A general scheme for Q-S synchronization of chaotic systems

This paper addresses the Q-S synchronization between chaotic and/or hyper-chaotic systems. Based on the Lyapunov stability theorem, a general scheme for Q-S synchronization of chaotic and/or hyper-chaotic systems is proposed. The drive and response systems discussed in this paper can be strictly different dynamical systems (including different dimensional systems). Four illustrative numerical simulations are also given to demonstrate the effectiveness of the proposed general scheme for Q-S synchronization.     

Adaptive generalized function projective lag synchronization of different chaotic systems with fully uncertain parameters

In this paper, a novel projective synchronization scheme called adaptive generalized function projective lag synchronization (AGFPLS) is proposed. In the AGFPLS method, the states of two different chaotic systems with fully uncertain parameters are asymptotically lag synchronized up to a desired scaling function matrix. By means of the Lyapunov stability theory, an adaptive controller with corresponding parameter update rule is designed for achieving AGFPLS between two diverse chaotic systems and estimating the unknown parameters. This technique is employed to realize AGFPLS between uncertain Lü chaotic system and uncertain Liu chaotic system, and between Chen hyperchaotic system and Lorenz hyperchaotic system with fully uncertain parameters, respectively. Furthermore, AGFPLS between two different uncertain chaotic systems can still be achieved effectively with the existence of noise perturbation. The corresponding numerical simulations are performed to demonstrate the validity and robustness of the presented synchronization method.     

Adaptive Q–S synchronization between coupled chaotic systems with stochastic perturbation and delay

This work investigates the adaptive Q–S synchronization of coupled chaotic (or hyper-chaotic) systems with stochastic perturbation, delay and unknown parameters. The sufficient conditions for achieving Q–S synchronization of two stochastic chaotic systems are derived based on the invariance principle of stochastic differential equation. By the adaptive control technique, the control laws and the corresponding parameter update laws are proposed such that the stochastic Q–S synchronization of non-identical chaotic (or hyper-chaotic) systems is to be obtained. Finally, two illustrative numerical simulations are also given to demonstrate the effectiveness of the proposed scheme.     

Modified projective synchronization of chaotic systems with disturbances via active sliding mode control

We apply the active sliding mode control technique to realize the modified projective synchronization of the chaotic systems. The disturbances are considered both in the drive system and the response system. The sufficient conditions for the modified projective synchronization both the non-identical and identical chaotic systems are presented. The corresponding numerical simulations are provided to illuminate the effectiveness of the proposed active sliding mode controllers.     

Phase and anti-phase synchronization of fractional order chaotic systems via active control

This paper is devoted to investigate the phase and anti-phase synchronization between two identical and non-identical fractional order chaotic systems using techniques from active control theory. The techniques are applied to fractional order chaotic Lü and Liu systems. Numerical results demonstrate the effectiveness and feasibility of the proposed control techniques.     

Synchronization of different dimensional chaotic systems with time varying parameters, disturbances and input nonlinearities

In this paper, the problems of robust exponential generalized and robust exponential Q-S chaos synchronization are investigated between different dimensional chaotic systems. We consider the more practical and realistic cases when unknown time varying parameters with uncertainties, environmental disturbances, and nonlinearity of input control signals are present. The adaptive technique is employed to design the appropriate controllers and the validity of the proposed controllers are proved using Lyapunov stability theorem. Furthermore, numerical simulations are performed to show the efficiency of the presented scheme.     

受扰不确定混沌系统的降阶修正函数投影同步

研究了具有不同阶数的受扰不确定混沌系统的降阶修正函数投影同步问题.基于Lyapunov稳定性理论和自适应控制方法,设计了统一的非线性状态反馈控制器和参数更新规则,使得混沌响应系统按照相应的函数尺度因子矩阵和混沌驱动系统的部分状态变量实现同步.方法考虑了实际系统中的模型不确定性和外界扰动,具有较强的实用性和鲁棒性.数值仿真证明了控制方法的有效性.     

