A kind of binary scaling function projective lag synchronization of chaotic systems with stochastic perturbation

This paper investigates a kind of binary function projective lag synchronization of uncertain chaotic systems with stochastic perturbation. In comparison with those of the existing scaling function synchronization, we assume that the given scaling function can be the binary boundary function, even an n-ary boundary function. Based on the LaSalle-type invariance principle for stochastic differential equation, the adaptive control law is derived to make the state of two chaotic systems function projective lag synchronized. Some numerical is also given to show the effectiveness of the proposed method.     

Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control

In this paper, we apply the nonsingular terminal sliding mode control technique to realize the novel combination-combination synchronization between combination of two chaotic systems as drive system and combination of two chaotic systems as response system with unknown parameters in a finite time. On the basic of the adaptive laws and finite-time stability theory, an adaptive combination sliding mode controller is proposed to ensure the occurrence of the sliding motion in a given finite time for four different chaotic systems. In theory, it is proved that the sliding mode technique can realize fast convergence for four different chaotic systems in the finite time. Some criteria and corollaries are derived for finite-time combination-combination synchronization of four different chaotic systems. Numerical simulation results are shown to verify the effectiveness and correctness of the combination-combination synchronization.     

Synchronization of stochastic reaction–diffusion systems via boundary control

This paper studies the problem of mean square asymptotical synchronization and \(H_\infty \) synchronization for coupled stochastic reaction–diffusion systems (SRDSs) via boundary control. Based on the deduced synchronization error dynamic, we design boundary controllers to achieve mean square asymptotical synchronization. By virtue of Lyapunov functional method and Wirtinger’s inequality, sufficient conditions are obtained for ensuring mean square asymptotical synchronization. When coupled SRDSs are subject to external disturbance, mean square \(H_\infty \) synchronization is investigated and corresponding criterion is presented under a designed boundary controller. In addition to focusing on systems with Neumann boundary conditions, we also briefly study coupled SRDSs with mixed boundary conditions and sufficient conditions are provided to achieve the desired performance. Numerical examples are used to verify the effectiveness of our theoretical results.     

Adaptive synchronization of stochastic neural networks with mixed time delays and reaction–diffusion terms

In this paper, the problem of exponential synchronization is investigated for a class of stochastic perturbed chaotic neural networks with both mixed time delays and reaction?Cdiffusion terms. By employing Lyapunov?CKrasovskii functional and stochastic analysis approaches, an adaptive controller is designed to guarantee the exponential synchronization of proposed neural networks in the mean square. In particular, the mixed time delays in this paper synchronously consist of constant delay in the leakage term (i.e., ??leakage delay??), discrete time-varying delay and distributed time-varying delay which are more general than those discussed in the previous literature. Furthermore, our synchronization criteria are easily verified and do not need to solve any linear matrix inequality. Therefore, the results obtained in this paper generalize and improve those given in the previous literature. Finally, the extensive simulations are performed to show the effectiveness and feasibility of the obtained method.     

Finite-time real combination synchronization of three complex-variable chaotic systems with unknown parameters via sliding mode control

The problem of real combination synchronization between three complex-variable chaotic systems with unknown parameters is investigated by nonsingular terminal sliding mode control in a finite time. Based on the adaptive laws and finite-time stability theory, a nonsingular terminal sliding mode control is designed to ensure the real combination synchronization of three complex-variable chaotic systems in a given finite time. It is theoretically gained that the introduced sliding mode technique has finite-time convergence and stability in both arriving and sliding mode phases. Numerical simulation results are given to show the effectiveness and reliability of the finite-time real combination synchronization.     

Synchronization of nonlinear chaotic electromechanical gyrostat systems with uncertainties

This paper solves the problem of robust synchronization of nonlinear chaotic gyrostat systems in a given finite time. The parameters of both master and slave chaotic gyrostat systems are assumed to be unknown in advance. In addition, the gyrostat systems are disturbed by unknown model uncertainties and external disturbances. Suitable update laws are proposed to estimate the unknown parameters. Based on the finite-time control idea and update laws, appropriate control laws are designed to ensure the stabilization of the closed-loop system in finite time. The precise value of the convergence time is given. A numerical simulation demonstrates the applicability and efficiency of the proposed finite-time synchronization strategy.     

Synchronization and chaos control by quorum sensing mechanism

Diverse rhythms are generated by thousands of oscillators that somehow manage to operate synchronously. By using mathematical and computational modeling, we consider the synchronization and chaos control among chaotic oscillators coupled indirectly but through a quorum sensing mechanism. Some sufficient criteria for synchronization under quorum sensing are given based on traditional Lyapunov function method. The Melnikov function method is used to theoretically explain how to suppress chaotic Lorenz systems to different types of periodic oscillators in quorum sensing mechanics. Numerical studies for classical Lorenz and Rössler systems illustrate the theoretical results.     

