Duffing方程的静态与动态分岔特性研究

对于Duffing方程的静态和全局动态分岔,通过研究平均方程的全局行为得到了出现各种分岔的条件,揭示了Duffing方程周期解的变化过程及其具有的非线性动力学性质。     

双重内共振系统非线性模态分岔的奇异性分析

利用多尺度法构造的一类1:2:5双重内共振系统的耦合非线性模态的分岔是一个两变量的分岔问题.利用Maple计算机代数可以通过消元将耦合的模态分岔方程分离为两个单变量的分岔方程.对分离后的单变量分岔方程进行奇异性分析,发现随着系统参数的变化,非线性模态的分岔既可以是一种模态向另一种模态的转化,也可以是一种模态的突然出现与消失.最后给出了两变量分岔问题可以利用消元后得到的单变量分岔方程和耦合方程进行处理的一种方法.     

弹性约束浅拱的非线性动力响应与分岔分析

浅拱采用竖向、转动方向弹性约束时,自振频率和模态与理想的铰支/固结边界存在差异,不同约束刚度将改变外激励下的非线性响应及各种分岔产生的参数域．由浅拱基本假定建立无量纲动力学方程, 采用在频率和模态中考虑约束刚度大小的方法,通过Galerkin全离散和多尺度摄动分析导出极坐标、直角坐标形式的平均方程, 其中方程系数与约束刚度一一对应．用数值方法分析了周期激励下竖向弹性约束系统最低两阶模态之间1∶2内共振时的动力行为, 所得结果与有限元的对比以及平均方程系数的收敛性证明了所采用方法是可行的．随着激励幅值、频率的变化存在若干分岔点,分岔发生时的参数分布与约束刚度值有关,在由分岔点连接的不稳定区或共振区附近,存在一系列稳态解、周期解、准周期解和混沌解窗口,且随参数的变化可观测到倍周期分岔．     

流体诱发水平悬臂输液管的内共振和模态转换(Ⅱ)

基于得到的水平悬臂输液管非线性动力学控制方程,详细研究了由流速最小临界值诱发的3∶1内共振．通过观察内共振调谐参数、主共振调谐参数和外激励幅值的变化,发现在内共振临界流速附近,流速导致系统出现模态转换、鞍结分岔、Hopf分岔、余维2分岔和倍周期分岔等非线性动力学行为,对应的管道系统的周期运动失稳出现跳跃、颤振和更加复杂的动力学行为．通过理论结果与数值模拟比较,表明了理论分析的有效性和正确性．     

两自由度非对称三次系统非线性模态的奇异性质

利用非线性模态子空间的不变性和摄动技术,研究两自由度非对称三次系统在奇异条件下系统的性质.重点考虑子系统之间线性耦合退化时的奇异性质.对于非共振情形,所得到的解析结果表明,系统出现单模态运动以及振动局部化现象,这种现象的强弱不但与非线性耦合刚度有关,而且与非对称参数有关.并解析地得到了参数的门槛值;对于1:1共振情形,模态随非线性耦合刚度和非对称参数的变化会出现分岔,得到了参数分岔集以及模态的分岔曲线.     

两自由度非线性振动系统主参数激励下的分岔分析

本文给出了参数激励作用下两自由度非线性振动系统,在1:2内共振条件下主参数激励低阶模态的非线性响应.采用多尺度法得到其振幅和相位的调制方程,分析发现平凡解通过树枝分岔产生耦合模态解,采用Melnikov方法研究全局分岔行为,确定了产生Smale马蹄型混沌的参数值.     

多频激励Duffing系统的分岔和混沌

本文通过引入非线性频率，利用Ｆｌｏｑｕｅｔ理论及解通过转迁集时的特性，研究了不可通约两周期激励作用下的Ｄｕｆｉｎｇ方程在一次近似下的各种分岔模式及其转迁集，并指出其通向混沌可能的途径·     

电池系统建模中Butler-Volmer方程的同伦分析求解

Butler-Volmer方程是电化学系统中描述电极动力学过程的本构方程,具有强非线性．为了对这一方程(耦合两个Ohm方程)进行解析求解,在同伦分析方法的框架下,发展了满足简单条件的广义非线性算子的算法,以取代原同伦分析中的非线性算子．该广义非线性算子的构造保证了高阶形变方程的线性特征．这一方法的有效性通过一些算例得到了验证．最后通过同伦分析方法对Butler-Volmer方程进行了求解,结果显示过电位和电流密度的级数解析解与数值解吻合很好,并有很好的收敛效率．     

