Analytical and numerical studies of noise-induced synchronization of chaotic systems

We study the effect that the injection of a common source of noise has on the trajectories of chaotic systems, addressing some contradictory results present in the literature. We present particular examples of one-dimensional maps and the Lorenz system, both in the chaotic region, and give numerical evidence showing that the addition of a common noise to different trajectories, which start from different initial conditions, leads eventually to their perfect synchronization. When synchronization occurs, the largest Lyapunov exponent becomes negative. For a simple map we are able to show this phenomenon analytically. Finally, we analyze the structural stability of the phenomenon. (c) 2001 American Institute of Physics.     

Synchronization of self-sustained oscillators by common white noise

We study the stability of self-sustained oscillations under the influence of external noise. For small-noise amplitude a phase approximation for the Langevin dynamics is valid. A stationary distribution of the phase is used for an analytic calculation of the maximal Lyapunov exponent. We demonstrate that for small noise the exponent is negative, which corresponds to synchronization of oscillators.     

Phase synchronization of chaotic rotators

We demonstrate the existence of phase synchronization of two chaotic rotators. Contrary to phase synchronization of chaotic oscillators, here the Lyapunov exponents corresponding to both phases remain positive even in the synchronous regime. Such frequency locked dynamics with different ratios of frequencies are studied for driven continuous-time rotators and for discrete circle maps. We show that this transition to phase synchronization occurs via a crisis transition to a band-structured attractor.     

简并光学参量振荡器混沌相同步与反相同步

从广泛意义上研究了简并光学参量振荡器的混沌同步. 数值结果表明, 当最大条件李指数小于零时, 处于混沌态的两个简并光学参量振荡器通过单向耦合可以实现相同步或反相同步. 当两个简并光学参量振荡器同时实现正耦合或负耦合时为相同步, 当其中一个为正耦合而另一个为负耦合时实现反相同步.     

Symmetry-breaking in local Lyapunov exponents

Integrable dynamical systems, namely those having as many independent conserved quantities as freedoms, have all Lyapunov exponents equal to zero. Locally, the instantaneous or finite time Lyapunov exponents are nonzero, but owing to a symmetry, their global averages vanish. When the system becomes nonintegrable, this symmetry is broken. A parallel to this phenomenon occurs in mappings which derive from quasiperiodic Schr?dinger problems in 1-dimension. For values of the energy such that the eigenstate is extended, the Lyapunov exponent is zero, while if the eigenstate is localized, the Lyapunov exponent becomes negative. This occurs by a breaking of the quasiperiodic symmetry of local Lyapunov exponents, and corresponds to a breaking of a symmetry of the wavefunction in extended and critical states. Received 25 October 2001 / Received in final form 8 December 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: r.ramaswamy@mail.jnu.ac.in     

Transitions to Long-Resident States in Coupled Chaotic Oscillators

The behaviour of coupled chaotic oscillators before complete synchronization is investigated. Long-time residence of trajectories appears besides one of the saddle foci. The tendency that orbits of the two oscillators get closer becomes faster with the increasing coupling strength. The diffusion of phase difference between the two oscillators is first enhanced and then suppressed. There are exact correspondences among these phenomena. The mechanism of these correspondences is explored. These phenomena uncover the route to synchronization of coupled chaotic oscillators.     

Robust pre-specified time synchronization of chaotic systems by employing time-varying switching surfaces in the sliding mode control scheme

In the conventional chaos synchronization methods, the time at which two chaotic systems are synchronized, is usually unknown and depends on initial conditions. In this work based on Lyapunov stability theory a sliding mode controller with time-varying switching surfaces is proposed to achieve chaos synchronization at a pre-specified time for the first time. The proposed controller is able to synchronize chaotic systems precisely at any time when we want. Moreover, by choosing the time-varying switching surfaces in a way that the reaching phase is eliminated, the synchronization becomes robust to uncertainties and exogenous disturbances. Simulation results are presented to show the effectiveness of the proposed method of stabilizing and synchronizing chaotic systems with complete robustness to uncertainty and disturbances exactly at a pre-specified time.     

