分数阶非线性方程近似解析解的新解法

将变分迭代法、同伦扰动法和Laplace变换相结合应用于分数阶非线性发展方程近似解的求解,其中Laplace变换可准确方便地求得分数阶的Lagrange乘子,而He的多项式可简单地处理方程中出现的非线性项,将新的处理方法应用到分数阶耦合的MKdV方程,结果表明该方法具有较高的精度和收敛性。     

同伦扰动法求解分数阶非线性扩散方程近似解

将Caputo分数阶微分算子引入到带有初值条件的扩散方程中,建立了时空分数阶方程。利用同伦扰动法并借助于Mathematica软件的符号计算功能,求解了分数阶非线性扩散方程的近似解,整数阶方程的结果作为特例被包含。     

分数阶Duffing振子的动力学研究

在经典Dtfffing振子中引入分数微分型阻尼项,推导了高效率的数值计算格式,对其表现出来的特有的非线性现象进行讨论．研究表明：分数微分型阻尼的分数阶值较小时,振子将出现倍周期分岔并导致混沌．在不同的外激励频率下,分数微分型Duffing振子会呈现对称性破缺、分岔、混沌等强烈的非线性现象;在一定参数范围内,分数微分型Duffing振子较经典Duffing振子,在较小的激励下即可进入混沌．     

Duffing振子与分数傅里叶变换的Chirp水印检测性能对比研究

研究Duffing 振子和分数傅里叶变换在Chirp类水印检测中的性能比较。首先分析目前分数傅里叶变换检测Chirp类水印的不足, 然后将嵌入在载体低频小波域的非周期Chirp信号通过分块平滑转换为单频周期信号, 利用Duffing振子阵列检测器检测微弱的周期信号。实验表明, 当信噪比为-41 dB时, Duffing振子仍然能有效检测到水印的存在, 此时分数傅里叶变换失效; 而当信噪比较高时, 分数傅里叶变换计算较Duffing振子检测简单。     

一类分数阶非线性振子方程的特性研究

将分数阶微分算子引入到黏弹性介质中的阻尼振动中建立分数阶非线性振动方程。利用Adomian分解方法借助Mathematica软件的符号计算功能求解了该类分数阶阻尼振动方程的近似解,研究了振子运动与方程中分数阶导数的关系。     

分数阶Rayleigh--Duffing-like系统的自适应追踪广义投影同步

本文针对分数阶Rayleigh-Duffing—like系统，设计了控制器和未知参数的辨识规则，实现了含有未知参数和随机扰动的分数阶Rayleigh—Duffing—like系统同给定信号的广义投影同步．数值仿真表明了所设计的控制器及未知参数辨识规则的有效性．     

分数阶对偶Burger方程的精确解

将分数阶复变换方法和tanh函数方法相结合,得到了一种用来求解时-空分数阶非线性微分方程精确解的复变换-tanh函数方法。借助于软件Mathematica的符号计算功能,使用该方法求解了分数阶对偶Burger方程,得到了分数阶对偶Burger方程的新的精确解。     

基于Duffing振子的DSSS信号载波检测方法

基于Duffing混沌振子检测微弱信号方法,提出一种DSSS/BPSK信号载波检测的新方法.该方法先介绍了Duffing振子检测微弱信号的技术.接着利用Duffing振子对小周期信号的敏感性和对噪声的免疫力,对DSSS/BPSK信号进行非线性平方变换能够检测出淹没在强噪声背景中的正弦信号.仿真结果表明.该方法能够在信噪比很低的情况下检测出DSSS/BPSK信号.而且性能良好.     

基于同伦分析法的轨道电路暂态仿真分析北大核心CSCD

为了进行轨道电路暂态分析,采用同伦分析法对轨道电路传输线方程求解。将Caputo分数阶导数作为线性算子,建立轨道电路传输线方程的高阶形变方程,求解得轨道电路传输线方程三阶近似解。根据不同的道床电阻确定其收敛区域,对轨道电路电压波暂态响应进行仿真分析。仿真结果表明,将分数阶导数作为线性算子的同伦分析法可以准确分析轨道电路传输线电压波传输特征和过程,为轨道电路暂态分析提供了一种新的方法。     

分数阶非线性方程精确解的新方法

将分数阶复变换方法和[(G/G)]方法相结合得到了一种辅助方程方法,用来求解分数阶非线性微分方程。利用该方法并借助于软件Mathematica的符号计算功能求解了分数阶Calogero KDV方程,得到了该方程新的精确解。     

