环形耦合Duffing振子间的同步突变

以环形耦合Duffing振子系统为研究对象,分析了耦合振子间的同步演化过程.发现在弱耦合条件下,如果所有振子受到同一周期策动力的驱动,那么系统在经历倍周期分岔、混沌态、大尺度周期态的相变时,各振子的运动轨迹之间将出现由同步到不同步再到同步的两次突变现象.利用其中任何一次同步突变现象可以实现系统相变的快速判别,并由此补充了利用倍周期分岔与混沌态的这一相变对微弱周期信号进行检测的方法. 关键词： Duffing振子 同步突变 相变 微弱信号检测     

多重耦合振子系统的同步动力学

耦合相振子的同步研究对理解复杂系统自组织协同的涌现具有重要的理论意义.相比于传统耦合振子的两体成对耦合,多重耦合近年来得到广泛的关注.当相振子间的多重耦合机制起主要作用时,系统会涌现一系列去同步突变,这一新颖的动力学特性对理解复杂系统群体动力学提供了重要的理论启示.本文研究了平均场的三重耦合Kuramoto系统的同步动力学,发现了去同步转变具有不可逆性,并利用平均场自洽方法和无序态线性稳定性分析揭示了不可逆去同步突变的动力学机制.进一步研究发现,随着振子自然频率分布半宽度的变化,系统会经历一系列去同步驻波态的转变.在相变临界点,系统在高维相空间会通过鞍结分岔导致同步态失稳而塌缩至稳定的低维不变环面.本文的研究揭示了多重耦合函数作用的振子系统的各种协同态及其相变机制,同时可为理解其他复杂系统(如超网络结构)协同态的动力学转变提供理论借鉴.     

耦合方式与初始条件结构对分数阶双稳态振子环形网络同步的影响

在由分数阶双稳态振子通过最近邻耦合构成的环形网络中研究了振子的同步与耦合方式以及初始条件结构的关系.通过选择初始条件结构、耦合方式和强度,可以控制网络呈现振幅死亡同步态、振幅死亡非同步态、混沌同步态和混沌非同步态等多种动力学行为.参数平面区域ε3-ε2内的最大条件Lyapunov指数和最大Lyapunov指数的等高线进一步表明,y与z方向的耦合竞争对网络的动力学行为的影响结果敏感地依赖于网络的初始条件结构.     

基于耦合Duffing振子的局部放电信号检测方法研究

以双向环形耦合Duffing振子系统为对象, 研究脉冲信号激励下耦合振子间动力学行为变化特征时, 发现其与单向环形耦合Duffing振子系统类似, 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. 基于这种现象, 提出了一种微弱脉冲信号检测的新方法, 用于检测强噪声背景中的局部放电脉冲信号. 实验测试表明, 利用本文方法对不同放电电极的局部放电脉冲信号进行检测时, 在低信噪比条件下可取得良好的检测效果, 进而扩展了现有的Duffing振子对非周期信号的检测范围及应用领域. 关键词： 耦合Duffing 振子 微弱信号检测 瞬态同步突变 局部放电脉冲信号     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

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     

二次耦合光力学系统的一类高维可控自持振荡行为

光力学系统通常的耦合是光压耦合, 是光场强度和纳米振子位移的一次耦合, 但在光场很强和振子振幅较大的光力学系统中, 非线性的耦合效应会变得非常明显和重要, 而且其所产生的非线性效应对制造具有特殊功能的光力学器件具有重要意义. 本文在二次耦合模型的基础上研究了光腔和振子之间通过二次耦合作用达到能 量平衡状态时系统所产生的自持振荡现象, 给出了二次耦合光力学系统的一般模型, 并通过数值方法研究了系统的定态行为和远离定态的极限环动力学行为, 标定了系统定态响应的稳定区域到极限环行为的分岔点. 发现在调节输入场参数(改变耦合系数)以及光腔和振子的弛豫系数时, 系统的相空间会出现一些稳定的高维自持振荡极限环. 通过数值分析发现该四维极限环在三维相空间的投影都趋于稳定的三维周期轨道, 并且该极限环轨道会随外部调控参数的改变发生扭动, 出现类似二维李萨如图样的稳定纽结结构. 该现象表明: 通过光场与振子的能量耦合, 利用一定强度的外部驱动可以有效控制振子的定态响应和振动, 可以让微振子锁定在具有一定振幅和频率的自发振动上, 为开发物理器件提供了可靠的光力学控制系统. 关键词： 光力系统 二次耦合 自持振荡 极限环     

