非线性微分方程奇异摄动系统及边值问题

项目名称： 非线性微分方程奇异摄动系统及边值问题

项目编号： No.11471146

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 杜增吉

作者单位： 江苏师范大学

项目金额： 68万元

中文摘要： 非线性微分方程奇异摄动系统和边值问题是当前一个非常活跃的课题,具有重要的应用背景，本项目拟开展如下研究： 1、研究复杂奇异摄动微分系统、快慢动力系统和高维动力系统的同异宿轨道、分支及混沌现象，讨论非线性时滞微分方程奇异摄动系统某些复杂性质。2、运用几何奇异摄动理论和动力系统分支理论，研究含有小时滞的KdV方程和Schr?dinger方程行波解的存在性以及精确行波解的性质。3、运用非线性分析理论和中心流形定理等研究具有扰动的多时滞种群捕食竞争系统和扰动反应扩散方程的稳定性、分岔周期解稳定性和全局渐近行为等复杂性质。4、运用奇异摄动方法、Morse 理论研究非线性微分方程奇异摄动非局部边值问题渐近解的存在性、唯一性等。 本项目将对非线性微分方程奇异摄动系统和边值问题的研究发展起促进作用，具有重要的意义。

中文关键词： 非线性微分方程；奇异摄动系统；边值问题；定性理论

英文摘要： Nonlinear singularly perturbed systems and boundary value problems of differential equations are very active subjects in mathematics, which have important application in the background. We will study the following research in the project： Firstly, we will study homoclinic orbit, heteroclinic orbits, bifurcations and chaotic phenomena of complex singularly perturbed differential systems, fast-flow systems and high-dimensional dynamical systems. We also study nonlinear singularly perturbed systems with time delays and reveal some complex properties. Secondly, by applying geometric singular perturbation theory and bifurcations theory, we study the existence of traveling wave solutions and the properties of exact traveling wave solutions for KdV equations and Schr?dinger equation with small delays. Thirdly, by employing nonlinear analysis theory and center manifold theory, we study the complex dynamical properties, such as stability, bifurcations of periodic solution and global asymptotic stability for perturbed multispecies competition-predator systems with multiple time delays and reaction-diffusion equations with perturbations. Finally, by using singularly perturbation methods and Morse theory, we study the existence and uniqueness of perturbed solutions for nonlocal singularly perturbed boundary value problems of differential equations. This project will develop nonlinear singularly perturbed systems and boundary value problems of differential equations and has great significance.

英文关键词： nonlinear differential equations;singularly perturbed systems;boundary value problems;qualitative theory