Optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems

Based on the Lyapunov stability and adaptive synchronization theory, optimization design of adaptive controllers and parameter observers with controllable gain coefficient are investigated in detail. The linear errors of corresponding variables and parameters are used to construct different appropriate positive Lyapunov functions V and the parameter observers and adaptive controllers are approached analytically by simplifying the differential inequality dV/dt?0. Particularly, an optional gain coefficient is selected in the parameter observers and positive Lyapunov function, which decides the transient period to identify the unknown parameters and reach synchronization. The scheme is used to study the synchronization of two non-identical hyperchaotic Rössler systems. The theoretical and numerical results confirm that the four unknown parameters in the drive system are estimated exactly and the two hyperchaotic systems reach complete synchronization when the controllers and parameter observers work on the driven system. To confirm the model independence of this scheme, an alternative hyperchaotic system is investigated, whereby the results confirm that the five unknown parameters are identified rapidly and exactly, and that the two hyperchaotic systems reach complete synchronization as well.     

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.     

Tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance

In this paper, the tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance are investigated. Based on the LaSalle’s invariant set theorem, a robust adaptive controller is contrived to acquire tracking control and generalized projective synchronization and parameter identification simultaneously. It is proved theoretically that the proposed scheme can allow us to drive the hyperchaotic system to any desired reference signals, including hyperchaotic signals, chaotic signals, periodic orbits or fixed value by the given scaling factor. The presented simulation results further demonstrate that the proposed method is effective and robust.     

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 synchronization between two different hyperchaotic systems

This work presents chaos synchronization between two different hyperchaotic systems using adaptive control. The sufficient conditions for achieving synchronization of two high dimensional chaotic systems are derived based on Lyapunov stability theory, and an adaptive control law and a parameter update rule for unknown parameters are given such that generalized Henon–Heiles system is controlled to be hyperchaotic Chen system. Theoretical analysis and numerical simulations are shown to verify the results.     

Complete synchronization,phase synchronization and parameters estimation in a realistic chaotic system

The two-parameter phase space in certain nonlinear system is investigated and the chaotic region of parameters are measured to show its chaotic properties. Within the chaotic parameter region, the complete synchronization, phase synchronization and parameters estimation are discussed in detail by using adaptive synchronization scheme and Lyapunov stability theory. Two changeable gain coefficients are introduced into the controllable positive Lyapunov function and thus the parameter observers. It is found that complete synchronization or phase synchronization occurs with different controllers being used though the parameter observers are the same. Phase synchronization is observed when zero eigenvalue of Jacobi matrix, which is composed of the errors of corresponding variables in the drive and driven chaotic systems. The optimized selection of controllers can induce transition of phase synchronization and complete synchronization.     

Generation of hyperchaos from the Chen–Lee system via sinusoidal perturbation

A system with more than one positive Lyapunov exponent can be classified as a hyperchaotic system. In this study, a sinusoidal perturbation was designed for generating hyperchaos from the Chen–Lee chaotic system. The hyperchaos was identified by the existence of two positive Lyapunov exponents and bifurcation diagrams. The system is hyperchaotic in several different regions of the parameters c, ε, and ω. It was found that this method not only can enhance or suppress chaotic behavior, but also induces chaos in non-chaotic parameter ranges. In addition, two interesting dynamical behaviors, Hopf bifurcation and intermittency, were also found in this study.     

Analysis and control of an SEIR epidemic system with nonlinear transmission rate

In this paper, the dynamical behaviors of an SEIR epidemic system governed by differential and algebraic equations with seasonal forcing in transmission rate are studied. The cases of only one varying parameter, two varying parameters and three varying parameters are considered to analyze the dynamical behaviors of the system. For the case of one varying parameter, the periodic, chaotic and hyperchaotic dynamical behaviors are investigated via the bifurcation diagrams, Lyapunov exponent spectrum diagram and Poincare section. For the cases of two and three varying parameters, a Lyapunov diagram is applied. A tracking controller is designed to eliminate the hyperchaotic dynamical behavior of the system, such that the disease gradually disappears. In particular, the stability and bifurcation of the system for the case which is the degree of seasonality β₁=0 are considered. Then taking isolation control, the aim of elimination of the disease can be reached. Finally, numerical simulations are given to illustrate the validity of the proposed results.     

Projective synchronization between different fractional‐order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique

In the present article, the authors have proposed a modified projective adaptive synchronization technique for fractional‐order chaotic systems. The adaptive projective synchronization controller and identification parameters law are developed on the basis of Lyapunov direct stability theory. The proposed method is successfully applied for the projective synchronization between fractional‐order hyperchaotic Lü system as drive system and fractional‐order hyperchaotic Lorenz chaotic system as response system. A comparison between the effects on synchronization time due to the presence of fractional‐order time derivatives for modified projective synchronization method and proposed modified adaptive projective synchronization technique is the key feature of the present article. Numerical simulation results, which are carried out using Adams–Boshforth–Moulton method show that the proposed technique is effective, convenient and also faster for projective synchronization of fractional‐order nonlinear dynamical systems. Copyright © 2013 John Wiley & Sons, Ltd.     