Switched modified function projective synchronization of hyperchaotic Qi system with uncertain parameters

This work is involved with switched modified function projective synchronization of two identical Qi hyperchaotic systems using adaptive control method. Switched synchronization of chaotic systems in which a state variable of the drive system synchronize with a different state variable of the response system is a promising type of synchronization as it provides greater security in secure communication. Modified function projective synchronization with the unpredictability of scaling functions can enhance security. Recently formulated hyperchaotic Qi system in the hyperchaotic mode has an extremely broad frequency bandwidth of high magnitudes, verifying its unusual random nature and indicating its great potential for some relevant engineering applications such as secure communications. By Lyapunove stability theory, the adaptive control law and the parameter update law are derived to make the state of two chaotic systems modified function projective synchronized. Synchronization under the effect of noise is also considered. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.     

Projective synchronization of different fractional-order chaotic systems with non-identical orders

This paper investigates the projective synchronization (PS) of different fractional order chaotic systems while the derivative orders of the states in drive and response systems are unequal. Based on some essential properties on fractional calculus and the stability theorems of fractional-order systems, we propose a general method to achieve the PS in such cases. The fractional operators are introduced into the controller to transform the problem into synchronization problem between chaotic systems with identical orders, and the nonlinear feedback controller is proposed based on the concept of active control technique. The method is both theoretically rigorous and practically feasible. We present two examples that illustrate the effectiveness and applications of the method, which include the PS between two 3-D commensurate fractional-order chaotic systems and the PS between two 4-D fractional-order hyperchaotic systems with incommensurate and commensurate orders, respectively. Abundant numerical simulations are given which agree well with the analytical results. Our investigations show that PS can also be achieved between different chaotic systems with non-identical orders. We have further reviewed and compared some relevant methods on this topic reported in several recent papers. A discussion on the physical implementation of the proposed method is also presented in this paper.     

Simple adaptive variable structure control for unknown chaotic systems

This paper proposes two novel adaptive variable structure tracking controllers for a large class of chaotic systems with unknown dynamics in presence of both external disturbances and input nonlinearities. The pros and cons of each proposed methodology is also represented. In order to eliminate the chattering effect in the former controlled system, two corresponding fuzzy adaptive controllers are presented. Besides, synchronization of two non-identical uncertain chaotic systems is investigated using our proposed methods in both full and reduced-order forms. It can be seen that not only our proposed control schemes can be applied to a wide class of uncertain chaotic systems but also it is simple to implement in practical application. Finally, the proposed methods are applied to some famous chaotic systems to verify the effectiveness of the proposed methods.     

Complex modified function projective synchronization of complex chaotic systems with known and unknown complex parameters

Much progress has been made in the research of modified function projective synchronization (MFPS) for real (complex) chaotic systems with real parameters. The scaling functions are always chosen as real-valued ones in previous MFPS schemes for chaotic systems evolving in the same or inverse directions simultaneously. However, MFPS with different complex-valued scaling functions (CMFPS) has not been previously reported, where complex-variable chaotic (hyperchaotic) systems (CVCSs) evolve in different directions with a time-dependent intersection angle. Therefore, CMFPS is discussed for CVCSs with known and unknown complex parameters in this paper. By constructing appropriate Lyapunov functions defined on complex field, and employing adaptive control technique, a set of simple and practical sufficient conditions for achieving CMFPS are derived, and complex update laws for estimating unknown parameters are also given. The corresponding theoretical proofs and computer simulations are worked out to demonstrate the effectiveness and feasibility of the proposed schemes.     

Adaptive generalized function projective synchronization of uncertain chaotic systems

This paper mainly investigates adaptive generalized function projective synchronization of two different uncertain chaotic systems, which is a further extension of many existing projection synchronization schemes, such as modified projection synchronization, function projective synchronization and so on. On the basis of Lyapunov stability theory, an adaptive controller for the synchronization of two different chaotic systems is designed, and some parameter update laws for estimating the unknown parameters of the systems are also gained. This technique is applied to achieve synchronization between Lorenz and Rössler chaotic systems. The numerical simulations demonstrate the validity and feasibility of the proposed method.     

The function cascade synchronization method and applications

In this paper, a new function cascade synchronization method of chaos system is proposed to achieve generalized projective synchronization for chaotic systems. Based on Laypunov stability, the proposed synchronization technique is applied to three famous chaotic systems: the unified chaotic system, Liu system and Rössler system, which can make the states of two identical chaotic systems asymptotically synchronized by choosing different special suitable error functions. Numerical simulations are presented to show the effectiveness.     

