Phase Synchronization of Coupled Rossler Oscillators: Amplitude Effect

Phase synchronization of two linearly coupled Rossler oscillators with parameter misfits is explored.It is found that depending on parameter mismatches,the synchronization of phases exhibits different manners.The synchronization regime can be divided into three regimes.For small mismatches,the amplitude-insensitive regime gives the phase-dominant synchronization; When the parameter misfit increases,the amplitudes and phases of oscillators are correlated,and the amplitudes will dominate the synchronous dynamics for very large mismatches.The lag time among phases exhibits a power law when phase synchronization is achieved.     

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.     

Spatiotemporal dynamics in a ring of N mutually coupled self-sustained systems

In this paper, we consider the spatiotemporal dynamics in a ring of N mutually coupled self-sustained oscillators in the regular state. When there are no parameter mismatches, the good coupling parameters leading to full, partial, and no synchronization are derived using the properties of the variational equations of stability. The effects of the spatial dimension of the ring on the stability boundaries of the synchronized states are performed. Numerical simulations validate and complement the results of analytical investigations. The influences of coupling parameter mismatch on the forecasted stability boundaries are also highlighted.     

Phase transitions in low-dimensional ψ 4 systems

The phase transition in a planar array of weakly coupled Ginzburg–Landau chains with real order parameter is studied, using an original variant of the two-level approximation. The results are extended to the quantum phase transition in a chain of quantum double well oscillators, coupled with an elastic interaction, using the matrix transfer method.     

Pattern formations in chaotic spatio-temporal systems

Pattern formations in chaotic spatio-temporal systems modelled by coupled chaotic oscillators are investigated. We focus on various symmetry breakings and different kinds of chaos synchronization-desynchronization transitions, which lead to certain types of spontaneous spatial orderings and the emergence of some typical ordered patterns, such as rotating wave patterns with splay phase ordering (orientational symmetry breaking) and partially synchronous standing wave patterns with in-phase ordering (translational symmetry breaking). General pictures of the global behaviors of pattern formations and transitions in coupled chaotic oscillators are provided.     

Hidden symmetry in chains of biological coupled oscillators

We experimentally investigated spatiotemporal patterns in chains of coupled biological oscillators with boundaries and found hidden symmetric patterns that are not straightforwardly derived from explicit geometrical symmetry of the systems. We propose a model of coupled oscillators in chains with a hidden oscillator interconnecting its boundaries. The model can explain all observed patterns including the hidden symmetric ones, while other models such as discrete analogs of Neumann boundary conditions in continuous systems cannot.     

Synchronization in arrays of coupled self-induced friction oscillators

We investigate synchronization phenomena in systems of self-induced dry friction oscillators with kinematic excitation coupled by linear springs. Friction force is modelled according to exponential model. Initially, a single degree of freedom mass-spring system on a moving belt is considered to check the type of motion of the system (periodic, non-periodic). Then the system is coupled in chain of identical oscillators starting from two, up to four oscillators. A reference probe of two coupled oscillators is applied in order to detect synchronization thresholds for both periodic and non-periodic motion of the system. The master stability function is applied to predict the synchronization thresholds for longer chains of oscillators basing on two oscillator probe. It is shown that synchronization is possible both for three and four coupled oscillators under certain circumstances. Our results confirmed that this technique can be also applied for the systems with discontinuities.     

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.     

脉冲激励下环形耦合Duffing振子间的瞬态同步突变现象

研究非周期信号激励下Duffing振子动力学行为变化特征时,发现处于倍周期分岔的环形耦合Duffing振子系统,在一定的参数条件下,脉冲信号能引起其中一个振子与其他振子运动轨迹间出现短暂失同步的现象即瞬态同步突变现象.利用这种现象可以快速检测出强噪声背景中的微弱脉冲信号,从而扩展了现有的Duffing振子对非周期信号的检测范围及应用领域. 关键词： 瞬态同步突变 微弱信号检测 脉冲信号 Duffing振子     

Structure of resonances and formation of stationary points in symmetrical chains of bilinear oscillators

Dynamics of strongly nonlinear systems can in many cases be modelled by bilinear oscillators, which are the oscillators whose springs have different stiffnesses in compression and tension. This underpins the analysis of a wide range of phenomena, from oscillations of fragmented structures, connections and mooring lines to deformation of geological media. Single bilinear oscillators were studied previously and the presence of multiple resonances both super- and sub-harmonic was found. Less attention was paid to systems of multiple bilinear oscillators that describe many natural and engineering processes such as for example the behaviour of fragmented solids. Here we fill this gap concentrating on the simplest case – 1D symmetrical chains of bilinear oscillators. We show that the presence and structure of resonances in a symmetric chain of bilinear oscillators with fixed ends depends upon the number of oscillating masses. Two elementary chains act as the basic ones: a single mass bilinear chain (a mass connected to the fixed points by two bilinear springs) that behaves as a linear oscillator with a single resonance and a two mass chain that is a coupled bilinear oscillator (two masses connected by three bilinear springs). The latter has multiple resonances. We demonstrate that longer chains either do not have resonances or get decomposed, in the resonance, into either the single mass or two mass elementary chains with stationary masses in between. The resonance frequencies are inherited from the basic chains of decomposition. We show that if the number of masses is odd the chain can be decomposed into the single mass bilinear chains separated by stationary masses. It then inherits the resonances of the single mass bilinear chain. The chains with the number of masses minus 2 divisible by 3 can be decomposed into the two mass bilinear chains separated by stationary masses and inherit the resonances of the two mass chains. The chains whose lengths satisfy both criteria (such as chains with 5, 11, 17 … masses) allow both types of resonances.     

