研究强非线性振动系统同宿分岔问题的规范形方法

以改进的规范形理论为基础,采用强非线性振动问题的分析方法,拓展了原有弱非线性振动系统同宿分岔判据的适用范围.首先在复规范形求解过程中引入待定固有频率,计算了一类单自由度强非线性振动系统的周期解.然后分别依据系统的待定固有频率趋于零和周期轨道趋近于鞍点两条途径获得了强非线性振动条件下系统同宿分岔的解析判据.最后通过与原有解析结果和数值结果相比较验证了本文方法的有效性. 关键词： 规范形 同宿分岔 强非线性 周期解     

一类复杂流行病学模型的混沌研究

研究了一类周期变化的非线性复杂发病率的广义流行病学模型(SIR(susceptible, infected, recovered)模型). 通过一系列坐标变换将原模型转化为Hamilton系统,运用Melnikov方法证明了该系统存在混沌运动,给出了发生同宿分岔的条件,并用数值模拟验证了上述结果. 关键词： SIR(susceptible infected recovered)模型 混沌运动 Melnikov方法 同宿分岔     

一类相对转动非线性动力系统的混沌运动

研究一类具有同宿轨道、异宿轨道的相对转动非线性动力系统的混沌运动. 建立具有非线性刚度、非线性阻尼和外扰激励作用的一类两质量相对转动非线性动力系统的动力学方程. 利用Melnikov方法讨论了系统的全局分岔和系统进入混沌状态的可能途径,给出了系统发生混沌的必要条件,并利用最大Lyapunov指数图,分岔图,Poincare截面图和相轨迹图进一步分析了系统的混沌行为. 关键词： 相对转动 非线性动力系统 混沌 Melnikov方法     

一类非线性相对转动系统的组合共振分岔与混沌

研究一类具有异宿轨道的非线性相对转动系统的分岔与混沌运动. 应用耗散系统的拉格朗日方程建立一类组合谐波激励作用下非线性相对转动系统的动力学方程. 利用多尺度法求解相对转动系统发生组合共振时满足的分岔响应方程并进行奇异性分析, 得到了系统稳态响应的转迁集. 根据相对转动系统异宿轨道参数方程, 求解了异宿轨道的Melnikov函数, 并给出了系统发生Smale马蹄变换意义下混沌的临界条件. 最后采用数值方法, 通过分岔图, 最大Lyapunov指数图, 相轨迹图和庞加莱截面图研究系统参数对混沌运动的影响.     

连续时间系统同宿轨的搜索算法及其应用

同宿轨的求解是非线性系统领域的核心问题之一, 特别是对动力系统分岔与混沌的研究有重要意义. 根据同宿轨的几何特点, 采用轨线逼近的方式, 通过定义逼近轨线与鞍点的距离, 将同宿轨的求解转化为求距离最小值的无约束非线性优化问题. 为了提高优化结果的完整性, 还提出了基于区间细分的搜索算法和实现方法, 并找出了Lorenz系统, Shimizu-Morioka系统和超混沌Lorenz系统等的多个同宿轨道和对应参数, 验证了本文方法的有效性. 关键词： 混沌 同宿轨 非线性系统 数值计算     

参外联合激励复合非线性振子的分岔分析

讨论了参外联合激励复合非线性振子的动力学行为.对其定常解在一阶近似下的方程进行了局部分岔分析,给出了简单分岔和Hopf分岔发生的条件,并通过对近似方程和原系统的数值模拟加以验证.分析了多种参数对该振子动力学行为演化过程的影响.根据全局分岔理论探讨了该振子在不同条件下发生同宿、异宿分岔的必要条件,其结论与数值计算的结果大致符合. 关键词： 复合非线性振子 局部分岔 全局分岔 混沌     

时滞位置反馈对一类非线性相对转动系统混沌运动和安全盆侵蚀的控制

本文以一类典型的相对转动振动系统为研究对象,研究激励引起的系统混沌运动和安全域侵蚀,并对系统施加时滞位置反馈来抑制这两类复杂动力学行为.首先,利用Melnikov函数法获得相对转动系统的混沌运动及安全盆侵蚀的激励振幅阈值;其次,通过讨论时滞反馈系统的Hopf分岔条件获得适用于Melnikov函数法的控制参数取值范围,进而利用Melnikov函数法获得时滞受控系统的全局分岔必要条件;最后,利用四阶Rung-Kutta法和点映射法数值模拟了时滞受控系统动力学行为随参数的演变,从而验证解析结果的有效性.研究发现:在正的增益系数和较短的时滞量下,时滞位置反馈能够有效抑制相对转动系统的混沌运动和安全盆侵蚀现象.     