Observer-based impulsive chaotic synchronization of discrete-time switched systems

This paper investigates impulsive chaotic synchronization of discrete-time switched systems with state-dependent switching strategy. The parameter-dependent Lyapunov function (PDLF) technique is used to establish stability criteria for a class of switched systems consisting of both stable and unstable subsystems. With these criteria, sufficient conditions are given to achieve observer-based impulsive chaotic synchronization. Examples are presented to illustrate the criteria.     

Application of Takagi–Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization

In this study, we investigate a class of chaotic synchronization and anti-synchronization with stochastic parameters. A controller is composed of a compensation controller and a fuzzy controller which is designed based on fractional stability theory. Three typical examples, including the synchronization between an integer-order Chen system and a fractional-order Lü system, the anti-synchronization of different 4D fractional-order hyperchaotic systems with non-identical orders, and the synchronization between a 3D integer-order chaotic system and a 4D fractional-order hyperchaos system, are presented to illustrate the effectiveness of the controller. The numerical simulation results and theoretical analysis both demonstrate the effectiveness of the proposed approach. Overall, this study presents new insights concerning the concepts of synchronization and anti-synchronization, synchronization and control, the relationship of fractional and integer order nonlinear systems.     

Projective synchronization between two different time-delayed chaotic systems using active control approach

In this paper, we investigate the projective synchronization between two different time-delayed chaotic systems. A suitable controller is chosen using the active control approach. We relax some limitations of previous work, where projective synchronization of different chaotic systems can be achieved only in finite dimensional chaotic systems, so we can achieve projective synchronization of different chaotic systems in infinite dimensional chaotic systems. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective synchronization between two different time-delayed chaotic systems. The validity of the proposed method is demonstrated and verified by observing the projective synchronization between two well-known time-delayed chaotic systems; the Ikeda system and Mackey–Glass system. Numerical simulations fully support the analytical approach.     

Dynamical randomicity and predictive analysis in cubic chaotic system

The dynamical randomicity and grey prediction in cubic chaotic system will be studied in this paper. These stochastic symbolic sequences bear three features. The distribution of frequency, inter-occurrence times, the first passage time, the rth passage time and the ordinal passage time are discussed separately. By using transfer probability of Markov chain (MC), one obtains analytic expressions of generating functions in three-probabilities stochastic wander model, which can be applied to all three symbolic systems. The visitation density function of cubic map will also be resolved. Especially, after introducing grey system theory, one is mainly using GM(1,1) model to forecast data sequences, and the usual forecast precision is approximately 90%. In the symbolic prediction of cubic chaotic dynamical system, the precision of grey prediction certainly will decrease as the length of symbolic sequence is increasing. But in this place we have found a generating rule that may realize chaotic synchronization at least in short and medium terms, and we can analyze and forecast in this way.     

高斯色噪声和谐波激励共同作用下耦合SD振子的混沌研究

耦合SD振子作为一种典型的负刚度振子, 在工程设计中有广泛应用. 同时高斯色噪声广泛存在于外界环境中, 并可能诱发系统产生复杂的非线性动力学行为, 因此其随机动力学是非线性动力学研究的热点和难点问题. 本文研究了高斯色噪声和谐波激励共同作用下双稳态耦合SD振子的混沌动力学, 由于耦合SD振子的刚度项为超越函数形式, 无法直接给出系统同宿轨道的解析表达式, 给混沌阈值的分析造成了很大的困难. 为此, 本文首先采用分段线性近似拟合该振子的刚度项, 发展了高斯色噪声和谐波激励共同作用下的非光滑系统的随机梅尔尼科夫方法. 其次, 基于随机梅尔尼科夫过程, 利用均方准则和相流函数理论分别得到了弱噪声和强噪声情况下该振子混沌阈值的解析表达式, 讨论了噪声强度对混沌动力学的影响. 研究结果表明, 随着噪声强度的增大混沌区域增大, 即增大噪声强度更容易诱发耦合SD振子产生混沌. 当阻尼一定时, 弱噪声情况下混沌阈值随噪声强度的增加而减小; 但是强噪声情况下噪声强度对混沌阈值的影响正好相反. 最后, 数值结果表明, 利用文中的方法研究高斯色噪声和谐波激励共同作用下耦合SD振子的混沌是有效的.本文的结果为随机非光滑系统的混沌动力学研究提供了一定的理论指导.      

Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control

This paper addresses a unified mathematical expression describing a class of chaotic systems, for which the problem of synchronization and anti-synchronization between different chaotic systems with fully uncertain parameters and different structure are studied. Based on the Lyapunov stability theory, a novel, simple, and systemic adaptive synchronization controller is designated, the analytic expression of the controller and the adaptive laws of parameters are developed. Moreover, the proposed scheme can be extended to anti-synchronize a class of chaotic systems. Two chaotic systems with different structure and fully uncertain parameters are employed as the examples to show the effectiveness of the proposed adaptive synchronization and anti-synchronization schemes. Additionally, the robustness and noise immunity of the adaptive synchronization scheme is investigated by measuring the mean squared error of the systems.     

