General methods for modified projective synchronization of hyperchaotic systems with known or unknown parameters

This work is concerned with the general methods for modified projective synchronization of hyperchaotic systems. A systematic method of active control is developed to synchronize two hyperchaotic systems with known parameters. Moreover, by combining the adaptive control and linear feedback methods, general sufficient conditions for the modified projective synchronization of identical or different chaotic systems with fully unknown or partially unknown parameters are presented. Meanwhile, the speed of parameters identification can be regulated by adjusting adaptive gain matrix. Numerical simulations verify the effectiveness of the proposed methods.     

Modified function projective combination synchronization of hyperchaotic systems

In this work, a novel combination synchronization scheme in which synchronization of a new combination hyperchaotic drive system formed by combining state variables of the original drive system with appropriate scaling factors with a response hyperchaotic system is considered. A self-combination system is constructed from hyperchaotic Lorenz system by combining state variables of the Lorenz system with appropriate scaling factors. Modified function projective synchronization between the newly constructed combination hyperchaotic Lorenz system and hyperchaotic Lu system is investigated using adaptive method. By Lyapunov stability theory, the adaptive control law and the parameter update law are derived to make the state of two systems as modified function projective synchronized. Numerical simulations are done to show the validity and effectiveness of the proposed synchronization scheme.     

不同超混沌系统的自适应修正函数投影

研究了具有未知参数的Lorenz-Stenflo(LS)系统和一种新型超混沌系统以及LS系统和超混沌Chen系统的修正函数投影同步问题.利用Lyapunov稳定性理论和主动控制方法,设计自适应控制器和参数更新规则,实现不同超混沌系统的修正函数投影同步,同时估计出系统的未知参数.数值仿真验证了该自适应控制器的有效性.     

Adaptive Functional Projective Lag Synchronization of a Hyperchaotic Rössler System

We explain the functional projective lag synchronization of a hyperchaotic Rössler system with four unknown parameters, where the output of the master system lags behind the output of the slave system proportionally. Based on Lyapunov stability theory, an active control method and adaptive control law are employed to make the states of two hyperchaotic Rössler systems asymptotically synchronized. Finally, some numerical examples are provided to show the effectiveness of our results.     

Comparison of feedback control methods for a hyperchaotic Lorenz system

More and more attention has been payed to the hyperchaotic system for the huge potential applications of hyperchaotic system such as secure communication and more complex structure than chaotic system. So at present the controlling of the hyperchaotic system simply and effectively is a frontier topic of nonlinear science. In this Letter, for the latest hyperchaotic Lorenz system, four feedback control methods were studied with analytic solution and necessary numerical simulations. It is found that the enhancing feedback control approach is the best choice of the given four feedback control methods for its relatively simple external inputs and relatively small necessary feedback coefficient after comparison. The conclusion is a helpful for the choice of control methods of any other chaotic and hyperchaotic systems.     

超混沌Chen系统和超混沌Roessler系统的异结构同步

用两种不同的方法——主动控制同步法和自适应控制同步法实现超混沌Chen系统和超混沌Roessler系统的异结构同步，各自设计了不同的控制器，使得响应系统与驱动系统同步．当参数已知时，采用主动控制法，方法简单有效且不需要构造Lyapunov函数，实现同步的时间短；当系统参数未知或结构不确定时，基于Lyapunov稳定性理论，给出自适应同步控制器的系统设计过程和参数自适应律，使得系统达到同步同时识别未知参数，数值模拟验证了两种方法的有效性。     

Projective synchronization of a new hyperchaotic Lorenz system

This Letter mainly concerns projective synchronization (PS) of a new hyperchaotic Lorenz system. PS with both identical and different scaling factors between two hyperchaotic Lorenz systems are realized. A general sufficient condition for PS in a certain class of chaotic (hyperchaotic) system with uncertainties is obtained by using adaptive control. Numerical simulations are performed to verify and illustrate the analytical results.     

Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system

The stability conditions in fractional order hyperchaotic systems are derived. These conditions are applied to a novel fractional order hyperchaotic system. The proposed system is also shown to exhibit hyperchaos for orders less than 4. Based on the Routh-Hurwitz conditions, the conditions for controlling hyperchaos via feedback control are also obtained. A specific condition for controlling only fractional order hyperchaotic systems is achieved. Numerical simulations are used to verify the theoretical analysis.     

On the synchronization of a hyperchaotic system based on adaptive method

This Letter researches the adaptive synchronization of a hyperchaotic system. Based on Lyapunov method, a general adaptive method via a driving vector and an original adaptive method via a single driving variable are presented to achieve the chaos synchronization. Especially, the latter makes the structure of adaptive controller simple and has more practical value compared with the former. Some typical numerical examples are included to demonstrate the effectiveness of the method proposed.     

统一超混沌系统的投影同步

利用两种方法研究了统一超混沌系统的同步问题.首先以全状态混合投影自适应同步方法,基于Lyapunov稳定性理论,设计了自适应控制器,理论证明了该控制器可以实现参数已知的统一超混沌系统的全状态混合映射同步.其次使用主动控制同步方法,设计了同步控制器,实现了统一超混沌系统的完全同步,最后数值仿真实验进一步验证了所提出方案的有效性. 关键词： 统一超混沌系统 自适应控制器 全状态混合投影同步 主动控制同步     

Function projective synchronization of different chaotic systems with uncertain parameters

This Letter investigates the function projective synchronization of different chaotic systems with unknown parameters. By Lyapunov stability theory, the adaptive control law and the parameter update law are derived to make the states of two different chaotic systems asymptotically synchronized up to a desired scaling function. Numerical simulations on Lorenz system and Newton-Leipnik system are presented to verify the effectiveness of the proposed scheme.     

