一种分数阶非线性系统稳定性判定定理的问题及分析

分数阶非线性系统稳定性理论的研究对于分数阶混沌系统同步控制的应用具有重要价值,将分数阶非线性系统稳定性判断转化为相应整数阶非线性系统稳定性判断的探讨很有意义.通过实例表明了:对于时变系数矩阵,如果整数阶系统稳定,其对应的阶次小于1的分数阶系统也稳定的判定定理是错误的,并分析了问题产生的原因.     

基于最小二乘支持向量机的分数阶混沌系统控制

提出了基于最小二乘支持向量机(LS-SVM)的分数阶混沌系统控制方法.基于分数阶线性系统稳定理论,通过线性分离的方法将系统分解为稳定的线性部分和相应的非线性部分,再利用支持向量机良好的非线性函数逼近和泛化能力设计了主动控制器,对非线性部分进行补偿,从而将分数阶混沌系统控制到平衡点.分别以分数阶Liu系统和分数阶Chen系统为例进行了仿真研究,表明该方法是有效和可行的.     

Dynamic analysis of a new chaotic system with fractional order and its generalized projective synchronization

Based on the stability theory of the fractional order system,the dynamic behaviours of a new fractional order system are investigated theoretically.The lowest order we found to have chaos in the new three-dimensional system is 2.46,and the period routes to chaos in the new fractional order system are also found.The effectiveness of our analysis results is further verified by numerical simulations and positive largest Lyapunov exponent.Furthermore,a nonlinear feedback controller is designed to achieve the generalized projective synchronization of the fractional order chaotic system,and its validity is proved by Laplace transformation theory.     

基于在线误差修正自适应SVR的非线性不确定分数阶混沌系统滑模控制

提出了一种基于在线误差修正自适应SVR的滑模控制方法, 用于解决一类非线性不确定分数阶混沌系统的控制问题. 分别通过对混沌系统非线性函数的离线SVR估计和基于增量学习的状态跟踪误差在线SVR预测, 解决了不确定分数阶混沌系统模型难以预测的问题. 同时根据Lyapunov稳定性理论设计出SVR权值自适应调整律. 本文以分数阶Arneodo 系统为例进行仿真, 仿真结果表明了, 对于带有外界噪声扰动的非线性不确定分数阶混沌系统, 该方法可以在有限时间内将系统稳定至期望状态, 提高对非线性函数的预测精度, 改善控制性能.     

Function projective synchronization of fractional order satellite system and its stability analysis for incommensurate case

In this article, the stability analysis, chaos control and the function projective synchronization between fractional order identical satellite systems have been studied. Based on the stability theory of fractional order systems, the conditions of local stability of nonlinear three-dimensional commensurate and incommensurate fractional order systems are discussed. Feedback control method is used to control the chaos in the considered fractional order satellite system. Using the fractional calculus theory and computer simulation, it is found that the chaotic behavior exists in the fractional order satellite system and the lowest order of derivative where the chaos exits is 2.82. Adams-Bashforth-Moulton method is applied during numerical simulations and the results obtain are displayed through graphs.     

Uniform of four fractional-order nonlinear feedback synchronizations

This paper designs four fractional order nonlinear feedback synchronizations with the simple configuration, followed with their uniform. The closed system's stability is proved based on the fractional order stability theory. Resorted to the fractional order unified chaotic system, it is illustrated that the uniform includes the active, ordinary, dislocated, speed nonlinear feedback synchronizations and their mixed formulations. Numerical simulations show the effectiveness of the proposed methods.     

一种异结构分数阶混沌系统投影同步的新方法

基于Lyapunov稳定性理论和分数阶系统稳定理论以 及分数阶非线性系统性质,提出了一种用来判定分数阶混沌系统是 否稳定的新的判定定理,并把该理论运用于对分数阶混沌系统的控制与 同步,同时给出了数学证明过程,严格保证了该方法的正确性与一般适用性. 运用所提出的稳定性定理,实现了异结构分数阶混沌系统的投影同步. 对分数阶Lorenz混沌系统与分数阶Liu混沌系统实现了投影同步; 针对四维超混沌分数阶系统,也实现了异结构投影同步. 该稳定性定理避 免了求解分数阶平衡点以及Lyapunov指数的问题,从而可以方便地选 择出控制律,并且所得的控制器结构简单、适用范围广. 数值仿真的结果取得了预期的效果,进一步验证了这一稳定性定理的 正确性及普遍适用性.     