采用振幅耦合方法研究多旋转中心混沌振子的相同步

为了找到具有多个旋转中心的混沌系统的相同步与其动力学拓朴变化之间的对应关系，采用线性振幅线性耦合方法，研究了Lorenz系统和Duffing系统的相同步，首先对Lorenz系统和Duffing系统分别进行极坐标变换，在线性振幅耦合基础上计算了两个系统的平均旋转数和Lyapunov指数，发现，随耦合强度的增大，系统相同步与系统的Lyapunov指数跃变存在一一对应的关系，这表明具有多个旋转中心的混沌系统的相同步与系统动力学拓朴变化也存在着对应关系. 关键词： Lyapunov指数 振幅耦合 相同步     

Phase synchronization in bi-directionally coupled chaotic ratchets

We show the existence of phase synchronization in bi-directionally coupled deterministic chaotic ratchets. The coupled ratchets were simulated in their chaotic states. A transition from a regime where the phases rotate with different velocities to a synchronous state where the phase difference is bounded was observed as the coupling was increased. In addition, the region of synchronization in which the system is permanently phase locked was identified. In this regime, the transverse Lyapunov exponent corresponding to both phases remain positive. Our calculations show that the transition to a synchronized state occurs via a crisis transition to an attractor filling the whole phase space.     

耦合混沌振子系统完全同步的动力学行为

以耦合Duffing振子为对象，研究了混沌系统进入完全同步态时的一些动力学行为. 在对称耦合情况下，随着耦合系数的变化系统达到各个混沌振子的相轨道完全相同的同步态——完全同步态. 通过计算Lyapunov指数表明，此时系统的前两个横向Lyapunov指数相等，同时系统之间的时间关联表现出明显的规律性. 关键词： Duffing振子 混沌同步 Lyapunov指数     

On the difference between negative damping and eigenmode synchronization as two phonation onset mechanisms

Negative damping and eigenmode synchronization as two different mechanisms of phonation onset are distinguished. Although both mechanisms lead to a favorable phase relationship between the flow pressure and the vocal fold motion as required for a net energy transfer into the vocal folds, the underlying mechanisms for this favorable phase relationship are different. The negative damping mechanism relies on glottal aerodynamics or acoustics to establish before onset and maintain after onset the favorable phase relationship, and therefore has minimum requirements on vocal fold geometry and biomechanics. A single degree-of-freedom vocal fold model is all that is needed for self-oscillation in the presence of a negative damping mechanism. In contrast, the mechanism of eigenmode synchronization critically depends on the geometrical and biomechanical properties of the vocal folds (at least 2-degrees-of-freedom are required), and has little requirement on the glottal aerodynamics other than flow separation. The favorable phase relation is established once synchronization occurs, regardless of the phase relationship imposed by glottal aerodynamics before onset. Unlike that of the negative damping mechanism, initiation of eigenmode synchronization requires neither a velocity-dependent flow pressure nor an alternating convergent-divergent glottis. The clinical implications of the distinctions between these two mechanisms are discussed.     

A secure communication scheme based on the phase synchronization of chaotic systems

Phase synchronization of chaotic systems with both weak and strong couplings has recently been investigated extensively. Similar to complete synchronization, this type of synchronization can also be applied in secure communications. We develop a digital secure communication scheme that utilizes the instantaneous phase as the signal transmitted from the drive to the response subsystems. Simulation results show that the scheme is difficult to be broken by some traditional attacks. Moreover, it operates with a weak positive conditional Lyapunov exponent in the response subsystem.     

噪声在外加周期信号控制强湍中的作用研究

研究了噪声在控制中的作用.研究发现,较小强度的偏噪声信号能够使系统失去层流态,重新进入无序态,此时模式之间的广义相同步也失去了;而噪声强度较大时,系统则能够维持准周期的层流态,模式相位之间依然能够达到广义相同步. 关键词： 相同步 时空混沌 湍流 非线性漂移波     

Synchronization of periodic self-oscillations by common noise

We consider the effect of external noise on the stability properties of self-oscillations. A stochastic equation for the phase is derived at the limit of weak noise (in the appropriate sense). The stationary probability-density distribution is used for an analytic calculation of the Lyapunov exponent. We show that the exponent is always negative for the small noise level, which corresponds to synchronization of self-oscillations.     

