Noise-induced synchronization and coherence resonance of a Hodgkin-Huxley model of thermally sensitive neurons

We study nontrivial effects of noise on synchronization and coherence of a chaotic Hodgkin-Huxley model of thermally sensitive neurons. We demonstrate that identical neurons which are not coupled but subjected to a common fluctuating input (Gaussian noise) can achieve complete synchronization when the noise amplitude is larger than a threshold. For nonidentical neurons, noise can induce phase synchronization. Noise enhances synchronization of weakly coupled neurons. We also find that noise enhances the coherence of the spike trains. A saddle point embedded in the chaotic attractor is responsible for these nontrivial noise-induced effects. Relevance of our results to biological information processing is discussed.     

Generalized chaotic synchronization in coupled Ginzburg-Landau equations

Generalized synchronization is analyzed in unidirectionally coupled oscillatory systems exhibiting spatiotemporal chaotic behavior described by Ginzburg-Landau equations. Several types of coupling between the systems are analyzed. The largest spatial Lyapunov exponent is proposed as a new characteristic of the state of a distributed system, and its calculation is described for a distributed oscillatory system. Partial generalized synchronization is introduced as a new type of chaotic synchronization in spatially nonuniform distributed systems. The physical mechanisms responsible for the onset of generalized chaotic synchronization in spatially distributed oscillatory systems are elucidated. It is shown that the onset of generalized chaotic synchronization is described by a modified Ginzburg-Landau equation with additional dissipation irrespective of the type of coupling. The effect of noise on the onset of a generalized synchronization regime in coupled distributed systems is analyzed.     

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.     

Observability of lag synchronization of coupled chaotic oscillators

Lag synchronization is a recently discovered theoretical phenomenon where the dynamical variables of two coupled, nonidentical chaotic oscillators are synchronized with a time delay relative to each other. We investigate experimentally and numerically to what extent lag synchronization can be observed in physical systems where noise is inevitable. Our measurements and numerical computation suggest that lag synchronization is typically destroyed when the noise level is comparable to the amount of average system mismatch. At small noise levels, lag synchronization occurs in an intermittent fashion.     

Chaotic synchronization: a nonlinear predictive filtering approach

The synchronization of chaotic systems is a difficult task due to their sensitive dependence on the initial conditions. Perfect synchronization is almost impossible when noise is present in the system. One of the well known stochastic filtering algorithms that is used to synchronize chaotic systems in the presence of noise is the extended Kalman filter (EKF). However, for highly nonlinear systems, the approximation error introduced by the EKF has been shown to be relatively high. In this paper, a nonlinear predictive filter (NPF) is proposed for synchronizing chaotic systems. In this scheme, it is not required to approximate the underlying nonlinearity and hence there is no need to compute the Jacobian of the chaotic system. Numerical simulations are carried out to compare the performances of the NPF and EKF algorithms for synchronizing different sets of chaotic systems and/or maps. The well known Lorenz and Mackey-Glass systems as well as Ikeda map are used for numerical evaluation of the performance. Results clearly show that the NPF based approach is superior to the EKF based approach in terms of the normalized mean square error (NMSE), total NMSE, and the time taken for synchronization (measured in terms of the normalized instantaneous square error) for all the systems and/or maps considered.     

Chaotic synchronization of coupled ergodic maps

With few exceptions, studies of chaotic synchronization have focused on dissipative chaos. Though less well known, chaotic systems that lack dissipation may also synchronize. Motivated by an application in communication systems, we couple a family of ergodic maps on the N-torus and study the global stability of the synchronous state. While most trajectories synchronize at some time, there is a measure zero set that never synchronizes. We give explicit examples of these asynchronous orbits in dimensions two and four. On more typical trajectories, the synchronization error reaches arbitrarily small values and, in practice, converges. In dimension two we derive bounds on the average synchronization time for trajectories resulting from randomly chosen initial conditions. Numerical experiments suggest similar bounds exist in higher dimensions as well. Adding noise to the coupling signal destroys the invariance of the synchronous state and causes typical trajectories to desynchronize. We propose a modification of the standard coupling scheme that corrects this problem resulting in robust synchronization in the presence of noise.     

Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters

This paper studies the adaptive complete synchronization of chaotic and hyperchaotic systems with fully unknown parameters. In practical situations, some systems' parameters cannot be exactly known a priori, and the uncertainties often affect the stability of the process of synchronization of the chaotic oscillators. An adaptive scheme is proposed to compensate for the effects of parameters' uncertainty based on the structure of chaotic systems in this paper. Based on the Lyapunov stability theorem, an adaptive controller and a parameters update law can be designed for the synchronization of chaotic and hyperchaotic systems. The drive and response systems can be nonidentical, even with different order. Three illustrative examples are given to demonstrate the validity of this technique, and numerical simulations are also given to show the effectiveness of the proposed chaos synchronization method. In addition, this synchronization scheme is quite robust against the effect of noise.     

Pragmatical generalized synchronization of chaotic systems with uncertain parameters by adaptive control

A new kind of generalized synchronization of two chaotic systems with uncertain parameters is proposed. Based on a pragmatical asymptotical stability theorem and an assumption of equal probability for ergodic initial conditions, an adaptive control law is derived so that it can be proved strictly that the common null solution of error dynamics and of parameter dynamics is actually asymptotically stable, i.e. these two identical systems are in generalized synchronization and the estimated parameters approach the uncertain values. It is called pragmatical generalized synchronization. Finally, two numerical examples are studied for two Quantum-CNN oscillator chaotic systems to show the effectiveness of the proposed generalized synchronization strategy with a double Duffing chaotic system as a goal system.     

Further results on complete synchronization for noise-perturbed chaotic systems

In this Letter, a class of general systems which covers several famous chaotic systems is studied in complete synchronization with noise perturbation. Special nonlinear coupling techniques as well as LaSalle-type invariance principle of stochastic differential equations are employed to deduce our sufficient conditions for complete synchronization without involving the boundedness of chaotic systems. Furthermore, the correlative numerical simulations are given to illustrate the effectiveness of our theoretic results.     

Colored-noise-induced chaotic array synchronization

The effect of a time-correlated Gaussian noise on one-dimensional arrays consisting of diffusively coupled chaotic cells is analyzed. A resonance effect between the time scale of the chaotic attractor and the colored Gaussian noise has been found. As well, depending on the number of cells, coupling, and noise strength, an improvement of the synchronization or a poor synchronization between cells within the array can occur for some values of the time correlation. These nonlinear cooperative effects are studied in terms of a linear stability analysis around the uniform synchronized behavior.     

Complete and generalized synchronization in a class of noise perturbed chaotic systems

In the paper, in light of the LaSalle-type invariance principle for stochastic differential equations, chaos synchronization is investigated for a class of chaotic systems dissatisfying a globally Lipschitz condition with noise perturbation. Sufficient criteria for both complete synchronization and generalized synchronization are rigorously established and thus successfully applied to realize chaos synchronization in the coupled unified chaotic systems. Furthermore, concrete examples as well as their numerical simulations are provided to illustrate the possible application of the established criteria.     

单向耦合映象格子的时空混沌同步

以耦合映象格子为对象,研究了时空混沌系统的同步问题. 基于Lyapunov稳定性定理,通过恰当地选择驱动函数,实现了两个单向耦合映象格子的完全同步. 仿真模拟验证了这种同步方法的有效性. 讨论了控制参量对同步速率的影响. 仿真模拟还表明,在存在系统偏差并受到噪声影响的情况下,仍然可以实现两系统的同步,这种同步方法具有一定的抗干扰能力. 关键词： 时空混沌 同步 耦合映象格子     

分数阶统一混沌系统的自适应同步

提出了分数阶统一混沌系统的自适应同步方法, 给出了自适应同步控制器和参数自适应率. 设计的单一控制器简单且只含有一个驱动变量. 所用同步方法适用于一类分数阶混沌系统的同步, 且具有较强的抗噪声能力, 具有较高的实用价值. 数值模拟结果表明了该方法的有效性. 关键词： 分数阶 统一混沌系统 自适应同步     