基于状态观测器的分数阶时滞混沌系统同步研究

研究分数阶时滞混沌系统同步问题,基于状态观测器方法和分数阶系统稳定性理论,设计分数阶时滞混沌系统同步控制器,使得分数阶时滞混沌系统达到同步,同时给出了数学证明过程.该同步控制器采用驱动系统和响应系统的输出变量进行设计,无需驱动系统和响应系统的状态变量,简化了控制器的设计,提高了控制器的实用性.利用Lyapunov稳定性理论和分数阶线性矩阵不等式,研究并给出了同步控制器参数的选择条件.以分数阶时滞Chen混沌系统为例,设计基于状态观测器的同步控制器,实现了分数阶时滞Chen混沌系统同步,并将其应用于保密通信系统中.仿真结果证明了该同步方法的有效性.     

The stability control of fractional order unified chaotic system with sliding mode control theory

This paper studies the stability of the fractional order unified chaotic system with sliding mode control theory. The sliding manifold is constructed by the definition of fractional order derivative and integral for the fractional order unified chaotic system. By the existing proof of sliding manifold, the sliding mode controller is designed. To improve the convergence rate, the equivalent controller includes two parts: the continuous part and switching part. With Gronwall’s inequality and the boundness of chaotic attractor, the finite stabilization of the fractional order unified chaotic system is proved, and the controlling parameters can be obtained. Simulation results are made to verify the effectiveness of this method.     

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.     

基于反馈和多最小二乘支持向量机的分数阶混沌系统控制

根据分数阶线性系统的稳定理论,将混沌系统分成稳定的线性部分和相应的非线性部分.设计主动控制器,对非线性部分进行补偿,从而将分数阶混沌系统控制到平衡点.为了提高主动控制器的补偿能力,提出基于反馈的多最小二乘支持向量机(M-LS-SVM)拟合模型.通过减聚类方法将输入空间划分为一些小的局部空间,在每个局部空间中用LS-SVM建立子模型.为解决子模型相互之间的严重相关问题,提高模型的精度和鲁棒性,各个子模型的预测输出通过主元递归(PCR)方法连接.仿真实验表明该方法有助于提高补偿精度和系统响应指标. 关键词： 分数阶 混沌系统 多最小二乘支持向量机 反馈     

Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory

In recent decades, mathematical modeling and engineering applications of fractional-order calculus have been extensively utilized to provide efficient simulation tools in the field of solid mechanics. In this paper, a nonlinear fractional nonlocal Euler–Bernoulli beam model is established using the concept of fractional derivative and nonlocal elasticity theory to investigate the size-dependent geometrically nonlinear free vibration of fractional viscoelastic nanobeams. The non-classical fractional integro-differential Euler–Bernoulli beam model contains the nonlocal parameter, viscoelasticity coefficient and order of the fractional derivative to interpret the size effect, viscoelastic material and fractional behavior in the nanoscale fractional viscoelastic structures, respectively. In the solution procedure, the Galerkin method is employed to reduce the fractional integro-partial differential governing equation to a fractional ordinary differential equation in the time domain. Afterwards, the predictor–corrector method is used to solve the nonlinear fractional time-dependent equation. Finally, the influences of nonlocal parameter, order of fractional derivative and viscoelasticity coefficient on the nonlinear time response of fractional viscoelastic nanobeams are discussed in detail. Moreover, comparisons are made between the time responses of linear and nonlinear models.     