Synchronization of two hyperchaotic systems via adaptive control

This letter presents chaos synchronization problem of two different hyperchaotic systems when the parameters of drive and response systems are fully unknown or uncertain. Based on Lyapunov stability theory, an adaptive control law and a parameter update rule for unknown parameters are derived such that two different high dimensional chaotic systems are to be synchronized. Hyperchaotic Chen system and Second-harmonic generation (SHG) system are taken as an illustrative example to show the effectiveness of the proposed method.     

Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters

In this paper, we investigate the reduced-order anti-synchronization of uncertain chaotic systems. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third-order chaotic system can be anti-synchronized with the canonical projection of a fourth-order chaotic system even though their parameters are unknown. The techniques are successfully applied to two examples: hyperchaotic Lorenz system (fourth-order) and Lorenz system (third-order); Lü hyperchaotic system (fourth-order) and Chen system (third-order). Theoretical analysis and numerical simulations are shown to verify the results.     

A new hyperchaotic system and its circuit implementation

In this paper, a new hyperchaotic system is presented by adding a nonlinear controller to the three-dimensional autonomous chaotic system. The generated hyperchaotic system undergoes hyperchaos, chaos, and some different periodic orbits with control parameters changed. The complex dynamic behaviors are verified by means of Lyapunov exponent spectrum, bifurcation analysis, phase portraits and circuit realization. The Multisim results of the hyperchaotic circuit were well agreed with the simulation 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.     

The evolution of a novel four-dimensional autonomous system: Among 3-torus, limit cycle, 2-torus, chaos and hyperchaos

In this paper, a novel four-dimensional autonomous system in which each equation contains a quadratic cross-product term is constructed. It exhibits extremely rich dynamical behaviors, including 3-tori (triple tori), 2-tori (quasi-periodic), limit cycles (periodic), chaotic and hyperchaotic attractors. In particular, we observe 3-torus phenomena, which have been rarely reported in four-dimensional autonomous systems in previous work. With the parameter r varying in quite a wide range, the evolution process of the system begins from 3-tori, and after going through a series of periodic, quasi-periodic and chaotic attractors in so many different shapes coming into being alternately, it evolves into hyperchaos, finally it degenerates to periodic attractor. Moreover, when the system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of the hyperchaotic systems already reported, especially the largest Lyapunov exponents. We also observe a chaotic attractor of a very special shape. The complex dynamical behaviors of the system are further investigated by means of Lyapunov exponents spectrum, bifurcation diagram and phase portraits.     

基于非线性混沌时序动力系统的预测方法研究

主要研究由混沌时序所确定的非线性动力系统的预测方法．研究了非线性自相关混沌模型的结构,模型阶数的确立技术．将神经网络和小波理论相结合,研究了小波变换神经网络的结构,给出了小波神经网络的学习方法;提出了一种新的基于小波网络的参数辨识方法．该方法可以有选择地提取时序中的不同的时间、频率尺度,实现原时序的趋势或细节预测．通过对混沌时序进行预处理,并比较预处理后的预测结果,得到了一些有益的结果:用非线性自相关混沌模型采用小波网络对模型参数进行辨识,其辨识的准确程度较高,用该模型对混沌时序(包括含有噪声)的预测比较有效．     

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.     

Adaptive control and synchronization of a new modified hyperchaotic Lü system with uncertain parameters

This paper addresses problems of control and synchronization for a new modified hyperchaotic Lü system with uncertain parameters. This new modified uncertain hyperchaotic Lü system is stabilized to its unique unstable equilibrium by using adaptive control. Furthermore, an adaptive control law and a parameter estimation update law are derived to synchronize two identical modified hyperchaotic Lü systems with uncertain parameters. Numerical examples are proposed to demonstrate and verify the theoretical analysis.     

Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters

This paper brings attention to hyperchaos anti-synchronization between two identical and different hyperchaotic systems by using adaptive control. The sufficient conditions for achieving the anti-synchronization of two hyperchaotic systems are derived based on Lyapunov stability theory. An adaptive control law and a parameter update rule for unknown parameters are introduced such that the hyperchaotic Chen system is controlled to be the hyperchaotic Lü system. Theoretical analysis and numerical simulations are shown to verify the results.     

Circuit realization,chaos synchronization and estimation of parameters of a hyperchaotic system with unknown parameters

In this article, the adaptive chaos synchronization technique is implemented by an electronic circuit and applied to the hyperchaotic system proposed by Chen et al. We consider the more realistic and practical case where all the parameters of the master system are unknowns. We propose and implement an electronic circuit that performs the estimation of the unknown parameters and the updating of the parameters of the slave system automatically, and hence it achieves the synchronization. To the best of our knowledge, this is the first attempt to implement a circuit that estimates the values of the unknown parameters of chaotic system and achieves synchronization. The proposed circuit has a variety of suitable real applications related to chaos encryption and cryptography. The outputs of the implemented circuits and numerical simulation results are shown to view the performance of the synchronized system and the proposed circuit.     

Chaos reduced-order anti-synchronization of chaotic systems with fully unknown parameters

In this present paper, we elaborated the concept of the reduced-order anti-synchronization of uncertain chaotic systems. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third order chaotic system can be anti-synchronized with the projection of a fourth-order chaotic system even though their parameters are unknown. The techniques are successfully applied to several examples: hyperchaotic Chen system (fourth-order) and Lü system (third-order), theoretical analysis and numerical simulations are shown to verify the results.