耦合哈密顿系统中测度同步的研究

测度同步现象是耦合哈密顿系统的一种重要性质.对规则系统和混沌系统的测度同步性 质作了深入研究，重点讨论了耦合哈密顿系统处于混沌状态时，系统测度同步的特点及系统 的相位关系.提出了一种定量判断测度同步的简单方法，考虑了高斯白噪声对系统中测度同 步性质的影响. 关键词： 耦合哈密顿系统 测度同步 相锁定 高斯白噪声     

Complexity and synchronization in stochastic chaotic systems

We investigate the complexity of a hyperchaotic dynamical system perturbed by noise and various nonlinear speech and music signals. The complexity is measured by the weighted recurrence entropy of the hyperchaotic and stochastic systems. The synchronization phenomenon between two stochastic systems with complex coupling is also investigated. These criteria are tested on chaotic and perturbed systems by mean conditional recurrence and normalized synchronization error. Numerical results including surface plots, normalized synchronization errors, complexity variations etc show the effectiveness of the proposed analysis.     

Generating and enhancing lag synchronization of chaotic systems by white noise

In this paper, we study the crucial impact of white noise on lag synchronous regime in a pair of time-delay unidirectionally coupled systems. Our result demonstrates that merely via white-noise-based coupling lag synchronization could be achieved between the coupled systems (chaotic or not). And it is also demonstrated that a conventional lag synchronous regime can be enhanced by white noise. Sufficient conditions are further proved mathematically for noise-inducing and noise-enhancing lag synchronization, respectively. Additionally, the influence of parameter mismatch on the proposed lag synchronous regime is studied, by which we announce the robustness and validity of the new strategy. Two numerical examples are provided to illustrate the validity and some possible applications of the theoretical result.     

Identifying parameter by identical synchronization between different systems

In this paper, parameters of a given (chaotic) dynamical system are estimated from time series by using identical synchronization between two different systems. This technique is based on the invariance principle of differential equations, i.e., a dynamical Lyapunov function involving synchronization error and the estimation error of parameters. The control used in this synchronization consists of feedback and adaptive control loop associated with the update law of estimation parameters. Our estimation process indicates that one may identify dynamically all unknown parameters of a given (chaotic) system as long as time series of the system are available. Lorenz and Rossler systems are used to illustrate the validity of this technique. The corresponding numerical results and analysis on the effect of noise are also given.     

Random shortcuts induce phase synchronization in complex Chua systems

This paper studies how phase synchronization in complex networks depends on random shortcuts, using the piecewise-continuous chaotic Chua system as the nodes of the networks. It is found that for a given coupling strength, when the number of random shortcuts is greater than a threshold the phase synchronization is induced. Phase synchronization becomes evident and reaches its maximum as the number of random shortcuts is further increased. These phenomena imply that random shortcuts can induce and enhance the phase synchronization in complex Chua systems. Furthermore, the paper also investigates the effects of the coupling strength and it is found that stronger coupling makes it easier to obtain the complete phase synchronization.     

Anticipation in the synchronization of chaotic semiconductor lasers with optical feedback

The synchronization of chaotic semiconductor lasers with optical feedback is studied numerically in a one-way coupling configuration, in which a small amount of the intensity of one laser (master laser) is injected coherently into the other (slave laser). A regime of anticipated synchronization is found, in which the intensity of the slave laser is synchronized to the future chaotic intensity of the master laser. Anticipation is robust to small noise and parameter mismatches, but in this case the synchronization is not complete. It is also shown that anticipated synchronization occurs in coupled time-delay systems, when the coupling has a delay that is less than the delay of the systems.     

不确定自催化反应扩散时空混沌系统的延时同步

设计了一种延迟同步控制器实现了时空混沌系统之间的同步控制.基于Lyapunov稳定性定理,确定了延迟同步控制器的结构以及系统状态变量之间的误差方程.以自催化反应扩散时空混沌系统为例,仿真模拟验证了该控制器的有效性.进一步设计了参量辨识器,对不确定自催化反应扩散时空混沌系统中的参量进行了有效辨识.通过研究有界噪声影响下系统的同步效果,表明该同步方法具有较好的抗干扰能力. 关键词： 时空混沌 延时同步 参量辨识 有界噪声