高维非线性系统的分支问题与仿真研究

项目名称： 高维非线性系统的分支问题与仿真研究

项目编号： No.61473340

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 自动化技术、计算机技术

项目作者： 李先义

作者单位： 浙江科技学院

项目金额： 58万元

中文摘要： 非线性系统分支问题的研究,一直是动力系统研究热点中的一项困难的工作。比较而言，高维非线性系统更容易出现比低维系统更加复杂的分支问题，如同宿、异宿分支，zero-Hopf分支, Hopf-Hopf分支, Flip-Flip分支, Fold-Flip分支等等。这些分支所产生闭轨的稳定性判别问题等是极端复杂而又困难的工作。本项目主要采用中心流形理论、分支理论、临界点理论, 以及投影法、Poincare Compactificatin、计算机仿真等方法与手段,一方面研究连续高维非线性系统（主要是各种各样的广义Lorenz系统以及一些生物生态数学模型）的各类分支问题（包括局部分支、全局分支和无穷远分支）；另一方面将研究相应连续模型离散化的分支问题。通过把连续模型与其相应的离散模型的分支问题进行对比研究，从分支理论的角度去揭示两者之间的差异。本项目的研究将进一步完善和发展非线性系统的分支理论和方法。

中文关键词： 高维非线性系统；分支问题；稳定性；系统分析；计算机仿真

英文摘要： The investigations for bifurcation problems of nonlinear systems have been always difficult work among hot points of research in dynamical sysytems. Comparably speaking, it is more easy for nonlinear systems with higher dimension to produce more complicate bifurcation problems than systems with lower dimension, such as homoclinic heteroclinic bifurcatio, zero-Hopf bifurcation, Hopf-Hopf bifurcation, Flip-Flip bifurcation, Fold-Flip bifurcation, etc.The problems for the stability discriminant of close obits bifurcated are extremly complicate and difficult. In this project, by using the center manifod theory, the bifurcation theory, the critical point theory, and some methods and means such as the Project Method, the Poincare Compactification, and numerical simulation, etc, on the one hand one will study various bifurcation problems for continuous nonlinear systems with higher dimension, where the systems are mainly the generalized Lorenz systems and some mathematical models for biology and ecology, inluding local bifurcation, global bifurcation and infinte bifurcation. On the other hand one will consider the bifurcation problems for the corresponding discrete models. In light of the comparison investigations of bifurcation problems for continuous models and their corresponding discrete models, the difference between them will be disclosed from th point of view of bifurcation theory. The investigations in this project will further improve and develop the theory and method of bifurcation for nonliear systems.

英文关键词： Nonlinear system with higher dimension;Bifurcation problem;Stability;System analysis;Computer simulation