Global Error Minimization method for solving strongly nonlinear oscillator differential equations

A modified variational approach called Global Error Minimization (GEM) method is developed for obtaining an approximate closed-form analytical solution for nonlinear oscillator differential equations. The proposed method converts the nonlinear differential equation to an equivalent minimization problem. A trial solution is selected with unknown parameters. Next, the GEM method is used to solve the minimization problem and to obtain the unknown parameters. This will yield the approximate analytical solution of the nonlinear ordinary differential equations (ODEs). This approach is simple, accurate and straightforward to use in identifying the solution. To illustrate the effectiveness and convenience of the suggested procedure, a cubic Duffing equation with strong nonlinearity is considered. Comparisons are made between results obtained by the proposed GEM method, the exact solution and results from five recently published methods for addressing Duffing oscillators. The maximal relative error for the frequency obtained by the GEM method compared with the exact solution is 0.0004%, which indicates the remarkable precision of the GEM method.     

Improved accuracy of He’s energy balance method for analysis of conservative nonlinear oscillator

In this paper, the accuracy of He’s energy balance method for the analysis of conservative nonlinear oscillator is improved based on combining features of collocation method and Galerkin–Petrov method. In order to demonstrate the effectiveness of proposed method, Duffing oscillator with cubic nonlinearity, double-well Duffing oscillator, and nonlinear oscillation of pendulum attached to a rotating support are considered. Comparison of results with ones achieved utilizing other techniques shows improved energy balance method can very effectively reduce the error of simple energy balance method. Also, results show in large amplitude of oscillation, and improved energy balance method yields better accuracy rather than second-order energy balance method based on collocation and second-order energy balance method based on Galerkin method. Improved energy balance method can be successfully used for accurate analytical solution of other conservative nonlinear oscillator.     

Approximate finite‐time control for a class of uncertain nonlinear systems with dynamic compensation

In this work, a new robust nonlinear feedback control method with dynamic active compensation is proposed; the active control method has been applied to an integral series of finite‐time single‐input single‐output nonlinear system with uncertainty. In further tracking the closed‐loop stability and nonlinear uncertainty, an extended state observer has been employed to solve the immeasurability and nonlinear uncertainty within a nonlinear system. A singular perturbation theory has been used to solve for the finite‐time stability of the closed‐loop system; furthermore, numerical simulations showed that the robust nonlinear feedback controller tracked the uncertainty in a nonlinear Duffing‐type oscillator and has proven the effectiveness of the approximate finite‐time control strategy proposed. By using an approximate finite‐time control approach with active compensation, the uncertainty in a nonlinear system could be accurately estimated and controlled. Copyright © 2017 John Wiley & Sons, Ltd.     

Numerical studies of the performance of an optimally controlled nonlinear stochastic oscillator

This paper deals with the optimal control of a random nonlinear triangular wave oscillator. It is assumed that the oscillator is subjected to two different kinds of perturbation — the first kind is represented by a vector of independent standard Wiener processes and the second kind by a generalized type of a Poisson process.Sufficient conditions on the optimal controls are derived. These conditions require the existence of a smooth solution to a certain nonlinear partial integrodifferential equation. Numerical procedures for the solution of this equation are suggested. The performance of the controlled random oscillator is investigated via the numerical solutions to the nonlinear partial integrodifferential equation. Also, the performance of the random oscillator in the case where no control is applied is studied by means of the numerical solutions to a linear partial integrodifferential equation.     

Solutions of a fractional oscillator by using differential transform method

In this paper, we present an efficient algorithm for solving a fractional oscillator using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of a fractional oscillator. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method.     

基于非线性自回归时序模型的振动系统辨识

针对线性和弱非线性振动系统进行了研究,提出采用非线性自回归时序(GNAR)模型进行系统频率辨识和判断系统性或非线性基本特征的方法。首先根据摄动法求解非线性微分方程的理论,论证GNAR模型与线性和弱非线性系统之间的本质联系,推导出GNAR模型系数与线性和非线性系统频率之间的解析关系,然后给出由GNAR模型系数和结构判断系统是否存在非线性,及辨识系统频率和非线性项基本特征的方法。最后,以单自由度线性振动系统和无阻尼Duffing振动系统为算例验证该辨识方法的有效性和准确性。实验结果表明,基于GNAR模型的振动系统基本特征辨识方法具有较好的识别精度,能用于估计系统的动力学特性。