Adaptive synchronization of Rossler and Chen chaotic systems

A novel adaptive synchronization method is proposed for two identical Rossler and Chen systems with uncertain parameters. Based on Lyapunov stability theory, we derive an adaptive controller without the knowledge of the system parameters, which can make the states of two identical Rossler and Chen systems globally asymptotically synchronized. Especially, when some unknown uncertain parameters are positive, we can make the controller more simple and, besides, the controller is independent of those positive uncertain parameters. All results are proved using a well-known Lyapunov stability theorem. Numerical simulations are given to validate the proposed synchronization approach.     

Parameters identification and adaptive synchronization of chaotic systems with unknown parameters

Based on Lyapunov stabilization theory, an adaptive controller with parameters identification for a class of chaotic systems with unknown parameters is proposed in this Letter. The proposed control scheme is successfully applied to some typical chaotic systems, which can be spilt into two terms: one is the term with known states, the other is the symmetric matrix term with unknown parameters, such as Lorenz system. And with the proposed adaptive control law, the two unified systems with unknown parameter are also to be synchronized. Simulation results verify the proposed scheme's effectiveness.     

A unified approach to fuzzy modelling and robust synchronization of different hyperchaotic systems

In this paper, a Takagi Sugeno （T-S） fuzzy model-based method is proposed to deal with the problem of synchronization of two identical or different hyperchaotic systems. The T S fuzzy models with a small number of fuzzy IF-THEN rules are employed to represent many typical hyperchaotic systems exactly. The benefit of employing the T-S fuzzy models lies in mathematical simplicity of analysis. Based on the T-S fuzzy hyperchaotic models, two fuzzy controllers arc designed via parallel distributed compensation （PDC） and exact linearization （EL） techniques to synchronize two identical hyperchaotic systems with uncertain parameters and two different hyperchaotic systems, respectively. The sufficient conditions for the robust synchronization of two identical hyperchaotic systems with uncertain parameters and the asymptotic synchronization of two different hyperchaotic systems are derived by applying the Lyapunov stability theory. This method is a universal one of synchronizing two identical or different hyperchaotic systems. Numerical examples are given to demonstrate the validity of the proposed fuzzy model and hyperchaotic synchronization scheme.     

Adaptive synchronization between two different chaotic systems with unknown parameters

A unified mathematical expression describing a class of chaotic systems is presented, for which the problem of adaptive synchronization between two different chaotic systems with unknown parameters has been studied. Based on Lyapunov stability theory, an adaptive synchronization controller is designed and analytic expression of the controller and the adaptive laws of parameters are developed. The adaptive synchronizations between Lorenz and Chen systems, a modified Chua's circuit and Rössler systems are taken as two illustrative examples to show the effectiveness of the proposed method.     

不同结构混沌系统的自适应同步和反同步

针对不同结构混沌系统的同步与反同步问题进行了研究.在系统参数已知时,采用主动控制法实现混沌系统的同步与反同步,并将主动控制器的设计方法进行了推广.在参数未知时,基于Lyapunov稳定性理论和自适应控制方法,给出了自适应控制器和参数自适应律,实现了参数均未知且结构不同的驱动系统和响应系统的同步与反同步.在控制器的设计过程中,将驱动系统和响应系统进行互换,讨论了互换前后的控制器和自适应律之间的关系.数值仿真结果说明了所提出设计方法的有效性. 关键词： 混沌同步 反同步 主动控制法 自适应控制法     

Anti-synchronization of chaotic systems with uncertain parameters via adaptive control

In this Letter, an adaptive control scheme is developed to study the anti-synchronization behavior between two identical and different chaotic systems with unknown parameters. This adaptive anti-synchronization controller is designed based on Lyapunov stability theory and an analytic expression of the controller with its adaptive laws of parameters is shown. The adaptive anti-synchronization between two identical systems (Chen system) and different systems (Genesio and Lü systems) are taken as two illustrative examples to show the effectiveness of the proposed method. Theoretical analysis and numerical simulations are shown to verify the results.     

Adaptive synchronization of a class of continuous chaotic systems with uncertain parameters

This Letter investigates the problem of synchronization between two different chaotic systems. A generic adaptive synchronization scheme is proposed for a class of chaotic systems with uncertain parameters. Based on the Lyapunov stability theorem, an adaptive controller and the corresponding parameters update laws are designed to synchronize two chaotic systems. Numerical simulations are also given to show the effectiveness of the proposed method.     

参数不确定的分数阶混沌系统广义错位延时投影同步

为进一步增强通信系统中保密通信的安全性,结合广义错位投影同步和延时投影同步,提出了广义错位延时投影同步.以分数阶Chen系统和Lü系统为例,针对两系统参数都不确定,基于分数阶稳定性理论与自适应控制方法,设计了非线性控制器和参数自适应律,实现了广义错位延时同步,并辨识出驱动系统和响应系统中所有不确定参数.理论分析和数值仿真验证了该方法的可行性与有效性.     

Pragmatical generalized synchronization of chaotic systems with uncertain parameters by adaptive control

A new kind of generalized synchronization of two chaotic systems with uncertain parameters is proposed. Based on a pragmatical asymptotical stability theorem and an assumption of equal probability for ergodic initial conditions, an adaptive control law is derived so that it can be proved strictly that the common null solution of error dynamics and of parameter dynamics is actually asymptotically stable, i.e. these two identical systems are in generalized synchronization and the estimated parameters approach the uncertain values. It is called pragmatical generalized synchronization. Finally, two numerical examples are studied for two Quantum-CNN oscillator chaotic systems to show the effectiveness of the proposed generalized synchronization strategy with a double Duffing chaotic system as a goal system.     