一类含三势阱Mathieu-Duffing振子的相对转动系统的分岔和混沌

建立了一类具有三势阱Mathieu-Duffing振子的两质量相对转动系统的非线性动力学方程.应用多尺度法和奇异性理论分析该系统在非自治情况下的余维3分岔特性.利用Melnikov方法获得系统在Smale马蹄意义下混沌的阈值.最后通过数值仿真,研究了系统的混沌行为和安全盆分岔,得到安全盆被侵蚀的过程与系统通向混沌的过程之间密切联系.     

谐和激励与有界噪声作用下具有同宿和异宿轨道的Duffing振子的混沌运动

研究了具有同宿轨道、异宿轨道的双势阱Duffing振子在谐和激励与有界噪声摄动下的混沌运动.基于同宿分叉和异宿分叉，由Melnikov理论推导了系统出现混沌运动的必要条件及出现分形边界的充分条件.结果表明：当Wiener过程的强度参数大于某一临界值时，噪声增大了诱发混沌运动的有界噪声的临界幅值，相应地缩小了参数空间的混沌域，且产生混沌运动的临界幅值随着噪声强度的增大而增大.同时数值计算了最大Lyapunov指数，由最大Lyapunov指数为零从另一角度得到了系统出现混沌运动的有界噪声的临界幅值，发现在Wi 关键词： 混沌 同宿和异宿分叉 随机Melnikov方法 最大Lyapunov指数     

一类含Mathieu-Duffing振子的相对转动系统的分岔和混沌

建立了一类具有Mathieu-Duffing振子的两质量相对转动系统的非线性动力学方程. 应用多尺度法求解该系统发生主共振-基本参数共振的分岔响应方程,并通过奇异性分析得到系统稳态响应的转迁集. 利用Melnikov方法讨论系统在外激扰动和参激扰动变化下的全局分岔和系统进入混沌状态的可能途径,得到外激和参激幅值变化下系统可能出现多次通向混沌的道路,获得系统发生混沌的必要条件. 最后采用数值方法验证了理论研究的有效性. 关键词： 相对转动 Mathieu-Duffing振子 混沌 Melnikov方法     

Heteroclinic Bifurcation of Strongly Nonlinear Oscillator

Analytical prediction of heteroclinic bifurcation of the strongly nonlinear oscillator is presented by using the extended normal form method. We consider the approximate periodic solution of the system subject to the quintic nonlinearity by introducing the undetermined fundamental frequency. For the occurrence of heteroclinicity, the bifurcation criterion is accomplished. It depends on the contact of the limit cycle with the saddle equilibrium. As is illustrated, the explicit application shows that the new results coincide very well with the results of numerical simulation when disturbing parameter is of arbitrary magnitude.     

Active control of horseshoes chaos in a driven Rayleigh oscillator with fractional order deflection

The problem of suppressing chaos in the Rayleigh oscillator with fractional order deflection is considered. The explanation of Melnikov?s techniques shows that the dynamic performance and robustness of the system are highly dependent on the fractional order α. The feedback control system is considered as active control strategy. It is revealed with analytical results that periodic perturbation from the controller enhances the performance of the active control strategy. The proposed control strategy is more efficient for deflection order α∈[1.5,2.5] and under super resonant condition between the driven frequency and perturbation frequency. Numerical simulations demonstrate the effectiveness of Melnikov?s analysis.     

Dynamic chaotization of the electronic subsystem in graphene superlattice

d'Alembert equation written for the electromagnetic waves propagating in the graphene superlattice is discussed. The chaotic behavior of the electrons in graphene superlattice is studied by the Melnikov method. Dynamic chaos of electron in graphene superlattice is shown to appear for certain intervals of amplitudes of preset alternate current. The frequency dependence of the critical amplitude of alternate current above which would be a chaos is investigated.     

