Adaptive synchronization of neural networks with or without time-varying delay

In this paper, based on the invariant principle of functional differential equations, a simple, analytical, and rigorous adaptive feedback scheme is proposed for the synchronization of almost all kinds of coupled identical neural networks with time-varying delay, which can be chaotic, periodic, etc. We do not assume that the concrete values of the connection weight matrix and the delayed connection weight matrix are known. We show that two coupled identical neural networks with or without time-varying delay can achieve synchronization by enhancing the coupling strength dynamically. The update gain of coupling strength can be properly chosen to adjust the speed of achieving synchronization. Also, it is quite robust against the effect of noise and simple to implement in practice. In addition, numerical simulations are given to show the effectiveness of the proposed synchronization method.     

On the complete synchronization of the Kuramoto phase model

We discuss the asymptotic complete phase-frequency synchronization for the Kuramoto phase model with a finite size N. We present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states. Our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations. But they are independent of the system size N, hence they can be used for the mean-field limit. For the complete synchronization estimates, we estimate the time evolution of the phase and frequency diameters for configurations. The initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast. In contrast, for the non-identical oscillators, the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations. Our estimates are based on the monotonicity arguments of extremal phase and frequencies, which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states. We compare our analytical results with numerical simulations.     

Explosive synchronization of multi-layer frequency-weighted coupled complex systems

Synchronization is a phenomenon that is ubiquitous in engineering and natural ecosystems.The study of explosive synchronization on a single-layer network gives the critical transition coupling strength that causes explosive synchronization.However, no significant findings have been made on multi-layer complex networks.This paper proposes a frequency-weighted Kuramoto model on a two-layer network and the critical coupling strength of explosive synchronization is obtained by both theoretical analysis and numerical validation.It is found that the critical value is affected by the interaction strength between layers and the number of network oscillators.The explosive synchronization will be hindered by enhancing the interaction and promoted by increasing the number of network oscillators.Our results have importance across a range of engineering and biological research fields.     

一类动力学系统通过函数耦合实现混沌同步

非线性动力学系统的混沌同步, 一般采用单向线性耦合的控制方式, 对于函数耦合方式研究的比较少. 这就存在一个问题, 对于非线性动力学系统, 在线性耦合实现混沌同步后, 是否其他函数的耦合方式都可以实现混沌同步? 本文对于一类非线性动力学系统, 研究了其线性耦合同步与函数耦合同步的关系, 证明当线性耦合实现同步后, 函数在满足一定的条件下, 可以通过函数耦合实现系统的混沌同步. 最后对于Duffing系统采用两种函数耦合进行了仿真计算, 证明了结论的正确性.     

Effects of noise on the outer synchronization of two unidirectionally coupled complex dynamical networks

We study the effect of noise on the outer synchronization between two unidirectionally coupled complex networks and find analytically that outer synchronization could be achieved via white-noise-based coupling. It is also demonstrated that, if two networks have both conventional linear coupling and white-noise-based coupling, the critical deterministic coupling strength between two complex networks for synchronization transition decreases with an increase in the intensity of noise. We provide numerical results to illustrate the feasibility and effectiveness of the theoretical results.     

The effect of finite response–time in coupled dynamical systems

The paper investigates synchronization in unidirectionally coupled dynamical systems wherein the influence of drive on response is cumulative: coupling signals are integrated over a time interval τ. A major consequence of integrative coupling is that the onset of the generalized and phase synchronization occurs at higher coupling compared to the instantaneous (τ?=?0) case. The critical coupling strength at which synchronization sets in is found to increase with τ. The systems explored are the chaotic Rössler and limit cycle (the Landau–Stuart model) oscillators. For coupled Rössler oscillators the region of generalized synchrony in the phase space is intercepted by an asynchronous region which corresponds to anomalous generalized synchronization.     

Experimental investigation of high-quality synchronization of coupled oscillators

We describe two experiments in which we investigate the synchronization of coupled periodic oscillators. Each experimental system consists of two identical coupled electronic periodic oscillators that display bursts of desynchronization events similar to those observed previously in coupled chaotic systems. We measure the degree of synchronization as a function of coupling strength. In the first experiment, high-quality synchronization is achieved for all coupling strengths above a critical value. In the second experiment, no high-quality synchronization is observed. We compare our results to the predictions of the several proposed criteria for synchronization. We find that none of the criteria accurately predict the range of coupling strengths over which high-quality synchronization is observed. (c) 2000 American Institute of Physics.     

The role of degree-weighted couplings in the synchronous onset of Kuramoto oscillator networks

This paper investigates the role of asymmetrical degree-dependent weighted couplings in synchronization of a network of Kuramoto oscillators, where the conditions of coupling criticality for the onset of phase synchronization in degree-weighted complex networks are arrived at. The numerical simulations visualize that for networks having power-law or exponential degree distributions, asymmetrical degree-weighted couplings (with increasing weighting exponent β) increases the critical coupling to achieve the onset of phase synchronization in the networks.     

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 R?ssler, 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.     

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 two coupled Van der Pol oscillators with attractive and repulsive coupling

We consider a variant of two coupled Van der Pol oscillators with both attractive and repulsive mean-field interactions. In the presence of attractive coupling, the system is in the complete synchrony, while repulsive coupling shows anti-synchronization state leading to suppression of oscillations with increasing interaction strength. The coupled system with both attractive and repulsive interactions shows competitive tendencies of being complete synchronization and anti-synchronization resulting in the stabilization of the fixed point. We have also studied the effect of the damping coefficient of the VdP oscillator on the nature of the transition from oscillatory to a steady-state. These oscillators stabilize to unstable equilibrium point or coupling dependent inhomogeneous steady state via second or first-order transitions respectively depending upon the damping coefficient and coupling strength. These transitions are analyzed in the parameter plane by analytical and numerical studies of the two coupled Van der Pol oscillators.     