Function projective synchronization of two four-scroll hyperchaotic systems with unknown parameters

Function projective synchronization (FPS) of two novel hyperchaotic systems with four-scroll attractors which have been found up to the present is investigated. Adaptive control is employed in the situation that system parameters are unknown. Based on Lyapunov stability theory, an adaptive controller and a parameter update law are designed so that the two systems can be synchronized asymptotically by FPS. Numerical simulation is provided to show the effectiveness of the proposed adaptive controller and the parameter update law.     

Function projective synchronization of two-cell quantum-CNN chaotic oscillators by nonlinear adaptive controller

In this Letter, we investigate function projective synchronization of two-cell quantum-CNN chaotic oscillators using nonlinear adaptive controller. Based on Lyapunov stability theory, the nonlinear adaptive control law is derived to make the state of two chaotic systems function projective synchronized. Two numerical simulations are presented to illustrate the effectiveness of the proposed nonlinear adaptive control scheme, which is more effective than that in previous literature.     

参数不确定量子细胞神经网络与Lorenz超混沌系统的函数投影同步研究

研究了具有不确定参数的量子细胞神经网络系统与Lorenz超混沌系统的函数投影同步.设计了自适应追踪控制器来实现两个不同系统渐进同步到已知的比例函数,并运用Lyapunov稳定性方法进行了证明.理论分析了量子细胞神经网络系统不确定参数的系数向量线性无关性,实现了不确定参数的识别和估计.数值仿真结果验证了函数投影同步和参数估计的有效性.     

Adaptive control and synchronization of an uncertain new hyperchaotic Lorenz system

This paper is involved with the adaptive control and synchronization problems for an uncertain new hyperchaotic Lorenz system. Based on the Lyapunov stability theory, the adaptive control law is derived such that the trajectory of hyperchaotic Lorenz system with unknown parameters can be globally stabilized to an unstable equilibrium point of the uncontrolled system. Furthermore, an adaptive control approach is presented to the synchronizations between two identical hyperchaotic systems, particularly between two different uncertain hyperchaotic systems. Numerical simulations show the effectiveness of the presented method.     

Modified projective synchronization of a fractional-order hyperchaotic system with a single driving variable

In this paper, the modified projective synchronization (MPS) of a fractional-order hyperchaotic system is investigated. We design the response system corresponding to the drive system on the basis of projective synchronization theory, and determine the sufficient condition for the synchronization of the drive system and the response system based on fractional-order stability theory. The MPS of a fractional-order hyperchaotic system is achieved by transmitting a single variable. This scheme reduces the information transmission in order to achieve the synchronization, and extends the applicable scope of MPS. Numerical simulations further demonstrate the feasibility and the effectiveness of the proposed scheme.     

Adaptive function project synchronization of Rössler hyperchaotic system with uncertain parameters

This Letter addresses the function project synchronization problem of two Rössler hyperchaotic in the presence of unknown system parameters. Based on Lyapunov stability theory an adaptive control law is proposed to make the states of two identical Rössler hyperchaotic systems asymptotically synchronized. Numerical simulations are presented to show the effectiveness of the proposed schemes.     

Unstable manifolds for the hyperchaotic Rössler system

We analyse stability of the generalized four-variable Rössler oscillating system depending on selected control parameters, by using analytic and Hurwitz-Routh methods. In contrast to the usual three-dimensional Rössler and Lorenz systems, we show that always there exists at least one unstable direction, and the number of positive local Lyapunov exponents may be different for both fixed points. We have found two new types of Hopf bifurcation, in which the dimension of the unstable manifold can be increased or reduced by two. Hence there are many possibilities for hyperchaotic unstable manifolds of various dimensions. We have also calculated various ranges of the control parameters for which different unstable manifolds can be obtained. This allows a better characterization of stability of the attractors in the hyperchaotic regime.     

Adaptive function projective synchronization of two-cell Quantum-CNN chaotic oscillators with uncertain parameters

This work investigates function projective synchronization of two-cell Quantum-CNN chaotic oscillators using adaptive method. Quantum-CNN oscillators produce nano scale chaotic oscillations under certain conditions. By Lyapunove stability theory, the adaptive control law and the parameter update law are derived to make the state of two chaotic systems function projective synchronized. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.     

基于Takagi-Sugeno模糊模型的超混沌系统自适应投影同步及参数辨识

研究了不确定超混沌系统的自适应投影同步问题.基于Takagi-Sugeno（T-S）模糊模型,设计了一种自适应模糊控制器和参数更新规则.利用Lyapunov稳定性理论,证明了所提方案可以实现不确定超混沌系统的投影同步,并同时辨识出未知的系统参数.通过对超混沌Lorenz系统的数值仿真实验进一步验证了所提方案的有效性. 关键词： Takagi-Sugeno模糊模型 自适应 投影同步 超混沌