On the fractional-order extended Kalman filter and its application to chaotic cryptography in noisy environment

In this paper via a novel method of discretized continuous-time Kalman filter, the problem of synchronization and cryptography in fractional-order systems has been investigated in presence of noisy environment for process and output signals. The fractional-order Kalman filter equation, applicable for linear systems, and its extension called the extended Kalman filter, which can be used for nonlinear systems, are derived. The result is utilized for chaos synchronization with the aim of cryptography while the transmitter system is fractional-order, and both the transmitter and transmission channel are noisy. The fractional-order stochastic chaotic Chen system is then presented to apply the proposed method for chaotic signal cryptography. The results show the effectiveness of the proposed method.     

Chaos in a new fractional-order system without equilibrium points

Chaotic systems without equilibrium points represent an almost unexplored field of research, since they can have neither homoclinic nor heteroclinic orbits and the Shilnikov method cannot be used to demonstrate the presence of chaos. In this paper a new fractional-order chaotic system with no equilibrium points is presented. The proposed system can be considered “elegant” in the sense given by Sprott, since the corresponding system equations contain very few terms and the system parameters have a minimum of digits. When the system order is as low as 2.94, the dynamic behavior is analyzed using the predictor–corrector algorithm and the presence of chaos in the absence of equilibria is validated by applying three different methods. Finally, an example of observer-based synchronization applied to the proposed chaotic fractional-order system is illustrated.     

Chaos control and function projective synchronization of fractional-order systems through the backstepping method

We study the chaos control and the function projective synchronization of a fractional-order T-system and Lorenz chaotic system using the backstepping method. Based on stability theory, we consider the condition for the local stability of nonlinear three-dimensional commensurate fractional-order system. Using the feedback control method, we control the chaos in the considered fractional-order T-system. We simulate the function projective synchronization between the fractional-order T-system and Lorenz system numerically using MATLAB and depict the results with plots.     

Chaos and synchronization of the fractional-order Chua’s system

Chaotic synchronization of fractional-order Chua’s system is further studied. An algorithm for numerical solution of fractional-order differential equations is presented; the chaos in a fractional-order Chua system with some parameters is discussed. The scheme of synchronization system consist of fractional-order Chua’s system is constructed. The synchronization conditions are investigated theoretically. And the synchronization thresholds are discussed by utilizing bifurcation graphs.     

Chaos,feedback control and synchronization of a fractional-order modified Autonomous Van der Pol–Duffing circuit

In this work, stability analysis of the fractional-order modified Autonomous Van der Pol–Duffing (MAVPD) circuit is studied using the fractional Routh–Hurwitz criteria. A necessary condition for this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than 3. Furthermore, the fractional Routh–Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria. Based on the fractional Routh–Hurwitz conditions and using specific choice of linear controllers, it is shown that the fractional-order MAVPD system is controlled to its equilibrium points; however, its integer-order counterpart is not controlled. Moreover, chaos synchronization of MAVPD system is found only in the fractional-order case when using a specific choice of nonlinear control functions. This shows the effect of fractional order on chaos control and synchronization. Synchronization is also achieved using the unidirectional linear error feedback coupling approach. Numerical results show the effectiveness of the theoretical analysis.     

Van der pol情绪混沌模型的滑模同步

研究了整数阶分数阶van der pol情绪混沌模型的滑模同步问题,利用分数阶微积分给出了情绪模型的主从系统取得同步的充分条件,研究表明,一定条件下,Van der pol情绪模型的主从系统能够达到同步,数值仿真验证了该方法的可行性.     

Synchronization of chaotic neural networks with mixed time delays

In this paper, we investigate the synchronization problems of chaotic neural networks with mixed time delays. We first establish the sufficient conditions for synchronization of identical chaotic neural networks with mixed time delays via linear output feedback control. To overcome the difficulty that complete synchronization between nonidentical chaotic neural networks cannot be achieved only by utilizing output feedback control, we use a sliding mode control approach to study the synchronization of nonidentical chaotic neural networks with mixed time delays, where the parameters and functions are mismatched. Numerical simulations are carried out to illustrate the main results.     

