Experimental observation of partial amplitude death in coupled chaotic oscillators

The dynamics of coupled Lorenz circuits is investigated experimentally. The partial amplitude death reported in {\em Phys. Rev.} E {\bf 72}, 057201 (2005) is verified by physical experiments with electronic circuits. With the increase of coupling constant, the coupled circuits undergo the transition from the breakdown of both the reflection symmetry and the translational symmetry to the partial amplitude death. Its stability is also confirmed by analysing the effects of noise.     

耦合分数阶混沌振荡器的主-从同步

近年来对分数阶系统的动力学研究得到了较为广泛的关注。本文研究了基于主-从耦合同步法的同步技术并实现了两个耦合的分数阶振荡器的混沌同步。仿真结果表明：在适当的耦合强度的调节下，该方法可实现两个耦合分数阶混沌振荡器的准确同步，且分数阶混沌振荡器的同步率明显慢于整数阶混沌振荡器的同步率；而耦合分数阶混沌振荡器在实现同步的过程中，随着阶数的提高，同步误差曲线变得平滑，这表明，系统阶数的提高改善了耦合混沌振荡器实现同步的平稳性。     

Transition to Amplitude Death in Coupled System with Small Number of Nonlinear Oscillators

In this work, we investigate the amplitude death in coupled system with small number of nonlinear oscillators. We show how the transitions to the partial and the complete amplitude deathes happen. We also show that the partial amplitude death can be found in globally coupled oscillators either.     

Synchronization of oscillators in complex networks

Theory of identical or complete synchronization of identical oscillators in arbitrary networks is introduced. In addition, several graph theory concepts and results that augment the synchronization theory and a tie in closely to random, semirandom, and regular networks are introduced. Combined theories are used to explore and compare three types of semirandom networks for their efficacy in synchronizing oscillators. It is shown that the simplest k-cycle augmented by a few random edges or links are the most efficient network that will guarantee good synchronization.      

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     

Cooperative behaviors of coupled nonidentical oscillators with the same equilibrium points

The cooperative behaviors resulted from the interaction of coupled identical oscillators have been investigated intensively. However, the coupled oscillators in practice are nonidentical, and there exist mismatched parameters. It has been proved that under certain conditions, complete synchronization can take place in coupled nonidentical oscillators with the same equilibrium points, yet other cooperative behaviors are not addressed. In this paper, we further consider two coupled nonidentical oscillators with the same equilibrium points, where one oscillator is convergent while the other is chaotic,and explore their cooperative behaviors. We find that the coupling mode and the coupling strength can bring the coupled oscillators to different cooperation behaviors in unidirectional or undirected couplings. In the case of directed coupling,death islands appear in two-parameter spaces. The mechanism inducing these transitions is presented.     

Collective dynamics of delay-coupled limit cycle oscillators

Coupled limit cycle oscillators with instantaneous mutual coupling offer a useful but idealized mathematical paradigm for the study of collective behavior in a wide variety of biological, physical and chemical systems. In most real-life systems however the interaction is not instantaneous but is delayed due to finite propagation times of signals, reaction times of chemicals, individual neuron firing periods in neural networks etc. We present a brief overview of the effect of time-delayed coupling on the collective dynamics of such coupled systems. Simple model equations describing two oscillators with a discrete time-delayed coupling as well as those describing linear arrays of a large number of oscillators with time-delayed global or local couplings are studied. Analytic and numerical results pertaining to time delay induced changes in the onset and stability of amplitude death and phase-locked states are discussed. A number of recent experimental and theoretical studies reveal interesting new directions of research in this field and suggest exciting future areas of exploration and applications.     

Collective dynamics of delay-coupled limit cycle oscillators

Coupled limit cycle oscillators with instantaneous mutual coupling offer a useful but idealized mathematical paradigm for the study of collective behavior in a wide variety of biological, physical and chemical systems. In most real-life systems however the interaction is not instantaneous but is delayed due to finite propagation times of signals, reaction times of chemicals, individual neuron firing periods in neural networks etc. We present a brief overview of the effect of time-delayed coupling on the collective dynamics of such coupled systems. Simple model equations describing two oscillators with a discrete time-delayed coupling as well as those describing linear arrays of a large number of oscillators with time-delayed global or local couplings are studied. Analytic and numerical results pertaining to time delay induced changes in the onset and stability of amplitude death and phase-locked states are discussed. A number of recent experimental and theoretical studies reveal interesting new directions of research in this field and suggest exciting future areas of exploration and applications.     

Analytical calculation of the transition to complete phase synchronization in coupled oscillators

Here we present a system of coupled phase oscillators with nearest neighbors coupling, which we study for different boundary conditions. We concentrate at the transition to the total synchronization. We are able to develop exact solutions for the value of the coupling parameter when the system becomes completely synchronized, for the case of periodic boundary conditions as well as for a chain with fixed ends. We compare the results with those calculated numerically.      