The synchronization transition in correlated oscillator populations

The synchronization transition of correlated ensembles of coupled Kuramoto oscillators on sparse random networks is investigated. Extensive numerical simulations show that correlations between the native frequencies of adjacent oscillators on the network systematically shift the critical point as well as the critical exponents characterizing the transition. Negative correlations imply an onset of synchronization for smaller coupling, whereas positive correlations shift the critical coupling towards larger interaction strengths. For negatively correlated oscillators the transition still exhibits critical behaviour similar to that of the all-to-all coupled Kuramoto system, while positive correlations change the universality class of the transition depending on the correlation strength. Crucially, the paper demonstrates that the synchronization behaviour is not only determined by the coupling architecture, but also strongly influenced by the oscillator placement on the coupling network.     

Experimental investigation on synchronization of three co-rotating non-identical coupled exciters driven by three motors

This work is a continuation and verification of the original literature by means of experiment. According to the previously published paper, the criteria of synchronization and stability for three co-rotating three exciters in an analytical form, as well as their numerical results, are directly provided. A synchronization experiment with three exciters is carried out. The experimental data on phase differences, rotational velocities, system responses, coupling torques, and so on, are quantitatively discussed by dividing the operational processes of the system into four stages. The coupling torques, which are the key determining factors for implementing synchronization, are particularly examined to explain the synchronization regime, based on the average energy method. The quality of synchronization is provided, and the noise and uncertainties in the experimental data are analyzed. It is shown that the experimental results obtained concur with the theoretical results.     

异质神经元的排列对环形耦合神经元网络频率同步的影响

以一维环形耦合的非全同FitzHugh-Nagumo神经元网络为研究对象,讨论这种异质神经元在环上的不同排列对其频率同步的影响.研究结果显示,异质神经元的排列不同,对应的神经元网络达到频率同步所需的临界耦合强度也不完全相同.在平均意义下,异质性较小的神经元在环上的距离越近,神经元网络达到频率同步所需的临界耦合强度越大;相反,异质性较大的神经元在环上的距离越近,神经元网络达到同步所需的临界耦合强度越小.通过对频率同步过程的分析,进一步给出了产生这一现象的动力学机理.     

R~4 corrections to holographic Schwinger effect

We consider R4 corrections to the holographic Schwinger effect in an AdS black hole background and a confining D3-brane background. The potential between a test particle pair are performed for both backgrounds. We find there is no potential barrier in the critical electric field, which means that the system becomes catastrophically unstable. It is shown that for both backgrounds, increasing the inverse 't Hooft coupling parameter 1/λ enhances the Schwinger effect. We also discuss the possible relation between the Schwinger effect and the viscosity-entropy ratioη/s in strong coupling.     

Synchronization transition of limit-cycle system with homogeneous phase shifts

The behaviors of coupled oscillators, each of which has periodic motion with random natural frequency in the absence of coupling, are investigated when phase shifts are considered. In the system of coupled oscillators, phase shifts are the same between different oscillators. Synchronization and synchronization transition are revealed with different phase shifts. Phase shifts play an important role for this kind of system. When the phase shift α<0.5π, the synchronization state can be attained by increasing the coupling, and the system cannot reach the synchronization state while α≥q0.5π. A clear scaling between complete synchronization critical coupling strength K_pc and α-0.5π is found.     

New approach to synchronization in asymmetrically coupled networks

Global exponentially synchronization in asymmetrically coupled networks is investigated in this Letter. We extend eigenvalue based method to synchronization in symmetrically coupled network to synchronization in asymmetrically coupled network. A new stability criterion of eigenvalue based is derived. In this criterion, both a term that is the second largest eigenvalue of a symmetrical matrix and a term that is the largest value of sum of column of asymmetrical coupling matrix play a key role. Comparing with existing results, the advantage of our synchronization stability result is that it can analytical be applied to the asymmetrically coupled networks and overcome the complexity on calculating eigenvalues of coupling asymmetric matrix. Therefore, this condition is very convenient to use. Moreover, a necessary condition of this synchronization stability criterion is also given by the elements of the coupling asymmetric matrix, which can conveniently be used in judging the synchronization stability condition without calculating the eigenvalues of coupling matrix.     

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.     

Stability of ferrimagnetic multilayers

We study the field induced instability of the ground state of ferrimagnetic multilayers consisting of a stacking alternating two different uniaxial ferromagnetic layers. For multilayers with even number of layers N, we obtain analytical expressions for the critical fields in terms of the magnetic parameters (anisotropies, and interlayer exchange coupling), for any value of N. The critical fields are calculated from the energy fluctuations for small variations in the equilibrium magnetic profile. The form of the hysteresis curves is discussed, using the expressions of the critical fields.     

Delay-coupled discrete maps: Synchronization, bistability, and quasiperiodicity

The synchronization transition is studied in delay-coupled logistic maps. For low coupling, in-phase and out-of-phase synchronous dynamics coexist, and with increasing coupling there is a regime of quasiperiodicity before eventual attraction to a fixed point at a critical value of coupling that depends on the nonlinearity. The presence of a region of asynchrony separating two synchronized regimes—termed anomalous behaviour—has been observed earlier in continuous systems and is shown here to occur in delay mappings as well. There are regions of in-phase, anti-phase, and out-of-phase dynamics of periodic as well as chaotic attractors.