右端不连续泛函微分方程的复杂动力学行为及其应用

项目名称： 右端不连续泛函微分方程的复杂动力学行为及其应用

项目编号： No.11501221

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 汪东树

作者单位： 华侨大学

项目金额： 18万元

中文摘要： 本项目利用泛函微分包含将右端不连续泛函微分方程正则化，研究右端不连续泛函微分方程解的一些基本性质、复杂动力学行为及其应用。主要研究内容包括：右端不连续泛函微分方程初值问题解的局部存在性和整体存在性、解的有界性、平衡点（多个平衡点）、周期解（多个周期解）、概周期解的存在性与各种稳定性、解轨线的有限时间收敛性等复杂动力学行为。主要研究方法包括：综合运用集值分析理论、泛函微分包含理论、非光滑分析理论、非光滑临界点理论等现代数学工具，并发展一些右端不连续泛函微分方程定性和稳定性理论研究的新方法。并利用这些新发展的方法与理论，来研究神经网络、生物数学等领域中一些用右端不连续泛函微分方程所刻画的数学模型。这些研究不仅丰富和发展了右端不连续泛函微分方程的基本理论，而且为分析和解决众多具不连续因素影响的实际问题提供有效方法和理论依据。

中文关键词： 不连续系统；泛函微分包含；复杂动力学行为；集值分析；非光滑分析

英文摘要： In this thesis, in order to investigate some fundamental questions, complex dynamical behaviors and applications for functional differential equations with discontinuous right-hand sides, functional differential equations with discontinuous right-hand sides were regularized in view of functional differential inclusions. For functional differential equations with discontinuous right-hand sides, the main topics include, the local and global existence of solutions for initial valued problems, boundedness of solutions, the existence of equilibrium (multiple equalibria), periodic solution (multiple periodic solutions), and almost periodic solution, different kinds of stability for solutions, the convergence behaviors in finite time of the solutions, and so on. Our methods to be used involve the applications of set-valued analysis theory, functional differential inclusions theory, nonsmooth analysis theory and nonsmooth critical point theory, and some others. And some new methods are introduced to deal with the quantitative theory and stability theory of functional differential equations with discontinuous right-hand sides. By applying these new methods and theoretical results, we formulate different kinds of mathematical models in many areas, such as neural networks and mathmatical biology, which described by functional differential equations with discontinuous right-hand sides. This research not only enriches and develops some basic theory of differential equation with the discontinuous right-hand sides, but also provides effective method and theoretical basis to solve many practical problems with discontinuous character.

英文关键词： Discontinuous system;Functional differential inclusion;Complex dynamic behavior;Set-value analysis;Nonsmooth analysis