一类广义扰动KdV-Burgers方程的同伦近似解

通过构造一个同伦映射, 研究了一类广义扰动KdV-Burgers方程. 在引入典型无扰动任意次广义KdV-Burgers方程扭状孤立波解的基础上, 研究了扰动方程的具有任意精度的近似解,指出了近似解级数的收敛性, 最后利用不动点定理,进一步说明近似解的有效性,并对精度进行了讨论. 关键词： 广义扰动KdV-Burgers方程 同伦映射 渐近方法 近似解     

广义Landau-Ginzburg-Higgs方程孤子解的扰动理论

利用扰动方法研究了一类广义Landau-Ginzburg-Higgs方程.引入一个同伦映射，将原方程的解表示为渐近展开式，然后用相应的线性方程的解来近似地表示.讨论了方程的解与得到的近似解的关系. 关键词： 孤子 扰动 同伦映射     

一类非线性扰动发展方程的广义迭代解

利用广义变分迭代方法研究了一类非线性发展扰动方程.首先引入一个泛函.然后求其变分,最后构造方程解的迭代关系式.得到了问题的近似解和精确解析解. 关键词： 发展方程 扰动 变分迭代     

一类Landau-Ginzburg-Higgs扰动方程孤子的近似解

利用解析方法研究了一类Landau-Ginzburg-Higgs方程. 由广义变分迭代理论得到了相应方程的解,从而得到了对应方程孤子的近似解. 关键词： 孤子 扰动 变分迭代     

非线性广义Landau-Ginzburg-Higgs方程孤子解的变分迭代解法

利用扰动方法研究了一类非线性广义Landau-Ginzburg-Higgs方程.首先引入一个泛函，计算它的变分，求得Lagrange乘子.然后构造了原方程解的迭代关系式，得到了近似解. 关键词： 孤子 扰动 变分迭代     

一类量子等离子体类孤波的近似解析解

本文探讨了在一类量子等离子体系统,研究了该系统的非线性动力学扰动方程.利用改进的广义泛函变分迭代方法,求得了对应系统的类孤波近似解析解.     

一类广义Sine-Gordon扰动方程的解析解

利用同伦映射方法研究了一类广义Sine-Gordon方程. 首先引入一个同伦变换. 然后构造了原方程解的迭代关系式. 最后得到了问题的解析解. 关键词： 孤子 扰动 同伦映射     

用广义变分迭代理论求一类相对转动动力学方程的解

研究了一类具有非线性阻尼力和强迫周期力项的相对转动扰动动力学方程. 首先利用变分原理构造了广义变分迭代, 其次决定方程的初始近似, 最后通过迭代表示式得到了对应方程的任意次近似解. 关键词： 相对转动 动力系统 变分迭代     

一类扰动发展方程近似解

采用了一个简单而有效的技巧, 研究了一类扰动发展方程. 首先引入求解一个相应典型方程的行波孤波解. 然后利用渐近方法得到了原扰动发展方程的近似解. 利用泛函分析的不动点定理, 指出了近似解级数的收敛性, 并讨论了近似解的精度.     

强阻尼广义sine-Gordon方程特征问题的变分迭代法

研究了在数学、力学中广泛出现的一类非线性强阻尼广义sine-Gordon扰动微分方程问题. 首先, 引入行波变换, 求出退化方程的精确解. 再构造一个泛函, 创建了一个变分迭代算法, 最后, 求出原非线性强阻尼广义sine-Gordon扰动微分方程问题的近似行波解析解. 用变分迭代法可得到的各次近似解, 具有便于求解、精度高等特点. 求得的近似解析解弥补了单纯用数值方法的模拟解的不足.     

基于Kramers逃逸速率的Duffing振子广义调参随机共振研究

以二维Duffing振子的随机共振为研究对象,提出Duffing振子的广义调参随机共振. 以Kramers逃逸速率为基础,建立了Duffing振子随机共振的判别函数,阐述了Duffing振子在不同噪声强度及信号频率输入条件下的广义调参随机共振规律,并给出了Duffing振子广义调参随机共振的一般方法. 关键词： Duffing振子 随机共振 Kramers逃逸速率 广义参数调节     

Linearizability conditions for the Rayleigh-like oscillators

In this work we consider a family of nonlinear oscillators that is cubic with respect to the first derivative. Particular members of this family of equations often appear in numerous applications. We solve the linearization problem for this family of equations, where as equivalence transformations we use generalized nonlocal transformations. We explicitly find correlations on the coefficients of the considered family of equations that give the necessary and sufficient conditions for linearizability. We also demonstrate that each linearizable equation from the considered family admits an autonomous Liouvillian first integral, that is Liouvillian integrable. Furthermore, we demonstrate that linearizable equations from the considered family does not possess limit cycles. Finally, we illustrate our results by two new examples of the Liouvillian integrable nonlinear oscillators, namely by the Rayleigh–Duffing oscillator and the generalized Duffing–Van der Pol oscillator.     

Bifurcations of a parametrically excited oscillator with strong nonlinearity

A parametrically excited oscillator with strong nonlinearity, including van der Pol and Duffing types, is studied for static bifurcations. The applicable range of the modified Lindstedt－Poincaré method is extended to 1/2 subharmonic resonance systems. The bifurcation equation of a strongly nonlinear oscillator, which is transformed into a small parameter system, is determined by the multiple scales method. On the basis of the singularity theory, the transition set and the bifurcation diagram in various regions of the parameter plane are analysed.     

