Adaptive feedback control and synchronization of non-identical chaotic fractional order systems

This paper addresses the reliable synchronization problem between two non-identical chaotic fractional order systems. In this work, we present an adaptive feedback control scheme for the synchronization of two coupled chaotic fractional order systems with different fractional orders. Based on the stability results of linear fractional order systems and Laplace transform theory, using the master-slave synchronization scheme, sufficient conditions for chaos synchronization are derived. The designed controller ensures that fractional order chaotic oscillators that have non-identical fractional orders can be synchronized with suitable feedback controller applied to the response system. Numerical simulations are performed to assess the performance of the proposed adaptive controller in synchronizing chaotic systems.     

Synchronization of nonidentical chaotic fractional-order systems with different orders of fractional derivatives

This paper addresses the problem of synchronization of chaotic fractional-order systems with different orders of fractional derivatives. Based on the stability theory of fractional-order linear systems and the idea of tracking control, suitable controllers are correspondingly proposed for two cases: the first is synchronization between two identical chaotic fractional-order systems with different fractional orders, and the other is synchronization between two nonidentical fractional-order chaotic systems with different fractional orders. Three numerical examples illustrate that fast synchronization can be achieved even between a chaotic fractional-order system and a hyperchaotic fractional-order system.     

Fractional order fixed-time nonsingular terminal sliding mode synchronization and control of fractional order chaotic systems

This paper presents fractional order fixed-time nonsingular terminal sliding mode control for stabilization and synchronization of fractional order chaotic systems with uncertainties and disturbances. First, a novel fractional order terminal sliding mode surface is proposed to guarantee the fixed-time convergence of system states along the sliding surface. Second, a nonsingular terminal sliding mode controller is designed to force the system states to reach the sliding surface within fixed-time and remain on it forever. Furthermore, the fractional Lyapunov stability theory is used to prove the fixed-time stability and the robustness of the proposed control scheme and estimate the upper bound of convergence time. Next, the proposed control scheme is applied to the synchronization of two nonidentical fractional order Liu chaotic systems and chaos suppression of fractional order power system. Simulation results verify the effectiveness of the proposed control scheme. Finally, some application issues about the proposed scheme are discussed.

     

Achievement of chaotic synchronization trajectories of master-slave manipulators with feedback control strategy

This paper addresses a master-slave synchronization strategy for complex dynamic systems based on feedback control.This strategy is applied to 3-DOF planar manipulators in order to obtain synchronization in such complicated as chaotic motions of end-effectors.A chaotic curve is selected from Duffing equation as the trajectory of master end-effector and a piecewise approximation method is proposed to accurately represent this chaotic trajectory of end-effectors.The dynamical equations of master-slave mani...     

连续时间系统的混沌同步

本文讨论混沌连续时间系统的完全同步问题，提出一个构造混沌同步系统的新方法。这个方法基于线性系统的稳定性分析准则。通过对系统线性项与非线性项的适当分离，当系统的雅可比矩阵的所有特征值都具有负实部时，同步误差e(t)的线性系统是渐进稳定的，即可实现新系统和原系统的完全同步。新方法不需计算条件Lyapunov指数以作为判定同步的条件，因而比通用方法更为简单有效。新方法适用于自治或非自治系统，尤其适用于具有多于两个正Lyapunov指数的超混沌系统。甚至当初始同步误差极大时，也能实现理想的混沌同步。以Lorenz系统，耦合Duffing振子系统和超混沌Roessler系统作为算例。数值计算结果证实所提出方法的有效性和鲁棒性。     

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.     

Achievement of chaotic synchronization trajectories of master–slave manipulators with feedback control strategy

This paper addresses a master-slave synchro- nization strategy for complex dynamic systems based on feedback control. This strategy is applied to 3-DOF pla- nar manipulators in order to obtain synchronization in such complicated as chaotic motions of end-effectors. A chaotic curve is selected from Duffing equation as the trajectory of master end-effector and a piecewise approximation method is proposed to accurately represent this chaotic trajectory of end-effectors. The dynamical equations of master-slave manipulators with synchronization controller are derived, and the Lyapunov stability theory is used to determine the stability of this controlled synchronization system. In numer- ical experiments, the synchronous motions of end-effectors as well as three joint angles and torques of master-slave manipulators are studied under the control of the proposed synchronization strategy. It is found that the positive gain matrix affects the implementation of synchronization con- trol strategy. This synchronization control strategy proves the synchronization＇s feasibility and controllability for com- plicated motions generated by master-slave manipulators.     