Ring intermittency in coupled chaotic oscillators at the boundary of phase synchronization

A new type of intermittent behavior is described to occur near the boundary of the phase synchronization regime of coupled chaotic oscillators. This mechanism, called ring intermittency, arises for sufficiently high initial mismatches in the frequencies of the two coupled systems. The laws for both the distribution and the mean length of the laminar phases versus the coupling strength are analytically deduced. Very good agreement between the theoretical results and the numerically calculated data is shown. We discuss how this mechanism is expected to take place in other relevant physical circumstances.     

Synchronization properties and effects of parameter mismatches in unidirectionally coupled chaotic vertical-cavity surface-emitting lasers

We numerically investigate the effects of parameter mismatches on chaos synchronization in vertical-cavity surfaceemitting lasers (VCSELs). We assume injection-locked chaos synchronization in a unidirectionally coupled and openloop optical feedback system. The accuracy of chaos synchronization is greatly affected by the mismatches of the device parameters and operation conditions between the two lasers. In particular, the oscillation frequency of the laser is one of the important parameters in a system of injection-locked chaos synchronization. However, the variations of the device characteristics of VCSELs are very large compared with those of other types of semiconductor lasers. We study the effects of parameter mismatches related to the oscillation frequency of VCSELs on chaos synchronization. We proved that mismatches in terms of the birefringence and the injection current play crucial roles for the quality of chaos synchronization.     

Phase-locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies

We study phase-locking in a network of coupled nonlinear oscillators with local interactions and random intrinsic frequencies. The oscillators are located at the vertices of a graph and interact along the edges. They are coupled by sinusoidal functions of the phase differences across the edges, and their intrinsic frequencies are independent and identically distributed with finite mean and variance.We derive an exact expression for the probability of phase-locking in a linear chain of such oscillators and prove that this probability tends to zero as the number of oscillators grows without bound. However, if the coupling strength increases as the square root of the number of oscillators, the probability of phase-locking tends to a limiting distribution, the Kolmogorov-Smirnov distribution. This latter result is obtained by showing that the phase-locking problem is equivalent to a discretization of pinned Brownian motion.The results on chains of oscillators are extended to more general graphs. In particular, for a hypercubic lattice of any dimension, the probability of phase-locking tends to zero exponentially fast as the number of oscillators grows without bound. We also consider a less stringent type of synchronization, characterized by large clusters of oscillators mutually entrained at the same average frequency. It is shown that if such clusters exist, they necessarily have a sponge-like geometry.     

Resurgence of oscillation in coupled oscillators under delayed cyclic interaction

This paper investigates the emergence of amplitude death and revival of oscillations from the suppression states in a system of coupled dynamical units interacting through delayed cyclic mode. In order to resurrect the oscillation from amplitude death state, we introduce asymmetry and feedback parameter in the cyclic coupling forms as a result of which the death region shrinks due to higher asymmetry and lower feedback parameter values for coupled oscillatory systems. Some analytical conditions are derived for amplitude death and revival of oscillations in two coupled limit cycle oscillators and corresponding numerical simulations confirm the obtained theoretical results. We also report that the death state and revival of oscillations from quenched state are possible in the network of identical coupled oscillators. The proposed mechanism has also been examined using chaotic Lorenz oscillator.     

耦合非线性振子系统的同步研究

研究了考虑振子振幅效应的耦合极限环系统的同步.研究表明，耦合极限环系统的序参量随耦合强度的增加呈现非单调变化，并且出现若干不可微的点；平均频率随耦合强度的变化过程表现为同步分岔树结构；在临界点处出现了相速度的滑移、锁定和相速度差的开关阵发现象，开关阵发的平均周期具有很好的标度关系；振子的平均振幅随相同步的进程实际上是由均匀化逐渐分岔而达到非均匀化的过程，振子振幅的变化范围在临界点处突然减小. 关键词： 耦合极限环系统 同步 振幅效应     

Desynchronization bifurcation of coupled nonlinear dynamical systems

We analyze the desynchronization bifurcation in the coupled Ro?ssler oscillators. After the bifurcation the coupled oscillators move away from each other with a square root dependence on the parameter. We define system transverse Lyapunov exponents (STLE), and in the desynchronized state one is positive while the other is negative. We give a simple model of coupled integrable systems with quadratic nonlinearity that shows a similar phenomenon. We conclude that desynchronization is a pitchfork bifurcation of the transverse manifold. Cubic nonlinearity also shows the bifurcation, but in this case the STLEs are both negative.     

Extreme sensitivity to detuning for globally coupled phase oscillators

We discuss the sensitivity of a population of coupled oscillators to differences in their natural frequencies, i.e., to detuning. We argue that for three or more oscillators, one can get great sensitivity even if the coupling is strong. For N globally coupled phase oscillators we find there can be bifurcation to extreme sensitivity, where frequency locking can be destroyed by arbitrarily small detuning. This extreme sensitivity is absent for N = 2, appears at isolated parameter values for N = 3 and N = 4, and can appear robustly for open sets of parameter values for N > or = 5 oscillators.     

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.     

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.