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.     

混沌吸引子在两个周期振子耦合下的相同步

在分析了系统稳定的基础上，对非线性混沌吸引子在两个独立外周期振子耦合下的相同步进 行了研究.与一个周期振子耦合的情况不同，两个周期振子对混沌吸引子的耦合具有排他性 和竞争性，相同步在两个亚稳态交替出现，各自同步时间长度由外振子参数决定.确定了周 期外振子参数与同步时间长度的关系并与数值模拟计算结果进行了比较. 关键词： 混沌吸引子 相同步     

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.     

耦合帐篷映射混沌同步系统的筛形吸引域

讨论了两个线性耦合的标准帐篷映射混沌同步系统的同步混沌吸引子的吸引域,证明其是筛形吸引域.提出了筛形品质因子的概念,并据之给出了系统的同步混沌吸引子的吸引域发生 从局部筛形到全局筛形的转变的耦合参数临界值.修正了当系统出现筛形吸引域时的横截Lyapunov指数的解析表达式.指出考查筛形吸引域在混沌同步系统中有着重要意义. 关键词： 混沌同步 筛形吸引域 耦合帐篷映射     

忆阻自激振荡系统的隐藏吸引子及其动力学特性

由压控忆阻替换三维自激振荡系统的线性耦合电阻,实现了一种新型的四维忆阻自激振荡系统.该系统不存在任何平衡点,但可生成周期、准周期、混沌等隐藏吸引子;特别地,当初始条件不同时,系统出现了不同拓扑结构混沌吸引子或准周期极限环与混沌吸引子的共存现象,以及准周期极限环与多种拓扑结构混沌吸引子的多吸引子现象.理论分析、数值仿真和硬件实验的结果一致,表明了所提出的忆阻自激振荡系统有着十分丰富而复杂的隐藏动力学特性.     

通过同步实现"有序+有序=混沌"的例子

研究了连续的混沌系统是否存在"有序+有序=混沌"的现象,研究表明两个吸引子为周期运动的动力学系统经双向耦合达到同步后,同步后的系统可产生混沌态.采用特定参数下的Lorenz系统和Rssler系统作为例子,对连续的动力系统给出了一个"有序+有序=混沌"的例子. 关键词： 混沌 有序 耦合 同步     

关于非自治系统三类广义同步存在性的研究

研究了两个单向耦合的非自治系统三类广义同步的存在性.在响应系统的修正方程具有渐近稳定平衡点、渐近稳定周期轨道或渐近稳定拟周期轨道的情况下,满足一定的条件,可将广义同步化流形存在性问题转化为Lipschitz函数族的压缩不动点问题,并且理论证明了该广义同步化流形的指数吸引性.同时,以Duffing系统为例进行了数值仿真,其结果与理论推导相一致. 关键词： 广义同步化流形 压缩不动点 指数吸引性     

单向耦合下两个不同Lorenz系统的广义同步

对单向耦合下两个不同的Lorenz系统的广义同步进行了研究,利用辅助系统方法,基于稳定性理论和响应系统的有界性,得到了它们达到广义同步时的充分条件,并根据响应系统的修正系统具有零渐近稳定平衡点、非零渐近稳定平衡点和轨道渐近稳定周期解的情况,将广义同步分为第一类、第二类和第三类;利用Routh-Hurwitz定理,对修正系统平衡点的稳定性进行了分析,给出了单向耦合下两个不同Lorenz系统具有第一类、第二类广义同步的充分条件.数值仿真表明了该方法的有效性与可行性.     

耦合映射混沌同步系统在加性噪声中的筛形域性态

由于实际系统中噪声不可避免，噪声使得同步混沌吸引子A变成具有一定生存时间<τ>的准稳态吸引子A′.以加性噪声作用下的二维耦合映射混沌同步系统为例，给定系统实验时间长 度T，解析发现：仅当<τ>>2T时准稳态同步混沌吸引子的筛形吸引域才可被定性观察到；而 当<τ><2T时则不复存在，此时，根据原无噪声时的筛形吸引域特征的不同，筛形域不仅可 以转变成时变筛形结构，还可以转变成分形结构.同时利用数值模拟作了进一步验证.该结果 对于二维耦合映射混沌同步系统具有普遍意义. 关键词： 混沌同步 筛形吸引域 瞬态混沌 耦合映射 加性噪声     

通过被动介质耦合的两螺旋波的同步

采用Br模型研究了通过被动介质耦合的两二维可激发系统中螺旋波的同步,被动介质由可激发元素组成,这些元素之间不存在耦合.数值模拟结果表明,被动介质对螺旋波的同步有很大影响,当两系统中的初态螺旋波相同时,被动介质可导致稳定螺旋波发生漫游,螺旋波转变为螺旋波对或反靶波;当两系统中的初态螺旋波不同步时,在适当的参数下,两螺旋波可以实现同步、相同步,此外还观察到两螺旋波波头相互排斥、多螺旋波共存、同步的时空周期斑图、系统演化到静息态等现象.在被动介质中,一般可观察到波斑图,但是在某些情况下,被动介质会出现同步振荡现象.这些结果有助于人们理解心脏系统中出现的时空斑图.     

Synchronization of incommensurate non-identical fractional order chaotic systems using active control

Chaos synchronization in fractional order chaotic systems is receiving increasing attention due to its applications in secure communications. In this article we use an active control technique to synchronize incommensurate non-identical fractional order chaotic dynamical systems. The relation between system order and the synchronization time is discussed. It is observed that the synchronization can be achieved faster by increasing the system order. Further we provide an application of the proposed theory in secure communication.     