Synchronization of non-identical chaotic delayed fuzzy cellular neural networks based on sliding mode control

-In this paper, we investigate the synchronization problems of chaotic fuzzy cellular neural networks with time-varying delays. To overcome the difficulty that complete synchronization between non-identical chaotic neural networks cannot be achieved only by utilizing output feedback control, we use a sliding mode control approach to study the synchronization of non-identical chaotic fuzzy cellular neural networks with time-varying delays, where the parameters and activation functions are mismatched. This research demonstrates the effectiveness of application in secure communication. Numerical simulations are carried out to illustrate the main results.     

不确定混沌系统的混合投影同步

本文研究了一类不确定混沌(超混沌)系统的混合投影问题.利用自适应方法和Lyapunov稳定性理论,获得了两个恒同或不同混沌系统实现混沌投影同步的一般方法.最后,数值仿真的结果验证了方法的有效性和鲁棒性.     

Function vector synchronization based on fuzzy control for uncertain chaotic systems with dead‐zone nonlinearities

In this article, a fuzzy adaptive control scheme is designed to achieve a function vector synchronization behavior between two identical or different chaotic (or hyperchaotic) systems in the presence of unknown dynamic disturbances and input nonlinearities (dead‐zone and sector nonlinearities). This proposed synchronization scheme can be considered as a generalization of many existing projective synchronization schemes (namely the function projective synchronization, the modified projective synchronization, generalized projective synchronization, and so forth) in the sense that the master and slave outputs are assumed to be some general function vectors. To practically deal with the input nonlinearities, the adaptive fuzzy control system is designed in a variable‐structure framework. The fuzzy systems are used to appropriately approximate the uncertain nonlinear functions. A Lyapunov approach is used to prove the boundedness of all signals of the closed‐loop control system as well as the exponential convergence of the corresponding synchronization errors to an adjustable region. The synchronization between two identical systems (chaotic satellite systems) and two different systems (chaotic Chen and Lü systems) are taken as two illustrative examples to show the effectiveness of the proposed method. © 2015 Wiley Periodicals, Inc. Complexity 21: 234–249, 2016     

Synchronizing chaotic systems up to an arbitrary scaling matrix via a single signal

This paper introduces a novel type of synchronization, where two chaotic systems synchronize up to an arbitrary scaling matrix. In particular, each drive system state synchronizes with a linear combination of response system states by using a single synchronizing signal. The proposed observer-based method exploits a theorem that assures asymptotic synchronization for a wide class of continuous-time chaotic (hyperchaotic) systems. Two examples, involving Rössler’s system and a hyperchaotic oscillator, show that the proposed technique is a general framework to achieve any type of synchronization defined to date.     

Function projective synchronization for fractional-order chaotic systems

This letter investigates the function projective synchronization between fractional-order chaotic systems. Based on the stability theory of fractional-order systems and tracking control, a controller for the synchronization of two fractional-order chaotic systems is designed. This technique is applied to achieve synchronization between the fractional-order Lorenz systems with different orders, and achieve synchronization between the fractional-order Lorenz system and fractional-order Chen system. The numerical simulations demonstrate the validity and feasibility of the proposed method.     

Global finite-time synchronization of different dimensional chaotic systems

This paper addresses the problem of global finite-time synchronization of two different dimensional chaotic systems. Firstly, the definition of global finite-time synchronization of different dimensional chaotic systems are introduced. Based on the finite-time stability methods, the controller is designed such that the chaotic systems are globally synchronized in a finite time. Then, some uncertain parameters are adopted in the chaotic systems, new control law and dynamical parameter estimation are proposed to guarantee that the global finite-time synchronization can be obtained. By considering a dynamical parameter designed in the controller, the adaptive updated controller is also designed to achieve the desired results. At last, the results of two different dimensional chaotic systems are also extended to two different dimensional networked chaotic systems. Finally, three numerical examples are given to verify the validity of the proposed methods.     

Synchronization of chaos in non-identical parametrically excited systems

In this paper, we investigate the synchronization of chaotic systems consisting of non-identical parametrically excited oscillators. The active control technique is employed to design control functions based on Lyapunov stability theory and Routh–Hurwitz criteria so as to achieve global chaos synchronization between a parametrically excited gyroscope and each of the parametrically excited pendulum and Duffing oscillator. Numerical simulations are implemented to verify the results.