脑科学中若干非线性动力学问题

对近年发展起来的脑科学中的非线性问题作一介绍.这些问题得到脑科学界的广泛注意.它们是同步, 混沌与混沌周游, 噪声与随机共振.在很多不同背景下的神经生理实验表明脑皮层的振荡活动都存在同步与去同步现象.混沌在人与动物的脑中扮演着重要的角色, 混沌周游与Freeman模型被认为与联想记忆或记忆的动态连接有关.适当噪声强度导致信噪比的极大提高------随机共振是脑神经系统能检测到极其微弱信号的工作机制.噪声也导致同步态并使之稳定.此外,噪声的一些其他作用也在本文中提及.      

A new Lyapunov approach for global synchronization of non-autonomous chaotic systems

This paper proposes a new approach for finding the Lyapunov function to study the sufficient global synchronization criterion of master-slave non-autonomous chaotic systems via linear state error feedback control. The approach is first demonstrated in a synchronization scheme for the second-order non-autonomous chaotic systems and then generalized to the schemes for the nth-order non-autonomous chaotic systems. Some algebraic synchronization criteria for the second-order chaotic systems are obtained. The sharpness of the new criteria is compared with that of the existing criteria of the same type by numerical examples.     

Anticipating spike synchronization in nonidentical chaotic neurons

Anticipating synchronization is investigated in nonidentical chaotic systems unidirectionally coupled in a master-slave configuration without a time-delay feedback. We show that if the parameters of chaotic master and slave systems are mismatched in such a way that the mean frequency of a free slave system is greater than the mean frequency of a master system, then the phase synchronization regime can be achieved with the advanced phase of the slave system. In chaotic neural systems, this leads to the anticipating spike synchronization: unidirectionally coupled neurons synchronize in such a way that the slave neuron anticipates the chaotic spikes of the master neuron. We demonstrate our findings with coupled Rössler systems as well as with two different models of coupled neurons, namely, the Hindmarsh–Rose neurons and the adaptive exponential integrate-and-fire neurons.     

A novel active pinning control for synchronization and anti-synchronization of new uncertain unified chaotic systems

This paper discusses the synchronization and anti-synchronization of new uncertain unified chaotic systems (UUCS). Based on the idea of active control, a novel active Pinning control strategy is presented, which only needs a state of new UUCS. The proposed controller can achieve synchronization between a response system and a drive system, and ensure the synchronized robust stability of new UUCS. Numerical simulations of new UUCS show that the controller can make chaotic systems achieve synchronization or anti-synchronization in a quite short period and both are of good robust stability.     

Chaos synchronization of discrete-time dynamic systems with a limited capacity communication channel

This paper offers a new control strategy for discrete-time chaos synchronization where the drive system and the response system are coupled via a limited capacity communication channel (LCCC for short). One simple condition is presented to ensure synchronization between the two chaotic systems coupled by a LCCC. Based on this condition, an explicit coder–decoder pair for the coding algorithm is designed and the synchronization error between the two chaotic systems decays to zero exponentially based on this coding algorithm. Finally, the proposed control strategy is applied to the well-known H\′{e}non system, and numerical simulations illustrate the validity of the obtained result.     

Complex modified projective synchronization of two chaotic complex nonlinear systems

In this paper, we present a novel type of synchronization called complex modified projective synchronization (CMPS) and study it to a system of two chaotic complex nonlinear 3-dimensional flows, possessing chaotic attractors. Based on the Lyapunov function approach, a scheme is designed to achieve CMPS for such pairs of (either identical or different) complex systems. Analytical expressions for the complex control functions are derived using this scheme to achieve CMPS. This type of complex synchronization is considered as a generalization of several kinds of synchronization that have appeared in the recent literature. The master and slave chaotic complex systems achieved CMPS can be synchronized through the use of a complex scale matrix. The effectiveness of the obtained results is illustrated by a studying two examples of such coupled chaotic attractors in the complex domain. Numerical results are plotted to show the rapid convergence of modulus errors to zero, thus demonstrating that CMPS is efficiently achieved.     

A new contribution for the impulsive synchronization of fractional-order discrete-time chaotic systems

In this paper, we discuss and investigate the impulsive synchronization of fractional-order discrete-time chaotic systems. The proposed method is based on the impulsive synchronization theory used in the integer-order case on the one hand and the mathematical analysis of the fractional-order discrete-time systems on the other hand. Sufficient conditions for the stability of synchronization error system are given, and application example with numerical simulations is illustrated in order to verify that the proposed method is applicable and effective. Furthermore, in order to validate the proposed synchronization approach, we have also provided the experimental implementation results using Arduino Mega boards.