分数阶统一混沌系统的投影同步

改变系统参数计算了分数阶统一系统的最大Lyapunov指数和关联维数,研究了分数阶统一系统的动力学行为.基于线性系统的稳定判定准则,设计了一种同步方案,实现了分数阶统一混沌系统的投影同步.通过对分数阶Chen系统、分数阶Lü系统和分数阶类Lorenz系统投影同步的数值模拟,进一步验证了所提出方案的有效性.     

Projective synchronization in coupled fractional order chaotic Rossler system and its control

This paper proposes a method to achieve projective synchronization of the fractional order chaotic Rossler system. First, construct the fractional order Rossler system's corresponding approximate integer order system, then a control method based on a partially linear decomposition and negative feedback of state errors is utilized on the new integer order system. Mathematic analyses prove the feasibility and the numerical simulations show the effectiveness of the proposed method.     

基于线性分离的自治混沌系统的投影同步

分析了自治混沌系统的投影同步问题.基于线性系统的稳定判定准则，提出一种新的线性分离的同步方法，并采用该方法实现了Lorenz系统，Rssler系统和超混沌Chen系统的投影同步.数值模拟进一步验证了所提出方案的有效性. 关键词： 投影同步 自治混沌系统 线性分离     

Generalized projective synchronization of fractional order chaotic systems

In this paper, based on the idea of a nonlinear observer, a new method is proposed and applied to “generalized projective synchronization” for a class of fractional order chaotic systems via a transmitted signal. This synchronization approach is theoretically and numerically studied. By using the stability theory of linear fractional order systems, suitable conditions for achieving synchronization are given. Numerical simulations coincide with the theoretical analysis.     

Hybrid projective synchronization of chaotic fractional order systems with different dimensions

The hybrid projective synchronization of different dimensional fractional order chaotic systems is investigated in this paper. It is shown that the slave system can be synchronized with the projection of the master system generated through state transformation. Based on the stability theorem of linear fractional order systems, a suitable controller for achieving the synchronization is given. The hybrid projective synchronization between the fractional order chaotic system and hyperchaotic system is successfully achieved in both reduced order and increased order. The corresponding numerical results verify the effectiveness of the proposed method.     

两个耦合的分数阶Chen系统的混沌投影同步控制

研究了两个耦合的分数阶Chen系统的混沌投影同步控制问题.通过建立近似的整数阶模型，使用状态误差反馈控制策略控制投影同步比例因子达到理想值.理论上证明了该控制方法的可行性，数值仿真进一步验证了该方法的有效性. 关键词： 分数阶 混沌系统 投影同步 Chen系统     

Investigation of Q-S synchronization in coupled chaotic incommensurate fractional order systems

Chaos and synchronization in fractional order systems have received increasing attention in recent years. In this paper, the problem of Q-S synchronization for different dimensional incommensurate fractional order chaotic systems is investigated. Based on Laplace transform and stability theory of linear integer order differential systems, some synchronization schemes are designed to achieve Q-S synchronization between n-D and m-D incommensurate fractional order chaotic systems. Test problems and numerical simulations are used to show the effectiveness of the proposed approach.     

Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems

In this Letter, a new lag projective synchronization for fractional-order chaotic (hyperchaotic) systems is proposed, which includes complete synchronization, anti-synchronization, lag synchronization, generalized projective synchronization. It is shown that the slave system synchronizes the past state of the driver up to a scaling factor. A suitable controller for achieving the lag projective synchronization is designed based on the stability theory of linear fractional-order systems and the pole placement technique. Two examples are given to illustrate effectiveness of the scheme, in which the lag projective synchronizations between fractional-order chaotic Rössler system and fractional-order chaotic Lü system, between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system, respectively, are successfully achieved. Corresponding numerical simulations are also given to verify the analytical results.     

统一混沌系统的投影同步与控制

对单向耦合的统一混沌系统间的投影同步进行了研究与控制-分析了投影同步现象的产生机 理，给出一方案对投影同步的特征量进行控制-该方案所需控制代价较小, 为设计适当的混 沌信号源提供了一种理论依据，在利用混沌电路系统进行保密通信方面有着一定的应用前景 -仿真实例表明理论分析的正确性和实际控制方案的有效性- 关键词： 统一混沌系统 投影同步 混沌控制 单向耦合     

Chaos synchronization of the Chua system with a fractional order

Chaos synchronization of two identical Chua systems with the same fractional order is studied by utilizing the Pecora–Carroll (PC) method, the active–passive decomposition (PAD) method, the one-way coupling method and the bidirectional coupling one. The sufficient conditions for achieving synchronization between these two systems are derived via the Laplace transformation theory. Numerical simulations show the effectiveness of the theoretical analyses.     