多涡卷混沌系统的广义同步控制

基于反步自适应方法,提出了一种在初值不同以及驱动系统参数未知的情况下,保持驱动和响应两多涡卷混沌系统同步的控制律设计方法.该方法所设计的控制律能够同时适用于两混沌系统的完全同步、反同步和一类非线性广义同步,具有较高的实用价值.仿真结果表明了所设计控制律的有效性. 关键词： 多涡卷混沌 广义同步 反步 自适应     

Multistability in coupled different-dimensional dynamical systems

Multistability or coexistence of different chaotic attractors for a given set of parameters depending on the initial condition only is one of the most exciting phenomenon in dynamical systems. The schemes to design multistability systems via coupling two identical or non-identical but the same-dimensional systems have been proposed earlier. Coupled different-dimensional systems are very useful to describe the real-world physical and biological systems. In this paper, a scheme for designing a multistable system by coupling two different-dimensional dynamical systems has been proposed. Coupled Lorenz and Lorenz–Stenflo systems have been considered to illustrate the scheme. The efficiency of the scheme is shown numerically, by presenting phase diagrams, bifurcation diagrams and variation of maximum Lyapunov exponents.     

基于自适应主动及滑模控制的分数阶超混沌系统异结构反同步

以超混沌Chen系统和超混沌Lorenz系统为例,研究了慢时变参数超混沌系统的反同步问题.首先利用主动控制的思想,消去超混沌系统中的非线性部分,然后基于Lyapunov稳定性理论,合理地选取参数自适应控制律,很好的解决了时变参数的参数摄动问题,从而实现了两个超混沌系统的反同步.在此基础之上,又进一步研究了分数阶超混沌系统,使用滑模控制方法对其进行控制,理论上分析了该方法的可行性.数值模拟实验进一步验证了所提出方法的有效性. 关键词： 超混沌 分数阶 自适应 滑模     

Analytical and simulation studies for diode and triode ion beam extraction systems

This work is concerned with ion beam dynamics and compares the emittance to aberration ratios of two-and three-electrode extraction systems. The study is conducted with the aid of Version 7 of SIMION 3D ray-tracing software. The beam dependence on various parameters of the extraction systems is studied and the numerical results lead to qualitative conclusions.     

Reliability of linear coupling synchronization of hyperchaotic systems with unknown parameters

Complete synchronization could be reached between some chaotic and/or hyperchaotic systems under linear coupling. More generally, the conditional Lyapunov exponents are often calculated to confirm the stability of synchronization and reliability of linear controllers. In this paper, detailed proof and measurement of the reliability of linear controllers are given by constructing a Lyapunov function in the exponential form. It is confirmed that two hyperchaotic systems can reach complete synchronization when two linear controllers are imposed on the driven system unidirectionally and the unknown parameters in the driving systems are estimated completely. Finally, it gives the general guidance to reach complete synchronization under linear coupling for other chaotic and hyperchaotic systems with unknown parameters.     

Magnetic measurements in resonance systems

Existing methods of measuring uhf properties of ferromagnetic substances in a resonator (and in lumped resonance systems) become inapplicable even for small samples as the Q of the electromagnetic system is increased. By regarding the magnetization oscillations in the specimen as an oscillatory system with two degrees of freedom, the possibility of measuring ferromagnetic resonance parameters in high-Q electromagnetic systems is studied.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 87–91, August, 1971.     

Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order

This Letter further investigates the full state hybrid projective synchronization (FSHPS) of chaotic and hyper-chaotic systems with fully unknown parameters. Based on the Lyapunov stability theory, a unified adaptive controller and parameters update law can be designed for achieving the FSHPS of chaotic and/or hyper-chaotic systems with the same and different order. Especially, for two chaotic systems with different order, reduced order MFSHPS (an acronym for modified full state hybrid projective synchronization) and increased order MFSHPS are first studied in this Letter. Five groups numerical simulations are provided to verify the effectiveness of the proposed scheme. In addition, the proposed FSHPS scheme is quite robust against the effect of noise.     

Persistence of zero velocity fronts in reaction diffusion systems

Steady, nonpropagating, fronts in reaction diffusion systems usually exist only for special sets of control parameters. When varying one control parameter, the front velocity may become zero only at isolated values (where the Maxwell condition is satisfied, for potential systems). The experimental observation of fronts with a zero velocity over a finite interval of parameters, e.g., in catalytic experiments [Barelko et al., Chem. Eng. Sci., 33, 805 (1978)], therefore, seems paradoxical. We show that the velocity dependence on the control parameter may be such that velocity is very small over a finite interval, and much larger outside. This happens in a class of reaction diffusion systems with two components, with the extra assumptions that (i) the two diffusion coefficients are very different, and that (ii) the slowly diffusing variables has two stable states over a control parameter range. The ratio of the two velocity scales vanishes when the smallest diffusion coefficient goes to zero. A complete study of the effect is carried out in a model of catalytic reaction. (c) 2000 American Institute of Physics.     

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.     

Adaptive projective synchronization between different chaotic systems with parametric uncertainties and external disturbances

The article deals with adaptive projective synchronization between two different chaotic systems with parametric uncertainties and external disturbances. Based on Lyapunov stability theory, the projective synchronization between a pair of different chaotic systems with fully unknown parameters are derived. An adaptive control law and a parameter update rule for uncertain parameters are designed such that the chaotic response system controls the chaotic drive system. Numerical simulation results are performed to explain the effectiveness and feasibility of the techniques.