Chaos in fractional-order Genesio-Tesi system and its synchronization

In this paper we study the chaotic dynamics of fractional-order Genesio-Tesi system. Theoretically, a necessary condition for occurrence of chaos is obtained. Numerical investigations on the dynamics of this system have been carried out and properties of the system have been analyzed by means of Lyapunov exponents. It is shown that in case of commensurate system the lowest order of fractional-order Genesio-Tesi system to yield chaos is 2.79. Further, chaos synchronization of fractional-order Genesio-Tesi system is investigated via two different control strategies. Active control and sliding mode control are proposed and the stability of the controllers are studied. Numerical simulations have been carried out to verify the effectiveness of controllers.     

Chaos synchronization between two different fractional-order hyperchaotic systems

This work investigates chaos synchronization between two different fractional-order hyperchaotic system (FOHS)s. A novel FOHS is also proposed in this paper. The Chen FOHS is controlled to be a new FOHS and the Lü FOHS, respectively. The analytical conditions for the synchronization of these pairs of different FOHSs are derived by utilizing Laplace transform. Furthermore, synchronization between two different FOHSs is achieved by utilizing feedback control method in a quite short period and both remain in chaotic states. Numerical simulations are used to verify the theoretical analysis using different values of the fractional-order parameter.     

Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal

In this paper, a drive-response synchronization method with linear output error feedback is presented for synchronizing a class of fractional-order chaotic systems via a scalar transmitted signal. Based on stability theory of fractional-order systems and linear system theory, a necessary and sufficient condition for the existence of the feedback gain vector such that global synchronization between the fractional-order drive system and response system can be achieved and its design method are given. This synchronization approach that is simple, global and theoretically rigorous enables synchronization of fractional-order chaotic systems be achieved in a systematic way and does not require the computation of the conditional Lyapunov exponents. An example is used to illustrate the effectiveness of the proposed synchronization method.     

Chaos control and generalized projective synchronization of heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive variable structure control

This paper proposes the chaos control and the generalized projective synchronization methods for heavy symmetric gyroscope systems via Gaussian radial basis adaptive variable structure control. Because of the nonlinear terms of the gyroscope system, the system exhibits chaotic motions. Occasionally, the extreme sensitivity to initial states in a system operating in chaotic mode can be very destructive to the system because of unpredictable behavior. In order to improve the performance of a dynamic system or avoid the chaotic phenomena, it is necessary to control a chaotic system with a periodic motion beneficial for working with a particular condition. As chaotic signals are usually broadband and noise like, synchronized chaotic systems can be used as cipher generators for secure communication. This paper presents chaos synchronization of two identical chaotic motions of symmetric gyroscopes. In this paper, the switching surfaces are adopted to ensure the stability of the error dynamics in variable structure control. Using the neural variable structure control technique, control laws are established which guarantees the chaos control and the generalized projective synchronization of unknown gyroscope systems. In the neural variable structure control, Gaussian radial basis functions are utilized to on-line estimate the system dynamic functions. Also, the adaptation laws of the on-line estimator are derived in the sense of Lyapunov function. Thus, the unknown gyro systems can be guaranteed to be asymptotically stable. Also, the proposed method can achieve the control objectives. Numerical simulations are presented to verify the proposed control and synchronization methods. Finally, the effectiveness of the proposed methods is discussed.     

Chaotic fractional-order Coullet system: Synchronization and control approach

The main objective of this study is to investigate and control the chaotic behaviour of fractional-order Coullet system, including the necessary condition for appearance of chaos. An active control technique, in a master–slave structure for synchronization is suggested to control this chaotic system. The stability condition is obtained in both of theoretical analysis and simulation manner. The numerical simulation verifies the performance of the proposed controller.     

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.     