Chaos of the relativistic parametrically forced van der Pol oscillator

A manifestly relativistically covariant form of the van der Pol oscillator in 1 + 1 dimensions is studied. We show that the driven relativistic equations, for which x and t are coupled, relax very quickly to a pair of identical decoupled equations, due to a rapid vanishing of the “angular momentum” (the boost in 1 + 1 dimensions). A similar effect occurs in the damped driven covariant Duffing oscillator previously treated. This effect is an example of entrainment, or synchronization (phase locking), of coupled chaotic systems. The Lyapunov exponents are calculated using the very efficient method of Habib and Ryne. We show a Poincaré map that demonstrates this effect and maintains remarkable stability in spite of the inevitable accumulation of computer error in the chaotic region. For our choice of parameters, the positive Lyapunov exponent is about 0.242 almost independently of the integration method.     

Synchronization by dynamical relaying in electronic circuit arrays

We experimentally study the synchronization of two chaotic electronic circuits whose dynamics is relayed by a third parameter-matched circuit, to which they are coupled bidirectionally in a linear chain configuration. In a wide range of operating parameters, this setup leads to synchronization between the outer circuits, while the relaying element remains unsynchronized. The specifics of the synchronization differ with the coupling level: for low couplings a state of intermittent synchronization between the outer circuits coexists with one of antiphase synchronization. Synchronization becomes in phase for moderate couplings, and for strong coupling identical synchronization is observed between the outer elements, which are themselves synchronized in a generalized way with the relaying element. In the latter situation, the middle element displays a triple-scroll attractor that is not possible to obtain when the chaotic oscillator is isolated.     

两个非耦合Hindmarsh-Rose神经元同步的非线性特征研究

利用Hindmarsh-Rose(HR)神经元输出的膜电压作为刺激调整两个具有不同初始条件的非耦合HR神经元的电流输入，通过分析神经元放电峰峰间期(ISI)的分布揭示了两个神经元同步过程轨道演化的机理.在周期信号刺激下，两个具有相同参数原处于混沌状态的神经元可以 实现完全同步，且可以同步到不同于刺激信号频率的周期响应上；两个具有不同参数的神经 元可以实现相位同步，参数差别较小的两个神经元可以相位同步到与刺激信号不同频率的周 期响应上，参数差别较大的两个神经元只可能相位同步到与刺激信号相同频率的周期响应上 .混沌信号刺激两个神经元只可能同步到产生混沌信号神经元的放电模式上，可见混沌刺激 更有利于神经元信息编码与解码.分析两个被调整神经元系统的最大条件Lyapunov 指数(Lmc )与刺激强度k的关系表明当k达到某一阈值时两个系统的Lmc均为负值是两个系统实现同 步的必要条件.平均发放率相同的混沌刺激和周期刺激相比较混沌刺激更容易使两个神经元 实现同步，表明混沌刺激产生的效应更强，该结论与实验结果相符合. 关键词： 放电峰峰间期 同步 相位同步 条件Lyapunov 指数     

耦合系统的混沌相位同步

通过引入混沌运动的相位定义分析了线性和非线性耦合参数对两个主共振子系统之间的混沌相位同步的影响.讨论了在近似于主共振条件下,两子系统不同步、不完全相位同步和完全相位同步之间的演化过程,揭示了不同状态相互转化与Lyapunov指数变化之间的关系,指出随着线性耦合力的增加,相位同步效应增强,然而随着非线性耦合力的增加,相位同步效应减弱. 关键词： 相位同步 Rssler振子 耦合 Lyapunov指数     

Complex lag synchronization of two identical chaotic complex nonlinear systems

Much progress has been made in the research of synchronization for chaotic real or complex nonlinear systems. In this paper we introduce a new type of synchronization which can be studied only for chaotic complex nonlinear systems. This type of synchronization may be called complex lag synchronization (CLS). A definition of CLS is introduced and investigated for two identical chaotic complex nonlinear systems. Based on Lyapunov function a scheme is designed to achieve CLS of chaotic attractors of these systems. The effectiveness of the obtained results is illustrated by a simulation example. Numerical results are plotted to show state variables, modulus errors and phase errors of these chaotic attractors after synchronization to prove that CLS is achieved.     

相空间压缩实现时空混沌系统的广义同步

提出了一种通过相空间压缩实现时空混沌系统广义同步的方法. 以Fitzhugh-Nagumo反应扩散时空混沌系统为例,仿真模拟说明了该方法的有效性与实用性. 通过研究有界噪声作用下该系统的同步效果,表明这种同步方法具有较强的抗干扰能力. 此方法可以实现任意时空混沌系统的广义同步,具有普适性. 同步控制器结构简单、易于应用. 关键词： 时空混沌 广义同步 相空间压缩