复合Ginzburg-Landau方程的动力学行为分析

目前对非线性波动方程的研究大都仅限于静态波解,即所考虑的波解的波速、振幅、波宽都是不变的,考虑动态波解,以复合Ginzburg-Landau(CGLE)方程为研究对象,探讨其动力学行为.在假设示性函数的基础上,所研究的无穷维耗散系统转化为三维向量场,给出了简单分岔和Hopf分岔存在的条件,揭示了系统平衡点和极限环随系统参数的变化规律,分析了参数平面的不同区域中系统的相图特性,得到系统存在两种不同频率的周期解,此外还数值模拟了系统由倍周期分岔导致混沌的过程,揭示了系统的复杂性.     

匀速转动内悬臂梁系统的建模与分岔研究

应用广义Hamilton原理建立旋转刚环内接悬臂梁系统在材料本构关系为线性时Rayleigh梁大挠度非线性动力学模型,并讨论当假定梁为Euler-Bernoulli型时的蜕化结果.研究表明,上述两种模型对系统的临界分岔值及分岔解静态没有影响.据此,采用了假设模态法解析地研究了匀速转动的Euler-Bernoulli梁模型的分岔行为,得到两种在物理上存在的分岔类型.为校验解析分析的结果,使用了有限元及打靶法两种数值研究工具.计算结果肯定了研究结论.     

圆柱贮箱液体非线性晃动的多维模态分析方法

将Faltinsen等提出的多维模态理论应用到求解圆柱贮箱液体非线性晃动问题中．根据Narimanov-Moiseiev的三阶渐近假设关系,通过选取主导模态以及确定它们的阶次关系,将一般形式的无穷维模态系统降为五维渐近模态系统,即描述自由液面波高的广义坐标之间相互耦合的二阶非线性常微分方程组．通过对这个模态系统的数值积分,得到了与以前的理论分析和实验结果相吻合的非线性现象．研究结果表明,多维模态方法是用来求解液体非线性晃动动力学的一个很好的工具．在我们的下一步工作中,将继续发展这种方法,用来研究更为复杂的晃动问题．     

An invariant manifold approach to nonlinear normal modes of oscillation

Summary A method for determining the amplitude-dependent mode shapes and the corresponding modal dynamics of weakly nonlinear vibratory systems is described. The method is a combination of a Galerkin projection and invariant manifold techniques and is applied to a class of distributed parameter vibratory systems. In this paper the general theory for a class of conservative systems is outlined and applied to determine the nonlinear mode shapes and modal dynamics of a linear Euler-Bernoulli team attached to a nonlinear elastic foundation.     

Static collapse: A survey of methods and modes of behavior

The phenomenon of static collapse, henceforth called ‘buckling’, is first illustrated by the behavior of a fairly thick cylindrical shell, which under axial compression deforms at first axisymmetrically and later nonaxisymmetrically. Thus, static buckling encompasses two modes of behavior, nonlinear collapse at the maximum point in a load versus deflection curve and bifurcation buckling. Accurate prediction of critical loads corresponding to either mode in the plastic range of material behavior requires a simultaneous accounting for moderately large deflections and nonlinear, irreversible, path-dependent material. A survey is given of plastic buckling, which spans three areas: asymptotic analysis of postbifurcation behavior of perfect and imperfect simple structures, general nonlinear analysis of arbitrary structures, and nonlinear analysis for collapse at a maximum load and bifurcation buckling of shells of revolution. In the survey of general nonlinear structural analysis, some emphasis is given to strategies for solving the governing nonlinear equations incrementally. Numerous examples, generated primarily with the STAGS computer program, which was developed by Almroth and his colleagues, reveal many complex modes of buckling behavior.     

Parametric and Autoparametric Resonance

A large number of mathematical questions are related to problems of parametric and autoparametric resonance in engineering models. The linearized problems generally produce systems of differential equations with periodic coefficients with special stability and genericity questions. We start by reviewing linear systems while discussing normal form techniques and bifurcation results. The linear and nonlinear analysis is illustrated in three cases: rotor dynamics, autoparametric resonance of a parametric oscillator and autoparametric resonance of a self-excited oscillator. In all cases bifurcations, symmetry considerations and attraction to nonclassical limit sets play a part.     

