几类非线性系统的复杂动力学行为研究

发布时间：2018-12-17 00:45

【摘要】：薄板、梁等结构模型广泛存在于实际工程问题中,这些实际系统模型大多是由高维甚至无穷维非线性动力学方程所描述的,数值研究可以揭示其复杂的非线性动力学行为。但是处理高维非线性系统在理论方法和空间几何描述上却有难度。随着非线性动力系统理论从低维到高维的进一步发展,研究高维非线性系统的复杂动力学问题已经成为当今非线性科学中的重要课题。本文将规范型方法、全局摄动法、能量相位法、广义Melnikov方法等解析方法应用到几类高维非线性系统模型中,来揭示其复杂的动力学行为,包括稳定性、分岔、次谐分岔、混沌运动。论文的主要研究内容包括以下几个方面。(1)研究了受横向激励的功能梯度材料悬臂板模型在1:1内共振和1/2亚谐共振下的稳定性和局部分岔行为。主要讨论了四种类型临界平衡点的分岔问题,视(μ1,μ2)为扰动参数,得到了初始平衡解的稳定区域和分岔曲线,随着扰动参数的变化,系统可能发生静态分岔和Hopf分岔。数值模拟得到了各种情形下初始平衡解及分岔解在稳定区域内的相图,展示了该类功能梯度材料悬臂板模型在不同参数下的运动状态。(2)利用全局摄动法和能量相位法研究了对称铺设复合材料悬臂板的全局分岔和混沌动力学行为。得到了系统Shilnikov型单脉冲同宿轨和多脉冲同宿轨存在的充分条件,而这类轨道的存在可以导致系统发生Smale马蹄意义下的混沌。理论和数值的结果都表明阻尼和外激励幅值对系统的混沌运动有重要的影响。(3)研究了受面内激励和横向激励的桁架夹芯板在1:2内共振下的全局分岔和混沌动力学行为。利用全局摄动法得到系统存在Shilnikov型单脉冲同宿轨的若干参数条件;利用广义Melnikov方法得到了在共振情况下桁架夹芯板产生多脉冲混沌的Melnikov函数,从理论上给出系统产生多脉冲混沌的判据。(4)研究了两端固定受横向激励的梁在1:1内共振下非线性动力学行为。得到了系统在Hamilton共振带上,存在着几类奇异同宿轨和异宿轨。这些奇异轨道是由系统慢流形和快流形上的轨道交替形成的。(5)利用Melnikov方法,研究了弹性梁的次谐分岔和混沌行为。得到系统出现次谐分岔和超次谐分岔的参数条件,给出系统混沌区域和非混沌区域的分界曲线。
[Abstract]:Thin plate, beam and other structural models are widely used in practical engineering problems. Most of these practical system models are described by high dimensional or even infinite dimensional nonlinear dynamic equations. Numerical studies can reveal their complex nonlinear dynamic behaviors. However, it is difficult to deal with high-dimensional nonlinear systems in terms of theoretical methods and spatial geometric descriptions. With the further development of the theory of nonlinear dynamical systems from low dimension to high dimension, the study of complex dynamics of high dimensional nonlinear systems has become an important subject in nonlinear science. In this paper, analytical methods such as normal form method, global perturbation method, energy phase method and generalized Melnikov method are applied to several kinds of high dimensional nonlinear system models to reveal their complex dynamic behaviors, including stability, bifurcation, subharmonic bifurcation, etc. Chaotic motion. The main contents of this paper are as follows: (1) the stability and local bifurcation behavior of functionally graded material cantilever plate model subjected to transverse excitation at 1:1 and 1 / 2 subharmonic resonance are studied. The bifurcation problem of four types of critical equilibrium points is discussed in this paper. (渭 1, 渭 2) is regarded as the perturbation parameter. The stable region and bifurcation curve of the initial equilibrium solution are obtained. With the change of the disturbance parameters, the system may have static bifurcation and Hopf bifurcation. The phase diagrams of the initial equilibrium solution and the bifurcation solution in the stable region are obtained by numerical simulation. The motion states of this kind of functionally graded material cantilever plates under different parameters are demonstrated. (2) the global bifurcation and chaotic dynamics of symmetric composite cantilever plates are studied by using global perturbation method and energy phase method. Sufficient conditions for the existence of Shilnikov type monopulse homoclinic orbits and multipulse homoclinic orbits are obtained, and the existence of such orbits can lead to chaos in the sense of Smale horseshoe. The theoretical and numerical results show that the damping and the amplitude of external excitation have important effects on the chaotic motion of the system. (3) the global bifurcation of truss sandwich plates subjected to in-plane and transverse excitation at 1:2 resonance is studied. Chaotic dynamics. By using the global perturbation method, some parameter conditions for the existence of Shilnikov type monopulse homoclinic orbit are obtained. In this paper, the generalized Melnikov method is used to obtain the Melnikov function of multi-pulse chaos for truss sandwich panels under resonance. A criterion for the generation of multi-pulse chaos in the system is given theoretically. (4) the nonlinear dynamical behavior of a beam with transverse excitation at both ends under resonance within 1:1 is studied. It is found that there are several kinds of singular homoclinic orbits and heteroclinic orbits in the Hamilton resonance band. These singular orbits are formed by alternating the orbits on the slow manifold and the fast manifold. (5) the subharmonic bifurcation and chaotic behavior of the elastic beam are studied by using the Melnikov method. The parameter conditions of subharmonic bifurcation and hypersubharmonic bifurcation are obtained, and the boundary curves of chaotic region and non-chaotic region of the system are given.
【学位授予单位】：南京航空航天大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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