Shape synchronization of drive-response for a class of two-dimensional chaotic systems via continuous controllers

This paper investigates the drive-response synchronization in shape for a class of two-dimensional continuous systems of chaos. The shape of the chaotic attractor of the drive chaotic system is considered in this paper. Using the signed curvatures of plane curves to describe the shapes of trajectories for drive and response systems, the continuous controller for shape synchronization is synthesized based on the fundamental theorem on plane curves in classical differential geometry. The continuous controller synthesized can guarantee that the response system is synchronized with the drive chaotic system in shape. The shape synchronization is obtained in spite of different dimensions in drive and response systems. Finally, the Duffing oscillator is utilized as an illustrative example. Simulation results show that the method proposed in this paper is effective for the application of secure communication.     

Synchronization of a novel fractional order stretch-twist-fold (STF) flow chaotic system and its application to a new authenticated encryption scheme (AES)

In this paper, a new fractional order stretch-twist-fold (STF) flow dynamical system is proposed. The stability analysis of the proposed system equilibria is accomplished and we establish that the system is exhibited chaos even for order less than 3. The active control method is applied to enquire the hybrid phase synchronization between two identical fractional order STF flow chaotic systems. These synchronized systems are applied to formulate an authenticated encryption scheme newly for message (text and image) recovery. It is widely applied in the field of secure communication. Numerical simulations are presented to validate the effectiveness of the proposed theory.     

Practical finite-time synchronization of jerk systems: Theory and experiment

This paper addresses the problem of finite-time synchronization of jerk chaotic systems through a simple linear feedback control. The controller is designed such that practical finite-time synchronization could be achieved. As example, we use a new jerk system obtained thanks to the chaotification of the Duffing system using jerk architecture and simplification via a single silicon p-n junction diode. Mathematic proof, numerical and PSpice simulations, and practical results are presented to show the feasibility of the proposed scheme. The proposed method could be applied to all jerk-like systems.     

Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic R?ssler systems

This paper introduces an observer-based approach to achieve projective synchronization in fractional-order chaotic systems using a scalar synchronizing signal. The proposed method, which enables a linear fractional error system to be obtained, exploits the Kalman decomposition and a proper stability criterion in order to stabilize the error dynamics at the origin. The approach combines three desirable features, that is, the theoretical foundation of the method, the adoption of a scalar synchronizing signal, and the exact analytical solution of the fractional error system written in terms of Mittag-Leffler function. Finally, the projective synchronization of the fractional-order hyperchaotic R?ssler systems is illustrated in detail.     

Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography

In this paper, the stability conditions and chaotic behaviors of new different fractional orders of reverse butterfly-shaped dynamical system are analytically and numerically investigated. Designing an appropriate feedback controller, the fractional order chaotic system is synchronized. Applying the synchronized fractional order systems in digital cryptography, a well secured key system is obtained. The numerical simulations are given to validate the correctness of the proposed synchronized fractional order chaotic systems and proposed key system.     

A new fractional order hyperchaotic Rabinovich system and its dynamical behaviors

In the paper, the dynamical behaviors of a new fractional order hyperchaotic Rabinovich system are investigated, which include its local stability, hyperchaos, chaotic control and synchronization. Firstly, a new fractional order hyperchaotic Rabinovich system with Caputo derivative is proposed. Then, the hyperchaotic attractors of the commensurate and incommensurate fractional order hyperchaotic Rabinovich system are found. After that, four linear feedback controllers are designed to stabilize this fractional order system Finally, by using the active control method the synchronization is studied between the fractional order hyperchaotic and chaos controlled Rabinovich system In addition, the theoretical predictions are confirmed by numerical simulations.     