基于比较系统方法的分数阶混沌系统脉冲同步控制

针对一类分数阶混沌系统的同步问题, 提出基于比较系统理论的脉冲同步方法. 通过构造新的响应系统, 可将原分数阶同步误差系统转化为整数阶同步误差系统, 基于Lyapunov稳定性理论与脉冲微分方程理论, 给出一组新的分数阶混沌系统全局渐近同步判据. 特别地, 当脉冲间距与脉冲控制增益为常数时, 可获得更为简单和实用的同步判据. 与现有结果相比, 所得充分条件更为严格和实用. 通过对分数阶Chen系统同步问题的数值仿真研究, 验证了所提方法的有效性和可行性.     

Adaptive lag synchronization and parameter identification of fractional order chaotic systems

This paper proposes a simple scheme for the lag synchronization and the parameter identification of fractional order chaotic systems based on the new stability theory. The lag synchronization is achieved and the unknown parameters are identified by using the adaptive lag laws. Moreover, the scheme is analytical and is simple to implement in practice. The well-known fractional order chaotic Lü system is used to illustrate the validity of this theoretic method.     

Dynamical analysis of a new fractional-order Rabinovich system and its fractional matrix projective synchronization

This paper constructs a new physical system, i.e., the fractional-order Rabinovich system, and investigates its stability, chaotic behaviors, chaotic control and matrix projective synchronization. Firstly, two Lemmas of the new system's stability at three equilibrium points are given and proved. Next, the largest Lyapunov exponent, the corresponding bifurcation diagram and the chaotic behaviors are studied. Then, the linear and nonlinear feedback controllers are designed to realize the system's local asymptotical stability and the global asymptotical stability, respectively. It's particularly significant that, the fractional matrix projective synchronization between two Rabinovich systems is achieved and two kinds of proofs are provided for Theorem 4.1. Especially, under certain degenerative conditions, the fractional matrix projective synchronization can be reduced to the complete synchronization, anti-synchronization, projective synchronization and modified projective synchronization of the fractional-order Rabinovich systems. Finally, all the theoretical analysis is verified by numerical simulation.     

基于模糊观测器的Chua混沌系统投影同步

研究了Chua混沌系统的广义投影同步问题.基于Takagi-Sugeno(T-S)模糊模型,设计了一种模糊观测器.利用Lyapunov稳定性理论证明了所提方案的可行性和全局稳定性.数值仿真试验进一步验证了该方案可以实现Chua混沌系统的投影同步.方法设计简单,且可以推广到其他的一些混沌系统. 关键词： T-S模糊模型 Chua混沌系统 观测器 投影同步     

部分线性的分数阶混沌系统修正函数投影同步

针对一类部分线性的分数阶混沌系统修正函数投影同步问题, 可通过单变量耦合构造出这类系统的响应系统, 从而提出修正函数投影同步的设计方法. 根据Routh-Hurwitz条件, 给出耦合部分线性系统实现修正函数投影同步的方法, 并设计了控制器. 该方法中控制器和误差动态方程的选择都是确定的, 理论分析和数值仿真都说明了该方法在部分线性的分数阶混沌系统中应用具有可行性和有效性. 关键词： 分数阶混沌系统 部分线性 修正函数投影同步     

利用单驱动变量实现一类分数阶混沌系统的修正投影同步

通过设计一个合适的响应系统,提出了一种修正投影同步方案,该方法通过传递一个驱动变量实现了一类分数阶混沌系统的修正投影同步,由于只需在驱动系统和响应系统之间传递一个驱动变量,具有较高的实用价值,最后以分数阶统一混沌系统为例进行了数值仿真,理论和数值仿真表明了该方法的有效性.     

The feedback control of fractional order unified chaotic system

This paper studies the stability of the fractional order unified chaotic system. On the unstable equilibrium points, the ``equivalent passivity' method is used to design the nonlinear controller. With the definition of fractional derivatives and integrals, the Lyapunov function is constructed by which it is proved that the controlled fractional order system is stable. With Laplace transform theory, the equivalent integer order state equation from the fractional order nonlinear system is obtained, and the system output can be solved. The simulation results validate the effectiveness of the theory.     

Projective synchronization of a complex network with different fractional order chaos nodes

Based on the stability theory of the linear fractional order system, projective synchronization of a complex network is studied in the paper, and the coupling functions of the connected nodes are identified. With this method, the projective synchronization of the network with different fractional order chaos nodes can be achieved, besides, the number of the nodes does not affect the stability of the whole network. In the numerical simulations, the chaotic fractional order Lü system, Liu system and Coullet system are chosen as examples to show the effectiveness of the scheme.     

Chaotic attractors in incommensurate fractional order systems

In this paper, based on the stability theorems in fractional differential equations, a necessary condition is given to check the existence of 1-scroll, 2-scroll or multi-scroll chaotic attractors in a fractional order system. This condition is proposed for incommensurate order systems in general, but in the special case it converts to the condition given in the previous works for the commensurate fractional order systems. Though the presented condition is only a necessary (and not sufficient) condition for the existence of chaos it can be used as a powerful tool to distinguish for what parameters and orders of a given fractional order system, chaotic attractors can not be observed and for what parameters and orders, the system may generate chaos. It can also be used as a tool to confirm or reject results of a numerical simulation. Some of the numerical results reported in the previous literature are confirmed by this tool.