耦合Van der Pol-Duffing振子的动力学分析

非线性耦合VanderPol -Duffing振子在强共振情形下具有极其复杂的动力学行为 ,本文利用数值方法研究了各系统参数对振子系统动力学性态的影响 ,揭示了减幅、增幅、周期。

一个Van Der Pol-Duffing系统的混沌与控制

研究了Van der Pol-Duffing振子的混沌动力学行为,应用直接微扰法构造了系统的通解,由该通解获得了预测混沌出现的Melnikov判据.在非微扰情形,相图和相应Poincaré截面的演化结果表明:系统阻尼和外驱动力的变化都可以导致系统由倍周期分叉进入混沌状态,当频率参数取相同值时,系统混沌被完全抑制.

改进型Van der Pol-Duffing混沌振子的同步研究

研究了改进型Van der Pol-Duffing混沌振子的同步问题。当驱动系统的参数已知时,根据Lyapunov稳定性理论,设计了一个线性反馈控制器,使两个相同的改进型Van der Pol-Duffing混沌振子同步,并得出了保守性较小的同步条件;当驱动系统的参数未知时,利用自适应控制方法,选择了适当的自适应律,构造了两个简单的控制器,使响应系统与驱动系统同步,并同时实现了驱动系统中未知参数的辨识。通过数值仿真,表明了这些方法的有效性。

广义分数阶van der Pol-Duffing振子的动力学响应与隔振效果研究

用平均法研究了含分数阶导数项的van der Pol-Duffing振子的动力学行为和力传递率。得到了主共振时振子的一阶解析解、定常解幅频曲线和相频曲线的解析表达式,进一步通过与数值解作对比,验证了解析解的正确性,分析了不同参数对幅频曲线和力传递率的影响。结果表明:解析解与数值解吻合良好;在无量纲情况下,共振区分数阶项系数、非线性参数、分数阶阶次、阻尼比对幅频曲线和力传递率的共振峰值均有抑制作用;不同频率区段参数对隔振效果的影响不同,在低频隔振区非线性参数和幅值越小隔振效果越好,此外阻尼比对力传递率影响很小;在高频隔振区增大非线性参数、幅值和阻尼比有助于提高隔振效果。

控制周期激励Van der Pol-Duffing振子的混沌

对周期激励VanderPol-Duffing振子进行了研究:x··-μ(1-x2)x·-αx+βx3=fcosωt。首先运用相图分析、直接观察运动时间序列的方法发现,VanderPol-Duffing振子在一定条件下会出现混沌行为。在实际工程中,混沌行为往往会导致振荡或不规则运动,甚至主系统的彻底崩溃,因此有必要抑制系统的混沌行为。文中采用周期激振力法对系统中的混沌行为进行了控制,并结合lyapunov指数谱进行了分析,结果表明VanderPol-Duffing振子中的混沌运动得到了有效的控制。

基于耦合Van der Pol-Duffing系统的微弱信号检测研究

传统方法检测微弱信号具有一定的困难,利用混沌振子对微弱信号敏感以及对强噪声具有良好免疫力的特性,提出基于耦合Van der Pol-Duffing振子系统检测微弱信号的新方法。对比不同参数下耦合系统的动力学行为,通过分岔图和二分法确定临界阈值,保证阈值搜索速度和阈值精度。阐述基于相图的微弱信号检测原理,通过从混沌态到周期态的转变成功检测淹没在强噪声中的微弱信号,信噪比门限达到–30 d B。同时考察不同精度幅值下噪声对系统状态的影响,不同频率信号以及相移对检测的影响。仿真结果表明,该耦合系统在强噪声条件下对微弱信号敏感,用于检测微弱信号是可行的。

时滞耦合van der Pol-Duffing振子环的动力学分析

研究含时滞的大规模van der Pol-Duffing耦合振子系统的非线性动力学.通过讨论特征方程根分布情况确定系统的稳定性,并在耦合时滞和强度平面上给出振幅死亡区域.结合数值算例,揭示同步和异步周期振荡、概周期运动以及混沌吸引子等现象.基于非线性振子电路和时滞电路,构建电路实验平台,有效验证理论和数值结果.研究结果表明,时滞可以显著影响系统动力学特性,如诱发振幅死亡、稳定性切换以及复杂振荡等.