Oscillation quenching mechanisms: Amplitude vs. oscillation death

Oscillation quenching constitutes a fundamental emergent phenomenon in systems of coupled nonlinear oscillators. Its importance for various natural and man-made systems, ranging from climate, lasers, chemistry and a wide range of biological oscillators can be projected from two main aspects: (i) suppression of oscillations as a regulator of certain pathological cases and (ii) a general control mechanism for technical systems. We distinguish two structurally distinct oscillation quenching types: oscillation   (OD$OD$) and amplitude death   (AD$AD$) phenomena. In this review we aim to set clear boundaries between these two very different oscillation quenching manifestations and demonstrate the importance for their correct identification from the aspect of theory as well as of applications. Moreover, we pay special attention to the physiological interpretation of OD$OD$ and AD$AD$ in a large class of biological systems, further underlying their different properties. Several open issues and challenges that await further resolving are also highlighted.     

Glassy synchronization in a population of coupled oscillators

A phase model for a population of oscillators with random excitatory and inhibitory mean-field coupling and subject to external white noise random forces is proposed and studied. In the thermodynamic limit different stable phases for the oscillator population may be found: (i) an incoherent state where all possible values of an oscillator phase are equally probable, (ii) a synchronized state where the population has a nonzero collective phase; (iii) a glassy phase where the global synchronization is zero but the oscillators are in phase with the random disorder; and (iv) a mixed phase where the oscillators are partially synchronized and partially in phase with the disorder. These predictions are based upon bifurcation analysis of the reduced equation valid at the thermodynamic limit and confirmed by Brownian simulation.     

Collective stochastic resonance behaviors of two coupled harmonic oscillators driven by dichotomous fluctuating frequency

The collective behaviors of two coupled harmonic oscillators with dichotomous fluctuating frequency are investigated,including stability, synchronization, and stochastic resonance(SR). First, the synchronization condition of the system is obtained. When this condition is satisfied, the mean-field behavior is consistent with any single particle behavior in the system. On this basis, the stability condition and the exact steady-state solution of the system are derived. Comparative analysis shows that, the stability condition is stronger than the synchronization condition, that is to say, when the stability condition is satisfied, the system is both synchronous and stable. Simulation analysis indicates that increasing the coupling strength will reduce the synchronization time. In weak coupling region, there is an optimal coupling strength that maximizes the output amplitude gain(OAG), thus the coupling-induced SR behavior occurs. In strong coupling region, the two particles are bounded as a whole, so that the coupling effect gradually disappears.     

不同质量和频率的耦合谐振子体系哈密顿量的对角化

讨论了耦合项为λ１Ｘ１Ｘ２和λ２ｐ１ｐ２的两个不同质量和频率的耦合谐振子哈密顿量的对角化。     

各向异性n模耦合谐振子的精确求解

利用量子变换理论,极其简洁地给出了种向异性耦合谐振子的能谱,从而提出了一种普遍的方法。     

Complex dynamics analysis of impulsively coupled Duffing oscillators with ring structure

Impulsively coupled systems are high-dimensional non-smooth systems that can exhibit rich and complex dynamics.This paper studies the complex dynamics of a non-smooth system which is unidirectionally impulsively coupled by three Duffing oscillators in a ring structure.By constructing a proper Poincare map of the non-smooth system,an analytical expression of the Jacobian matrix of Poincare map is given.Two-parameter Hopf bifurcation sets are obtained by combining the shooting method and the Runge-Kutta method.When the period is fixed and the coupling strength changes,the system undergoes stable,periodic,quasi-periodic,and hyper-chaotic solutions,etc.Floquet theory is used to study the stability of the periodic solutions of the system and their bifurcations.     

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     

非对易空间的三模谐振子能谱

在非对易空间中,用不变本征算符方法(IEO),对非耦合、坐标耦合、动量耦合三种三模谐振子系统能谱进行求解,并将求解结果与一般对易空间的能谱进行比较分析.通过比较发现,当非对易参数为零时,所求能级差还原到了与普通空间相对应的一般量子系统哈密顿量能级差,验证了推导结果的正确性;同时讨论了耦合系数对非对易空间能谱的影响. 关键词： 不变本征算符 非对易空间 三模谐振子能谱 能级差     

无反馈作用下混沌系统的振幅死亡

考虑了一种无反馈的作用机理，通过分析定常强迫作用下R?ssler系统的分岔图和Lyapunov指数谱，发现定常强迫能够导致混沌系统产生振幅死亡.这种现象的产生类似于耦合的极限环系统和耦合的R?ssler系统，弱定常强迫作用下的R?ssler系统经由一个周期运动被驱动到系统自身的一个平衡点上.进一步对受迫系统时间序列的研究表明，当定常强迫强度超过一个临界值时，系统的状态不断在小振幅周期运动与静止之间交替出现. 关键词： 振幅死亡 定常强迫 R?ssler系统     

含时耦合谐振子系统的时间演化与双模压缩态

运用Lewis Riesenfeld不变量理论,通过适当地选取厄米不变量,得到了含驱动项和双模耦合项的含时耦合谐振子系统薛定谔方程的封闭解,给出了系统的演化算符及其产生双模光场的压缩态的条件,并得出系统压缩态的量子涨落与驱动项无关但与系统所处的初态有关的结论. 关键词： 含时耦合谐振子系统 Lewis Riesenfeld不变量理论 双模压缩态