Chaos synchronization for a class of nonequivalent systems with restrictive inputs via time-varying sliding mode

In this paper, the synchronization between Duffing and Van der Pol chaotic systems with control inputs constraint is investigated. In practice, the maximum admissible values of the control inputs are restrained. To solve this problem, based on time-varying sliding mode theory, we can obtain the best possible control quality without violating technical and environmental constraints by selecting the switching line parameters. Also a kind of extended state observer is used to compensate for the uncertainties of systems, using only the available synchronizing error. Then the controller becomes physically realizable based on the states of the observer, and can be used to synchronize between Duffing and Van der Pol chaotic systems. Finally, simulation results are presented to demonstrate the effectiveness of the proposed control scheme.     

分数阶对称金融模型复杂度分析及有限时间同步

分析了一类分数阶对称金融非线性系统的复杂度特性,利用有限时间同步理论设计控制器,实现了有限时间同步。根据分数阶系统定义和Adomain分解法对该系统的非线性项进行Adomain分解,结合分解系数定义系统的表达式,将其离散化。基于谱熵复杂度及C0复杂度的基本算法,利用Matlab仿真其复杂度曲线及复杂度图谱。为进一步探究对称金融非线性系统的动力学特性,利用有限时间同步理论设计误差控制器,实现有限时间同步,仿真结果表明该控制器可使系统在极短的时间内实现同步且鲁棒性好。     

Modified projective synchronization of stochastic fractional order chaotic systems with uncertain parameters

In this paper, the synchronization of fractional order chaotic systems with random and uncertain parameters is analyzed. Firstly, based on the orthogonal polynomial expansion, the fractional order Lü and Lorenz systems with random and uncertain parameters are reduced into the equivalent deterministic systems. Secondly, modified projective synchronization of equivalent deterministic Lü and Lorenz systems is explored. Lastly, the theoretical results are verified by the numerical simulations.     

Hybrid robust modified function projective lag synchronization in two different dimensional chaotic systems

In this article, a novel synchronization scheme, modified function projective lag synchronization (MFPLS) in two different dimensional chaotic systems with parameter perturbations, is proposed. In the proposed method, the states of two nonidentical chaotic systems with different orders are asymptotically lag synchronized up to a desired scaling function matrix by means of reduced order and increased order, respectively. Furthermore, based on the reality situation, the parameter perturbations are involved, which are assumed to appear in both drive and response systems. With the Lyapunov stability theory, an adaptive controller is designed to achieve MFPLS. Theoretical proof and numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme.     

A new chaotic system with fractional order and its projective synchronization

Based on Rikitake system, a new chaotic system is discussed. Some basic dynamical properties, such as equilibrium points, Lyapunov exponents, fractal dimension, Poincaré map, bifurcation diagrams and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed is a new chaotic system. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the new fractional-order three-dimensional system with order less than 3. The lowest order to yield chaos in this system is 2.733. The results are validated by the existence of one positive Lyapunov exponent and some phase diagrams. Further, based on the stability theory of the fractional-order system, projective synchronization of the new fractional-order chaotic system through designing the suitable nonlinear controller is investigated. The proposed method is rather simple and need not compute the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the presented synchronization scheme.     

基于Caputo导数的分数阶非线性振动系统响应计算

研究了含分数阶Caputo导数的非线性振动系统响应的数值计算方法。首先,由Caputo分数阶导数算子的叠加关系,得到含分数阶导数项非线性振动系统状态方程的标准形式。其次,基于Caputo导数与Riemann-Liouville导数和Grunwald-Letnikov导数间的关系,推导计算了Caputo导数的一般数值迭代格式。本文方法不要求状态方程中各分数阶导数阶数相等,弱化了已有算法中对分数阶导数阶数的限制,并可推广到多自由度的情形。随后,选择若干有解析解的算例验证了本文方法的正确性。最后,以多吸引子共存的分数阶Duffing振子系统为例,比较Caputo和GL两种算法所得结果,说明了用GL算法求解存在的问题。     

A modified adaptive control method for synchronization of some fractional chaotic systems with unknown parameters

A modified adaptive control method is developed in this article and the parameters identification method is then applied in fractional order systems with unknown parameters. The new modified control method based on Lyapunov stability theory is successfully applied to investigate the synchronization of pair of fractional order systems amongst Genesio–Tesi, Qi and Chen systems. By means of the Adams–Bosford–Moulton method, the numerical results show that the modified method is easy to implement and reliable for synchronizing the two different fractional order chaotic systems.