Voltage transformer ferroresonance analysis using multiple scales method and chaos theory

In this article, the Multiple Scales Method is used to analyze the chaotic behavior and different types of fixed points in ferroresonance of voltage transformers considering core loss. This phenomenon has nonlinear chaotic dynamics and includes subharmonic, quasi‐periodic, and also chaotic oscillations. In this article, the chaotic behavior and various ferroresonant oscillations modes of the voltage transformer is studied. This phenomenon consists of different types of bifurcations such as Period Doubling Bifurcation (PDB), Saddle Node Bifurcation (SNB), Hopf Bifurcation (HB) and chaos. The dynamic analysis of ferroresonant circuit is based on bifurcation theory. The bifurcation and phase plane diagrams are illustrated using a continuous method and linear and nonlinear models of core loss. To analyze ferroresonance phenomenon, the Lyapunov exponents are calculated via Multiple Scales Method obtaining Feigenbaum numbers. The bifurcation diagrams illustrate the variation of the control parameter. Therefore, the chaos is created and increased in the system. © 2013 Wiley Periodicals, Inc. Complexity 18: 34‐45, 2013     

Symmetry breaking, coupling management, and localized modes in dual-core discrete nonlinear-Schrödinger lattices

We introduce a system of two linearly coupled discrete nonlinear Schrödinger equations (DNLSEs), with the coupling constant subject to a rapid temporal modulation. The model can be realized in bimodal Bose-Einstein condensates (BEC). Using an averaging procedure based on the multiscale method, we derive a system of averaged (autonomous) equations, which take the form of coupled DNLSEs with additional nonlinear coupling terms of the four-wave-mixing type. We identify stability regions for fundamental onsite discrete symmetric solitons (single-site modes with equal norms in both components), as well as for two-site in-phase and twisted modes, the in-phase ones being completely unstable. The symmetry-breaking bifurcation, which destabilizes the fundamental symmetric solitons and gives rise to their asymmetric counterparts, is investigated too. It is demonstrated that the averaged equations provide a good approximation in all the cases. In particular, the symmetry-breaking bifurcation, which is of the pitchfork type in the framework of the averaged equations, corresponds to a Hopf bifurcation in terms of the original system.     

A bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting

In this paper, we propose a bioeconomic differential algebraic predator–prey model with Holling type II functional response and nonlinear prey harvesting. As the nonlinear prey harvesting is introduced, the proposed model displays a complex dynamics in the predator–prey plane. Taking into account of the economic factor, our predator–prey system is established by bioeconomic differential algebraic equations. The effect of economic profit on the proposed model is analyzed by viewing it as a bifurcation parameter. By jointly using the normal form of differential algebraic models and the bifurcation theory, the stability and bifurcations (singularity induced bifurcation, Hopf bifurcation) are discussed. These results obtained here reveal richer dynamics of the bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting, and suggest a guidance for harvesting in the practical word. Finally, numerical simulations are given to demonstrate the results.     

Nonlinear dynamics and bifurcations in external feedback control of microcantilevers in atomic force microscopy

We study nonlinear dynamics and bifurcations in external feedback control of vibrating microcantilevers in atomic force microscopy. The efficiency and validity of the control methodology for microcantilevers was demonstrated and abundant nonlinear phenomena were observed in a previous numerical study. Using the averaging method and center manifold theory, we analyze a degenerate bifurcation of codimension two to explain a key feature of the previous numerical results. Numerical examples are also given, in which theoretical results are compared with numerical computations for bifurcations and unstable manifolds.     

飞行器内裂纹转子系统的非线性动力学特性研究

在Jeffcott转子的开闭裂纹及方波模型基础上,建立了飞行器内裂纹转子系统的运动模型．数值研究表明:当飞行器以不同的等速度飞行时,转子轴与水平面之间夹角的变化将造成重力分量的变化,从而使转子运动在周期解、拟周期或浑沌状态之间变化,而且出现非线性现象的转速比、刚度变化比等参数的范围、进入和退出浑沌的路径、响应中的频率成份也会发生变化．飞行器的飞行速度变化还会改变裂纹转子响应的稳定性．飞行器等速飞行后的加速过程将引起转子振幅的突升及其后的下降,而且会使裂纹转子系统响应可能在不同的非线性状态下交替改变．