含平方项和5次幂项的Van der Pol-Duffing系统混沌控制

研究了同时含有平方项和高次幂项的Van der Pol-Duffing系统的混沌行为及混沌控制问题,数值仿真分析了该系统存在的典型非线性动力学行为.主要采用单初始点分岔分析方法、最大Lyapunov指数和Poincare映射方法,从不同侧面揭示了在周期激振力作用下系统的周期运动、混沌运动,以及运动形式的演化过程,并用x|x|控制方法实现了系统的混沌抑制问题.

PRINCIPAL RESPONSE OF VAN DER POL-DUFFING OSCILLATOR UNDER COMBINED DETERMINISTIC AND RANDOM PARAMETRIC EXCITATION

The principal resonance of Van der Pol-Duffing oscillator to combined deterministic and random parametric excitations is investigated. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response were studied. Jumps were shown to occur under some conditions. The effects of damping, detuning, bandwidth, and magnitudes of deterministic and random excitations are analyzed. The theoretical analysis were verified by numerical results.

一类Van der Pol-Duffing系统的变结构滑模控制

研究了一类含有平方项和5次幂项的Van der Pol-Duffing系统的跟踪控制问题.首先,基于Lyapunov指数理论和分岔理论分析了该系统的复杂动力学行为,包括周期运动、倍周期分岔、混沌运动等;然后,在系统参数已知和未知的两种情况下,基于Lyapunov稳定性定理分别构造了两类简单的变结构滑模控制器对该系统的混沌行为进行跟踪控制,并均跟踪控制到了预期的运动状态.最后,利用数值仿真验证了上述两类滑模控制器对该系统跟踪控制的有效性.

采用Van-der混沌振子抑制局部放电信号中周期性窄带干扰

局部放电在线监测的热点与难点之一是如何有效去除现场大量的干扰,从而真实地提取局部放电信号。为此,从非线性理论出发,基于Van-der混沌振子的运动特性,提出用Van-der混沌振子去除局部放电信号中周期性窄带干扰的新思路。该振子的时域波形对于与参考信号频差较小的周期信号具有敏感性,对噪声和与参考信号频差较大的周期信号具有免疫力。然后重新定义一种Floquet指数,利用该振子实现了多种混叠信号中周期性窄带信号幅值的检测。最后通过仿真实验和实测信号分析验证了利用该混沌振子检测周期性窄带干扰信号的可行性和有效性。

Chaotic Motions of the van der Pol-Duffing Oscillator Subjected to Periodic External and Parametric Excitations with Delayed Feedbacks

Chaotic dynamics of the van der Pol-Duffing oscillator subjected to periodic external and parametric excitations with delayed feedbacks are investigated both analytically and numerically in this manuscript.With the Melnikov method,the critical value of chaos arising from homoclinic or heteroclinic intersections is derived analytically.The feature of the critical curves separating chaotic and non-chaotic regions on the excitation frequency and the time delay is investigated analytically in detail.The monotonicity of the critical value to the excitation frequency and time delay is obtained rigorously.It is presented that there may exist a special frequency for this system.With this frequency,chaos in the sense of Melnikov may not occur for any excitation amplitudes.There also exists a uncontrollable time delay with which chaos always occurs for this system.Numerical simulations are carried out to verify the chaos threshold obtained by the analytical method.

扩展Duffing-Van der Pol系统混沌同步研究

基于混沌系统的主动控制同步方法,研究了具有不同初始条件以及不同参数条件的扩展Duffing-Van der Pol系统混沌轨道的同步问题.分析了主动控制同步方法的基本原理,并通过数值模拟验证了该同步方法的有效性.结果表明,主动同步控制器不仅具有简单的结构,而且能够达到良好的控制效果.