Reduced-order anti-synchronization of the projections of the fractional order hyperchaotic and chaotic systems

The article aims to study the reduced-order anti-synchronization between projections of fractional order hyperchaotic and chaotic systems using active control method. The technique is successfully applied for the pair of systems viz., fractional order hyperchaotic Lorenz system and fractional order chaotic Genesio-Tesi system. The sufficient conditions for achieving anti-synchronization between these two systems are derived via the Laplace transformation theory. The fractional derivative is described in Caputo sense. Applying the fractional calculus theory and computer simulation technique, it is found that hyperchaos and chaos exists in the fractional order Lorenz system and fractional order Genesio-Tesi system with order less than 4 and 3 respectively. The lowest fractional orders of hyperchaotic Lorenz system and chaotic Genesio-Tesi system are 3.92 and 2.79 respectively. Numerical simulation results which are carried out using Adams-Bashforth-Moulton method, shows that the method is reliable and effective for reduced order anti-synchronization.     

Exact Solutions for Fractional Differential-Difference Equations by (G′/G)-Expansion Method with Modified Riemann-Liouville Derivative

In this paper, the ($G′/G$)-expansion method is suggested to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann–Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential difference equation into its differential difference equation of integer order. With the aid of symbolic computation, we choose nonlinear lattice equations to illustrate the validity and advantages of the algorithm. It is shown that the proposed algorithm is effective and can be used for many other nonlinear lattice equations in mathematical physics and applied mathematics.     

Asymptotic stability with probability one of MDOF nonlinear oscillators with fractional derivative damping

In this paper, the asymptotic stability with probability one of multi-degree-of-freedom (MDOF) nonlinear oscillators with fractional derivative damping parametrically excited by Gaussian white noises is investigated. A stochastic averaging method and the Khasminskii’s procedure are employed to evaluate the largest Lyapunov exponent, whose sign determines the stability of the system. As an example, two coupled nonlinear oscillators with fractional derivative damping is worked out to demonstrate the proposed procedure and to examine the effect of fractional order on the stochastic stability of system. In particular, the case of factional order more than 1 is studied for the first time.     

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.     

分数阶对数耦合系统在非周期外力作用下的定向输运现象

本文讨论了分数阶对数耦合系统在非周期外力作用情况下, 耦合粒子链的定向输运现象. 由于粒子在黏性介质中的运动具有“记忆性”, 所以本文通过将系统建模为分数阶对数耦合模型来研究各个系统参数对粒子链运动状态的影响. 数值仿真表明: 1)对于此类系统, 只有在存在外力作用的情况下粒子链才能够产生定向输运现象, 并且粒子链平均流速随着外力的增大而增大. 2)对于分数阶阶数较小的系统, 阻尼记忆性对粒子链的运动状态有显著的影响, 具体表现为: 粒子链的平均流速存在上界(这个上界非常小), 无论外力、耦合力以及噪声强度如何变化, 粒子链的平均流速都不会超过这个上界. 当系统的阻尼力很大且外力为零时, 粒子链不会产生定向输运现象. 3) 当系统的阶数与外力较大时, 虽然粒子链能够产生定向流, 但是此时系统对耦合力与噪声具有免疫性. 4) 耦合力与噪声强度对粒子链运动的影响只在外力较小的情况下有所表现. 在这种情况下, 当系统阶数充分大时, 粒子链的平均流速随着耦合力与噪声强度的变化而变化, 并且伴随着定向流的产生.     

分数阶Newton-Leipnik系统的动力学分析

依据分数阶线性系统的稳定性理论,研究了具有双重混沌吸引子的Newton-Leipnik系统取不同分数阶时的动力学行为.研究表明该系统具有逆向Hopf分岔过程,即随着阶数的下降,分数阶Newton-Leipnik系统由双重混沌吸引子突变为单吸引子,其动力学行为将由混沌态历经短暂的周期态后收敛于稳定的平衡点.     

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.     

Nonuniversality of Critical Exponents in a Fractional Quenched Kardar–Parisi–Zhang Equation

In this article we address the problem of the depinning transition for driven interfaces in random media. We introduce a fractional Kardar–Parisi–Zhang equation with quenched noise, in which the normal diffusion term is replaced by a fractional Laplacian accounting for long-range interactions through quenched disorder. The critical values of the external driving force and nonlinear term coefficient evidently depend on the system size at the depinning transition. For a fixed value of the external driving force, the fractional order much determines the value of the nonlinear term coefficient that leads to a depinned interface. Near the depinning threshold, the critical exponent obtained numerically is nonuniversal, and weakly depends on the fractional order.