Chaos in a new system with fractional order

The dynamics of fractional-order systems have attracted a great deal of attentions in recent years. With fractional order, the dynamics of a system which includes comprehensive dynamical behaviors, such as fixed point, periodic motion, chaotic motion, and transient chaos is studied numerically in this paper. It is known that chaos exists in the fractional-order system with order less than 3. In this study, the lowest order found for this system to yield chaos is 2.43. The results are validated by the existence of a positive Lyapunov exponent. Period doubling routes to chaos in the fractional-order system are also obtained. Moreover, generation of a four-scroll chaotic attractor by the system is observed.     

Parameter identification and synchronization of fractional-order chaotic systems

The knowledge about parameters and order is very important for synchronization of fractional-order chaotic systems. In this article, identification of parameters and order of fractional-order chaotic systems is converted to an optimization problem. Particle swarm optimization algorithm is used to solve this optimization problem. Based on the above parameter identification, synchronization of the fractional-order Lorenz, Chen and a novel system (commensurate or incommensurate order) is derived using active control method. The new fractional-order chaotic system has four-scroll chaotic attractors. The existence and uniqueness of solutions for the new fractional-order system are also investigated theoretically. Simulation results signify the performance of the work.     

Designing synchronization schemes for chaotic fractional-order unified systems

Synchronization in chaotic fractional-order differential systems is studied both theoretically and numerically. Two schemes are designed to achieve chaos synchronization of so-called unified chaotic systems and the corresponding numerical algorithms are established. Some sufficient conditions on synchronization are also derived based on the Laplace transformation theory. Computer simulations are used for demonstration.     

新的分数阶混沌系统及其同步

提出一个新的分数阶混沌系统,该系统含有三个参数,三个非线性项.通过理论分析,给出了分数阶混沌系统存在混沌吸引子的必要条件,通过数值仿真给出了混沌吸引子的图像,接着设计自适应同步控制器和参数自适应律,实现分数阶混沌系统的同步,数值仿真的结果表明设计控制器很好的实现了驱动系统和响应系统的同步.     

基于事件触发的时滞Lur’e系统主从同步研究

研究了基于事件触发采样控制的时滞混沌Lur’e系统主从同步问题.首先考虑了系统中包含的传输时滞构造了系统时滞模型.然后,通过构造三重积分项的Lyapunov-Krasovskii泛函,并结合Wirtinger积分不等式和凸组合技术对Lyapunov-Krasovskii泛函的导数进行估计,给出了混沌系统主从同步的充分条件.所提出的事件触发机制应用于主从同步研究中,可以有效地减少采样数据传输,缓解网络带宽压力,提高网络带宽利用率.最后,通过时滞蔡氏电路的数值仿真,验证了所提出同步准则的有效性.     

Design of sliding mode controller for a class of fractional-order chaotic systems

In this paper, a sliding mode control law is designed to control chaos in a class of fractional-order chaotic systems. A class of unknown fractional-order systems is introduced. Based on the sliding mode control method, the states of the fractional-order system have been stabled, even if the system with uncertainty is in the presence of external disturbance. In addition, chaos control is implemented in the fractional-order Chen system, the fractional-order Lorenz system, and the same to the fractional-order financial system by utilizing this method. Effectiveness of the proposed control scheme is illustrated through numerical simulations.     

A new hyperchaotic system and its synchronization

In this paper, a four-dimensional (4D) continuous autonomous hyperchaotic system is introduced and analyzed. This hyperchaotic system is constructed by adding a linear controller to the 3D autonomous chaotic system with a reverse butterfly-shape attractor. Some of its basic dynamical properties, such as Lyapunov exponents, Poincare section, bifurcation diagram and the periodic orbits evolving into chaotic, hyperchaotic dynamical behavior by varying parameter d are studied. Furthermore, the full state hybrid projective synchronization (FSHPS) of new hyperchaotic system with unknown parameters including the unknown coefficients of nonlinear terms is studied by using adaptive control. Numerical simulations are presented to show the effective of the proposed chaos synchronization scheme.