dimensional lattices by exploring the sinusoidally coupled pendulum. Loss of stability of the breather solution is studied. We also find the existence of discrete breather in lattices with parameter mismatches. Breather phase synchronization is exhibited for the coupled chaotic oscillators.     

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.     

The dynamics of chimera states in heterogeneous Kuramoto networks

We study a variety of mixed synchronous/incoherent (“chimera”) states in several heterogeneous networks of coupled phase oscillators. For each network, the recently-discovered Ott-Antonsen ansatz is used to reduce the number of variables in the partial differential equation (PDE) governing the evolution of the probability density function by one, resulting in a time-evolution PDE for a variable with as many spatial dimensions as the network. Bifurcation analysis is performed on the steady states of these PDEs. The results emphasise the commonality of the dynamics of the different networks, and provide stability information that was previously inferred.     

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.     

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.     

Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators

For general networks of pulse-coupled oscillators, including regular, random, and more complex networks, we develop an exact stability analysis of synchronous states. As opposed to conventional stability analysis, here stability is determined by a multitude of linear operators. We treat this multioperator problem exactly and show that for inhibitory interactions the synchronous state is stable, independent of the parameters and the network connectivity. In randomly connected networks with strong interactions this synchronous state, displaying regular dynamics, coexists with a balanced state exhibiting irregular dynamics. External signals may switch the network between qualitatively distinct states.     

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 → ∞.     

The instability of chaotic synchronization in coupled Lorenz systems: from the Hopf to the Co-dimension two bifurcation

We investigate the Hopf bifurcation of the synchronous chaos in coupled Lorenz oscillators. We find that the system undergoes a phase transition along the Hopf instability of the synchronous chaos. The phase transition makes the traveling wave component with the phase difference φ(i)-φ(i+1)=2π/N between adjacent sites unstable. The phase transition also plays a role to relate the Hopf bifurcation with the co-dimension two bifurcation of the synchronous chaos.     

Role of multistability in the transition to chaotic phase synchronization

In this paper we describe the transition to phase synchronization for systems of coupled nonlinear oscillators that individually follow the Feigenbaum route to chaos. A nested structure of phase synchronized regions of different attractor families is observed. With this structure, the transition to nonsynchronous behavior is determined by the loss of stability for the most stable synchronous mode. It is shown that the appearance of hyperchaos and the transition from lag synchronization to phase synchronization are related to the merging of chaotic attractors from different families. Numerical examples using Rossler systems and model maps are given. (c) 1999 American Institute of Physics.     

Determination of a coupling function in multicoupled oscillators

A new method to determine a coupling function in a phase model is theoretically derived for coupled self-sustained oscillators and applied to Belousov-Zhabotinsky (BZ) oscillators. The synchronous behavior of two coupled BZ reactors is explained extremely well in terms of the coupling function thus obtained. This method is expected to be applicable to weakly coupled multioscillator systems, in which mutual coupling among nearly identical oscillators occurs in a similar manner. The importance of higher-order harmonic terms involved in the coupling function is also discussed.     

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.     

Clustering and Synchronization in an Array of Repulsively Coupled Phase Oscillators

Clustering and synchronization in an array of repulsively coupled phase oscillators are numerically investigated. It is found that oscillators are divided into several clusters according to the symmetry in the structure.Synchronization occurs between oscillators in each cluster, while those oscillators belonging to different clusters remain asynchronous. Such synchronization may collapse for all clusters when the dynamics of only one oscillator is altered properly. The synchronous state may return back after a short period of transient process. This is determined by the strength of the oscillator altered. Its application in the communication of one-to-several is suggested.     

Clustering and Synchronization in an Array of Repulsively Coupled Phase Oscillators

Clustering and synchronization in an array of repulsively coupled phase oscillators are numerically investigated. It is found that oscillators are divided into several clusters according to the symmetry in the structure. Synchronization occurs between oscillators in each cluster, while those oscillators belonging to different clusters remain asynchronous. Such synchronization may collapse for all clusters when the dynamics of only one oscillator is altered properly. The synchronous state may return back after a short period of transient process. This is determined by the strength of the oscillator altered. Its application in the communication of one-to-several is suggested.     

水下目标噪声测量中的同步钟测距

本测距系统采用两只型号相同的ＷＺＧ－５ＭＨｚ石英晶体振荡器，在同一个时基标准下进行同步测距，５ＭＨｚ石英晶体振荡器具有准确度高，稳定性好的特点，从而成功的解决了在舰船辐射噪声测量中，目标船与测量水听器间测距误差大的问题。本文较详细的介绍了同步钟测距原理，并对在舰船辐射噪声测量中噪声级与距离关系，目标船相对测量水听器的位置，测距误差等进行了描述。     

N-soliton quantum scattering in the sine-Gordon theory

The quantum treatment of soliton scattering in the sine-Gordon model, using the path integral collective coordinate method is generalized to N solitons. The solitions. The first quantum correction to the phase shift of N-soliton scattering is equal to the zero-point energy of an effective multi-soliton Hamiltonian. The energies of the oscillators of this Hamiltonian are shown to be equal to the stability angles of a complete set of solutions of the Schrödinger equation for small fluctuations around a classical N-soliton. Consequently, calculating the fluctuations and their stability angles by the inverse scattering method, we obtain the energies of the oscillators. The first quantum correction to the phase shift (the O(1) part in a development in powers of γ) is evaluated by summing the stability angles. This result is in agreement with the “exact” scattering amplitude conjectured by Faddeev, Kulish and Korepin.