Amplitude control of limit cycle in a van der Pol Duffing system

This paper applies washout filter technology to amplitude control of limit cycles emerging from Hopf bifurcation of the van der Pol-Duffing system. The controlling parameters for the appearance of Hopf bifurcation are given by the Routh-Hurwitz criteria. Noticeably, numerical simulation indicates that the controllers control the amplitude of limit cycles not only of the weakly nonlinear van der Pol-Duffing system but also of the strongly nonlinear van der Pol-Duffing system. In particular, the emergence of Hopf bifurcation can be controlled by a suitable choice of controlling parameters. Gain-amplitude curves of controlled systems are also drawn.

Synchronization of Impulsive Real and Complex Van der Pol Oscillators

Nonlinear systems involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems, for example, many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, population dynamics, etc., do exhibit impulsive effects. In a recent paper [1], both real and complex Van der Pol oscillators were introduced and shown to exhibit chaotic limit cycles and in [2] an active control and chaos synchronization was introduced. In this paper, impulsive synchronization for the real and complex Van der Pol oscillators is systematically investigated. We derive analytical expressions for impulsive control functions and show that the dynamics of error evolution is globally stable, by constructing appropriate Lyapunov functions. This means that, for a relatively large set of initial conditions, the differences between the master and slave systems vanish exponentially and synchronization is achieved. Numerical results are obtained to test the validity of the analytical expressions and illustrate the efficiency of these techniques for inducing chaos synchronization in our nonlinear oscillators.

基于忆阻器的van der Pol混沌系统的线性反馈同步控制

基于忆阻器的van der Pol混沌系统的混沌特性,采用了线性反馈同步的控制方法,给出了基于忆阻器的van der Pol混沌系统实现同步的控制参数的取值范围,并通过数值仿真论证了该方法的可行性。

Adaptive backstepping control and synchronization of a modified and chaotic Van der Pol-Duffing oscillator

In this paper,we propose a backstepping approach for the synchronization and control of modified Van-der Pol Duffing oscillator circuits.The method is such that one controller function that depends essentially on available circuit parameters that is sufficient to drive the two coupled circuits to a synchronized state as well achieve the global stabilization of the system to its regular dynamics.Numerical simulations are given to demonstrate the effectiveness of the technique.

一类非线性Van der Pol-Duffing振子的隐藏吸引子（英文）

研究了一类非线性Van der Pol-Duffing振子的隐藏吸引子.运用经典动力系统Hopf分支理论,研究该非线性系统的周期轨、Hopf分支和其他动力学行为,通过谐波线性化方法和一种新的数值算法,来定位隐藏吸引子,并通过数值模拟对该非线性系统存在隐藏吸引子进行验证.

Active Control of Chaotic Oscillations in Nonlinear Chemical Dynamics

Abstract: This work studies the active control of chemical oscillations governed by a forced modified Van der Pol-Duffing oscillator. We considered the dynamics of nonlinear chemical systems subjected to an external sinusoidal excitation. The approximative solution to the first order of the modified Van der Pol-Duffing oscillator is found using the Lindstedt’s perturbation method. The harmonic balance method is used to find the amplitudes of the oscillatory states of the system under control. The effects of the constraint parameter and the control parameter of the model on the amplitude of oscillations are presented. The effects of the active control on the behaviors of the model are analyzed and it appears that with the appropriate selection of the coupling parameter, the chaotic behavior of the model has given way to periodic movements. Numerical simulations are used to validate and complete the analytical results obtained.

参数激励耦合系统的复杂动力学行为分析

分析了耦合van der Pol振子参数共振条件下的复杂动力学行为.基于平均方程,得到了参数平面上的转迁集,这些转迁集将参数平面划分为不同的区域,在各个不同的区域对应于系统不同的解.随着参数的变化,从平衡点分岔出两类不同的周期解,根据不同的分岔特性,这两类周期解失稳后,将产生概周期解或3-D环面解,它们都会随参数的变化进一步导致混沌.发现在系统的混沌区域中,其混沌吸引子随参数的变化会突然发生变化,分解为两个对称的混沌吸引子.值得注意的是,系统首先是由于2-D环面解破裂产生混沌,该混沌吸引子破裂后演变为新的混沌吸引子,却由倒倍周期分岔走向3-D环面解,也即存在两条通向混沌的道路:倍周期分岔和环面破裂,而这两种道路产生的混沌吸引子在一定参数条件下会相互转换.