【作者】 刘丽静；

【导师】 王知人；

【作者基本信息】 燕山大学 ， 运筹学与控制论， 2008， 硕士

【摘要】 分岔和混沌是非线性系统最重要而又最基本的特性,几乎在所有涉及非线性科学的领域中,都存在着分岔现象和混沌运动。论文在分析和总结非线性微分动力系统分岔和混沌研究现状的基础上对电磁场中的细长压杆和导电梁式板的非线性行为进行了研究。论文主要研究了以下几方面的内容首先,总结了当前国内外学者对分岔现象及混沌运动问题的研究现状,阐述了产生叉形分岔、Hopf分岔和鞍结分岔三种平衡点分岔和闭轨分岔、同宿轨(异宿轨)分岔的条件。给出了Hamilton系统的同宿轨及异宿轨的计算方法。阐述了Melnikov方法,并给出了几种常用的Melnikov函数。其次,对电磁场中细长压杆进行受力分析,得到了电磁场作用下细长压杆的运动方程,并针对静力学、线性动力学和非线性动力学三种模型,进行了分岔特性研究。最后,用Melnikov方法研究了在横向磁场和机械载荷共同作用下的梁式板的非线性行为,首先对无扰动系统进行了分析,由无扰系统是Hamilton系统,根据其Hamilton能量函数求得了它的闭轨道参数方程,并给出了振幅和其系统能量之间的关系,并进一步计算了扰动系统出现次谐分岔的Melnikov函数,由Melnikov函数存在简单零点得知该系统存在Smale马蹄变换意义下的混沌,并进一步得到存在混沌运动时横向磁场强度与外界机械力幅之间的关系。更多还原

【Abstract】 Bifurcation and Chaos is the most important and most basic features in nonlinear systems,almost all involved in the field of nonlinear science, there is a bifurcation phenomena and chaotic motion. Based on the analysis and summary of nonlinear differential bifurcation and chaotic dynamical systems on the basis of the status quo in the electromagnetic conductivity slender bar and beam-plate nonlinear behavior were studied. This paper mainly on the following aspects of content.Firstly, summed up the current domestic and foreign scholars on the bifurcation phenomenon of chaotic motion and the issue of the status quo, have fork bifurcation expounded, Hopf bifurcation and saddle-node bifurcation balance bifurcation and three closed orbit bifurcation, homoclinic orbit (heteroclinic track) bifurcation conditions. Hamilton systems and gives the homoclinic tracks and the heteroclinic orbit calculation. Melnikov function briefly introduced the solution.Secondly, the electromagnetic field in a slender bar Analysis, have been under slender bar electromagnetic field equations of motion and against static mechanics, linear dynamics and nonlinear dynamics three models, a study of bifurcation.Finally, Melnikov method used in the horizontal magnetic field and mechanical loads under common beam-plate nonlinear behavior, first of all, on non-disturbance system were analyzed by the system is unperturbed Hamiltonian system, in accordance with its energy function Hamilton obtained its close orbital parameters equation, and gives the amplitude and its system the relationship between volume and further disturbances system in the calculation of the bifurcation of the market Melnikov function, the simple existence of Melnikov function of the system that existed Smale horseshoe transform the sense of chaos, and further chaos when transverse magnetic field presence with the outside world mechanical strength of the relationship between the rate.更多还原