Predicting optimal drive sweep rates for autoresonance in Duffing-type oscillators: A beat method using Teager-Kaiser instantaneous frequency

Sustained resonance in a linear oscillator is achievable with a drive whose constant frequency matches the resonant frequency of the oscillator. But in oscillators with nonlinear restoring forces such as the pendulum, Duffing and Duffing-Van der Pol oscillator, the resonant frequency changes as the amplitude changes, so a constant frequency drive results in a beat oscillation instead of sustained resonance. Duffing-type nonlinear oscillators can be driven into sustained resonance, called autoresonance, when the drive frequency is swept in time to match the changing resonant frequency of the oscillator. We find that near-optimal drive linear sweep rates for autoresonance can be estimated from the beat oscillation resulting from constant frequency excitation. Specifically, a least squares estimate of the Teager-Kaiser instantaneous frequency versus time for the beat response to a stationary drive provides a near-optimal estimate of the nonstationary drive linear sweep rate needed to sustain resonance in the pendulum, Duffing and Duffing-Van der Pol oscillators. We confirm these predictions with model-based numerical simulations. An advantage of the beat method of estimating optimal drive sweep rates for maximal autoresonant response is that no model is required so experimentally generated beat oscillation data can be used for systems where no model is available.     

STOCHASTIC AVERAGING OF STRONGLY NON-LINEAR OSCILLATORS UNDER BOUNDED NOISE EXCITATION

A stochastic averaging method for strongly non-linear oscillators under external and/or parametric excitation of bounded noise is proposed by using the so-called generalized harmonics functions. The method is then applied to study the primary resonance of Duffing oscillator with hardening spring under external excitation of bounded noise. The stochastic jump and its bifurcation of the system are observed and explained by using the stationary probability density of amplitude and phase. Subsequently, the method is applied to study the dynamical instability and parametric resonance of Duffing oscillator with hardening spring under parametric excitation of bounded noise. The primary unstable region is delineated by evaluating the Lyapunov exponent of linearized system, and the response and jump of non-linear system around the unstable region are examined by using the sample functions and stationary probability density of amplitude and phase.     

Regular nonlinear response of the driven Duffing oscillator to chaotic time series

Nonlinear response of the driven Duffng oscillator to periodic or quasi-periodic signals has been well studied.In this paper,we investigate the nonlinear response of the driven Duffng oscillator to non-periodic,more specifically,chaotic time series.Through numerical simulations,we find that the driven Duffng oscillator can also show regular nonlinear response to the chaotic time series with different degree of chaos as generated by the same chaotic series generating model,and there exists a relationship between the state of the driven Duffng oscillator and the chaoticity of the input signal of the driven Duffng oscillator.One real-world and two artificial chaotic time series are used to verify the new feature of Duffng oscillator.A potential application of the new feature of Duffng oscillator is also indicated.     

Duffing系统的主-超谐联合共振

非线性动力系统极易发生共振,在多频激励下可能发生联合共振或组合共振,目前关于非线性系统的主-超谐联合共振的研究少见报道.本文以Duffing系统为对象,研究系统在主-超谐联合共振时的周期运动和通往混沌的道路.应用多尺度法得到系统的近似解析解,并利用数值方法对解析解进行验证,结果吻合良好.基于Lyapunov第一方法得到稳态周期解的稳定性条件,并分析了非线性刚度对稳态周期解的幅值和稳定性的影响.此外,由于近似解只能描述周期运动,不足以描述系统的全局特性,因而应用Melnikov方法对系统进行全局分析,得到系统进入Smale马蹄意义下混沌的条件,依据该条件以及主-超谐联合共振的条件选取一组参数进行数值仿真.分岔图和最大Lyapunov指数显示出两个临界值:当激励幅值通过第一个临界值时,异宿轨道破裂,混沌吸引子突然出现,系统以激变方式进入混沌;激励幅值通过第二个临界值时,系统在混沌态下再次发生激变,进入另一种混沌态.利用Melnikov方法考察了第一个临界值在多种频率组合下的变化趋势,并用数值仿真验证了解析结果的正确性.     

Series resonance circuit with ferroelectric capacitor

A series resonance circuit under sinuousoidal driving is investigated experimentally. The inductance consists of an air coil. The capacitance is made up of a ferroelectric material that introduces its nonlinear dielectric properties into the circuit. The dynamical system linear coil-nonlinear capacaitor shows an interesting behaviour. The phase portrait differs in general from the ellipse of the harmonic oscillator. For appropriate external conditions period doubling sequences, chaos and therein enclosed periodic windows might occur. Starting from a cubic nonlinearity of the dielectric properties a Duffing equation is proposed as a model for periodic behaviour of the series resonance circuit. Simulations of experimentally recorded phase portraits yield good agreement between experiment and model.     

逆算符方法求解非线性动力学方程及其一些应用实例

首先简介逆算符方法及如何实现对它的数学机械化;然后用逆算符方法研究了三个典型的非线性方程:Lorentz方程,广义Duffing方程和双耦合广义Duffing方程。用四阶龙格-库塔方法进行比较,说明逆算符方法比龙格-库塔方法具有更高的精度和更快的收敛性。本文把逆算符方法应用于混沌行为的研究,并将此法在微机上实现了数学机械化。该法有很大的普适性,特别适用于对复杂问题的定量计算,大有应用和发展前途。 关键词：