Using the extended Melnikov method to study the multi-pulse global bifurcations and chaos of a cantilever beam

The aim of this paper is to investigate the multi-pulse global bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended Melnikov method in the resonant case. First, the extended Melnikov method for studying the Shilnikov-type multi-pulse homoclinic orbits and chaos in high-dimensional nonlinear systems is briefly introduced in the theoretical frame. Then, this method is utilized to investigate the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of the cantilever beam. How to employ this method to analyze the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications is demonstrated through this example. Finally, the results of numerical simulation are given and also show that the Shilnikov-type multi-pulse chaotic motions can occur for the nonlinear non-planar oscillations of the cantilever beam, which verifies the analytical prediction.     

Controlling of chaos by weak periodic perturbations in Duffing-van der Pol oscillator

This paper investigates the possibility of controlling horseshoe and asymptotic chaos in the Duffing-van der Pol oscillator by both periodic parametric perturbation and addition of second periodic force. Using Melnikov method the effect of weak perturbations on horseshoe chaos is studied. Parametric regimes where suppression of horseshoe occurs are predicted. Analytical predictions are demonstrated through direct numerical simulations. Starting from asymptotic chaos we show the recovery of periodic motion for a range of values of amplitude and frequency of the periodic perturbations. Interestingly, suppression of chaos is found in the parametric regimes where the Melnikov function does not change sign.     

准周期激励与应变超晶格的动力学稳定性

假设超晶格锯齿形沟道对粒子的作用等效为形状相似的周期场作用. 在经典力学框架内,引入正弦平方势,把粒子运动方程化为具有阻尼项和双频激励项的摆方程. 用Melnikov方法对单频激励系统的分叉与混沌进行分析;用Lyapunov方法对双频激励系统的稳定性进行讨论. 结果表明：在弱非线性情况下,双频激励系统存在局域不稳定,且这种不稳定将向全局扩展,直至混沌的出现;导致混沌的双频激励强度远小于单频激励强度;外加一个适当的超声场可望将这种敏感钝化,使系统的稳定性得到改善. 关键词： 超晶格 准周期激励 混沌 稳定性     

基于扩展型Duffing振子的局部放电信号检测方法研究

目前, 小波阈值去噪法、数字滤波法、傅里叶频域变换法等常用的微弱信号检测方法所能达到的最低检测信噪比为-10 dB, 而双向环形耦合Duffing振子能达到的最低检测信噪比为-20 dB. 但是, 现场检测时常常会出现更低信噪比的放电脉冲信号, 因此现有检测方法就很难满足信号检测的实际需求. 为了有效解决该难题, 研究了一种扩展型Duffing振子的微弱脉冲信号检测的新方法. 该方法的主要思想是使用广义时间尺度变换, 将Duffing振子模型变换为扩展型Duffing振子模型, 有效扩展了微弱信号的频率检测范围. 仿真结果表明, 扩展型Duffing振子不仅具有良好的噪声免疫特性, 而且能有效检测到信噪比低至-40 dB的局部放电微弱脉冲信号, 进一步扩展了现有Duffing振子微弱信号检测方法的检测范围和应用领域.     

On the nonlinear on–off dynamics of a butterfly valve actuated by an induced electromotive force

In this paper, we study the nonlinear dynamics of a butterfly valve actuated by the induced electromotive force (emf) of a permanent magnet, with a focus on the on–off dynamics of the valve and its nonlinear response under ambient perturbation. The complex interplay between the electromagnetic, hydrodynamic and mechanical forces leads to a fundamentally multiphysical, nonlinear dynamical model for the problem. First, we analyze the stability of the on–off conditions in terms of three critical dynamical parameters – the actuating DC voltage, inlet velocity and the opening angle. Next, the response of the system to perturbations around the equilibrium points is studied in terms of the frequency response using the method of multiple scales. Finally, evidence of fractality is established using Melnikov analysis and a plot of the basins of attraction. The results reported in the article, in addition to being of fundamental theoretical interest, are expected to impact practical design considerations of electromechanical butterfly valves.     

Chaotic behaviour from smooth and non-smooth optical solitons under external perturbation

Smooth and non-smooth optical solitons in the nonlinearly dispersive Schrödinger equation are given by phase portraits. The Melnikov technique is used to detect conditions for chaotic motion of this deterministic system and to analyse conditions for the suppression of chaos. Our results show that the system is in a state of Melnikov chaos by external disturbances. After the implementation of the controlled system, the optical solitons can transmit in a stable station for a long time. Numerical simulation also shows that maximum interference frequency of the system enables the dynamic behaviour to be more complex. The effect of controller parameter on phase portraits as well as on the numerical simulations of bifurcation diagram and maximum Lyapunov exponents are also investigated.