耦合分数阶双稳态振子的同步、反同步与振幅死亡

研究了耦合分数阶振子的同步、反同步和振幅死亡等问题. 基于P-R振子在特定参数下的双稳态特性, 利用最大条件Lyapunov指数、最大Lyapunov指数和分岔图等数值方法分析发现, 通过选取初始条件和耦合强度, 可以控制耦合振子呈现混沌同步、混沌反同步、全部振幅死亡同步、全部振幅死亡反同步和部 分振幅死亡等丰富的动力学现象. 基于蒙特卡罗方法的原理, 在初始条件相空间中随机选取耦合振子的初始位置, 计算不同耦合强度下耦合振子的全部振幅死亡态、部分振幅死亡态和非振幅死亡态的比例, 从统计学角度表征了耦合分数阶双稳态振子的动力学特征. 几种有代表性的双稳态振子的吸引域进一步证明了统计方法的计算结果. 关键词： 振幅死亡 吸引域 双稳态     

一类耦合非线性振子同步混沌Hopf分岔及其电路仿真

对于一类同时存在扩散耦合和梯度耦合的非线性振子系统, 通过空间傅里叶变换，得到具有不同波矢的各运动模式的相互独立的运动方程. 计算各横截模的Lyapunov指数, 可在耦合参数平面上确定同步混沌的稳定区域. 在稳定区域边界, 一对共轭横截模式失稳，导致同步混沌的Hopf分岔. 对耦合Lorenz振子系统进行了数值模拟，并设计了耦合Lorenz振子系统的电路, 进行耦合振子系统同步混沌Hopf分岔的电路仿真实验. 计算和仿真的结果表明，Hopf分岔的特征频率等于失稳横截模式的振荡频率. 关键词： 耦合非线性振子 同步混沌 横截模式 电路仿真     

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.     

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.     

Long-term fluctuations in globally coupled phase oscillators with general coupling: finite size effects

We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models such as the Ising model, the scaling property of D is D～O(1) for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of D is different for the coherent and incoherent regimes: D～O(1/N(a)) with a certain constant a>0 in the coherent regime and D～O(1) in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.     

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.     

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 → ∞.     

Explosive synchronization in clustered scale-free networks: Revealing the existence of chimera state

The collective dynamics of Kuramoto oscillators with a positive correlation between the incoherent and fully coherent domains in clustered scale-free networks is studied. Emergence of chimera states for the onsets of explosive synchronization transition is observed during an intermediate coupling regime when degree-frequency correlation is established for the hubs with the highest degrees. Diagnostic of the abrupt synchronization is revealed by the intrinsic spectral properties of the network graph Laplacian encoded in the heterogeneous phase space manifold, through extensive analytical investigation, presenting realistic MC simulations of nonlocal interactions in discrete time dynamics evolving on the network.     

Synchronization in a system of globally coupled oscillators with time delay

We study the synchronization phenomena in a system of globally coupled oscillators with time delay in the coupling. The self-consistency equations for the order parameter are derived, which depend explicitly on the amount of delay. Analysis of these equations reveals that the system in general exhibits discontinuous transitions in addition to the usual continuous transition, between the incoherent state and a multitude of coherent states with different synchronization frequencies. In particular, the phase diagram is obtained on the plane of the coupling strength and the delay time, and ubiquity of multistability as well as suppression of the synchronization frequency is manifested. Numerical simulations are also performed to give consistent results.     

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     

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.     

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.     

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.