【作者】 陈强；

【导师】 赵晓华；

【作者基本信息】 浙江师范大学 ， 应用数学， 2009， 硕士

【摘要】 随着人类认识、改造和利用自然的能力的不断提高,以及实际应用的需要,人们面临大量非线性问题的处理。Hamilton系统是非线性科学研究中的一个重要领域,它的产生与发展具有深刻的实际背景。Hamilton系统广泛存在于物理科学、生命科学及工程科学等众多领域,特别是经典力学、天体力学、航空航天科学以及生物工程的很多模型都以Hamilton系统的形式出现。经典意义下的Hamilton系统都是在偶数维相空间上定义的,其相空间具有很好的辛流形结构性质,在这种框架下发展的Hamilton系统理论已成为动力系统理论的重要组成部分。为了使Hamilton系统的观点和方法能应用于实际问题研究中广泛存在的奇数维系统,人们对经典的Hamilton系统进行了扩展,从而提出了广义Hamilton系统的概念。广义Hamilton系统作为经典Hamilton系统的推广,可以作为描述包括奇数维系统在内的更加广泛的非线性动力学问题的模型,在机械工程,光学,分子动力学等等中都有很多的应用;其中有很大的一类是具有球面叶层结构的广义Hamilton系统。在许多实际问题研究的模型中,这种具有球面叶层结构的Hamilton系统模型,其Hamilton函数常常以二次函数的形式居多。对它的研究可以归结为研究下面五类Hamilton函数对应的广义Hamilton系统:迄今为止,对前四种情况,相应的广义Hamilton系统在球面叶层上的平衡点分叉及其全局相图已被完全研究清楚.但对Hamilton函数中含四个参数的第五种情况,其对应的广义Hamilton系统在球面叶层上的分叉及相图还未见到相关研究.基于上述,本文利用微分方程定性理论与动力系统分叉理论(特别是Hamilton系统相图分析技巧)研究了第五类Hamilton函数对应的具有球面叶层的广义Hamilton系统,仔细分析了平衡点分叉及稳定性性质,获得了完整的全局相图分类。更多还原

【Abstract】 With the persistent improvement of the ability of human being of the recognizing ,reconstructing and utilizing the nature ,as well as the needs of practical apply, people have to face to deal with huge nonlinear problems. Hamiltonian system is an impor-tant field of nonlinear science study, whose creation and development have a profound practical background.As a kind of generalization of classical Hamiltonian system, generalized Hamilton system can be used to study many nonlinear dynamical models with any dimension including odd dimension, which come from divers science fields, such as mechanical engineering, optics, and molecular dynamics,et al.From the existing results of practical models , we note that generalized Hamilton system with sphere foliation structure and quadratic Hamiltonian function often appears as dynamical models of many real problems. It has been shown that such systems can be reduced the following five cases: Previous researches had devoted to the first four cases and obtained the complete re-sults on the corresponding Hamiltonian systems. Up to the present,however, to Hamil-ton function with four-parameters, the bifurcation properties and phase portraits of the corresponding systems are yet to be studied.Based on the facts mentioned above , by making use of the qualitative theories of differential equation and the bifurcation theory of dynamics system (especially Hamil-tonian system phase diagram analyze technique), this paper study the four-parameter Hamiltonian system (the fifth case) and analyze the bifurcation and stability of equilib-ria in detail, obtain the classification of phase portraits on sphere.更多还原