Cluster synchronization in a network of non-identical dynamic systems

Cluster synchronization in a network of non-identical dynamic systems is studied in this paper, using two-cluster synchronization for detailed analysis and discussion. The results show that the common intercluster coupling condition is not always needed for the diffusively coupled network. Several sufficient conditions are obtained by using the Schur unitary triangularization theorem, which extends previous results. Some numerical examples are presented for illustration.     

Dynamical hysteresis and spatial synchronization in coupled non-identical chaotic oscillators

We identify a novel phenomenon in distinct (namely non-identical) coupled chaotic systems, which we term dynamical hysteresis. This behavior, which appears to be universal, is defined in terms of the system dynamics (quantified for example through the Lyapunov exponents), and arises from the presence of at least two coexisting stable attractors over a finite range of coupling, with a change of stability outside this range. Further characterization via mutual synchronization indices reveals that one attractor corresponds to spatially synchronized oscillators, while the other corresponds to desynchronized oscillators. Dynamical hysteresis may thus help to understand critical aspects of the dynamical behavior of complex biological systems, e.g. seizures in the epileptic brain can be viewed as transitions between different dynamical phases caused by time dependence in the brain’s internal coupling.     

Projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks

We study projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks. We relax some limitations of previous work, where projective-anticipating and projective-lag synchronization can be achieved only on two coupled chaotic systems. In this paper, we realize projective-anticipating and projective-lag synchronization on complex dynamical networks composed of a large number of interconnected components. At the same time, although previous work studied projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In this paper, the dynamics of the nodes of the complex networks are time-delayed chaotic systems without the limitation of the partial linearity. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random dynamical networks, and we find both its existence and sufficient stability conditions. The validity of the proposed method is demonstrated and verified by examining specific examples using Ikeda and Mackey-Glass systems on Erdos-Renyi networks.     

两个不相同系统的广义同步化

采用Temam无穷维动力系统的惯性流形理论，证明了两个不相同的系统能实现广义同步化.在两个系统具有吸收集和吸引子的基础上，通过定义在一个函数类上的压缩映射的不动点得到了广义同步化流形，该流形是Lipschitz光滑流形，而且具有不变性与指数吸引性.数值仿真证实了所建立理论的正确性.     

Experimental and numerical study on the basin stability of the coupled metronomes

The basin stability is an effective parameter to measure the stability of multistable system under perturbations. In this paper, we try to explore the effects of the coupling strength on the basin stability of the coupled metronomes. In two coupled non-identical metronomes, the coupling strength linearly decreases the basin stability of in-phase synchronization while increases that of the anti-phase synchronization. In three coupled metronomes, there are rich coexisting collectively dynamics as in-phase, anti-phase synchronization, quasi-period states and period 4 states. The coupling strength may still change the basin stability of these coexisting dynamics states. The results are observed in experimental systems and numerical models. Our findings are significant on understanding the multistable dynamics under noisy environment.     

An approach for characterizing coupling in dynamical systems

The study of coupling in dynamical systems dates back to Christian Hyugens who, in 1665, discovered that pendulum clocks with the same length pendulum synchronize when they are near to each other. In that case the observed synchronous motion was out of phase. In this paper we propose a new approach for measuring the degree of coupling and synchronization of a dynamical system consisting of interacting subsystems. The measure is based on quantifying the active degrees of freedom (e.g. correlation dimension) of the coupled system and the constituent subsystems. The time-delay embedding scheme is extended to coupled systems and used for attractor reconstruction of the coupled dynamical system. We use the coupled Lorenz, Rossler and Hénon model systems with a coupling strength variable for evaluation of the proposed approach. Results show that we can measure the active degrees of freedom of the coupled dynamical systems and can quantify and distinguish the degree of synchronization or coupling in each of the dynamical systems studied. Furthermore, using this approach the direction of coupling can be determined.     

Anti-phase synchronization of two coupled mechanical metronomes

This paper mainly investigates the anti-phase synchronization of two coupled mechanical metronomes not only by means of numerical simulations, but also by experimental tests. It is found that the attractor basin of anti-phase synchronization enlarges as the rolling friction increases. Furthermore, this paper studies the relationship between different initial conditions and synchronization types. The impacts of rolling friction on in-phase and anti-phase synchronization times are also discovered. Finally, in-phase and anti-phase synchronization conditions of non-identical metronomes are discussed. These results indicate the potential complexity of the dynamics of coupled metronomes.     

Transmission of information and synchronization in a pair of coupled chaotic circuits: An experimental overview

We propose a rationale for experimentally studying the intricate relationship between the rate of information transmission and synchronization level in active networks, applying theoretical results recently proposed. We consider two non-identical coupled Chua’s circuit with non-identical coupling strengths in order to illustrate the proceeding for experimental scenarios of very few data points coming from highly non-coherent coupled systems, such that phase synchronization can only be detected by methods that do not rely explicitely on the calculation of the phase. A relevant finding is to show that for the coupled Chua’s circuit, the larger the level of synchronization the larger the rate of information exchanged between both circuits. We further validate our findings with data from numerical simulations, and discuss an extension to arbitrarily large active networks.     

Synchronization of period-doubling oscillations in vascular coupled nephrons

The mechanisms by which the individual functional unit (nephron) of the kidney regulates the incoming blood flow give rise to a number of nonlinear dynamic phenomena, including period-doubling bifurcations and intra-nephron synchronization between two different oscillatory modes. Interaction between the nephrons produces complicated and time-dependent inter-nephron synchronization patterns. In order to understand the processes by which a pair of vascular coupled nephrons synchronize, the paper presents a detailed analysis of the bifurcations that occur at the threshold of synchronization. We show that, besides infinite cascades of saddle-node bifurcations, these transitions involve mutually connected cascades of torus and homoclinic bifurcations. To illustrate the broader range of occurrence of this bifurcation structure for coupled period-doubling systems, we show that a similar structure arises in a system of two coupled, non-identical Ro?ssler oscillators.