非线性分段连续型微分系统数值方法的分支相容性研究

项目名称： 非线性分段连续型微分系统数值方法的分支相容性研究

项目编号： No.11201084

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 王琦

作者单位： 广东工业大学

项目金额： 22万元

中文摘要： 延迟微分系统的分支相容性是近年来数值分析领域的热点研究课题。作为延迟微分系统的一部分，分段连续型微分系统在控制科学、生物学和力学等领域有着广泛的应用，其数值方法的分支相容性具有毋庸置疑的重要性。本项目以带一个[t]的非线性分段连续型微分系统、混合型非线性分段连续微分系统和交替向前与滞后型非线性分段连续微分系统为研究对象，综合运用规范型理论、中心流形定理和改进的离散Hopf分支理论等工具，不仅考查平衡点的稳定性和Hopf分支点的存在性等系统本身的分支行为，而且研究数值方法保持系统分支结构的条件，如Neimark-Sacker分支参数的取值与数值逼近、分支方向及其稳定性等。该项目的研究将揭示取整函数影响分段连续型微分系统分支行为的内在机理，对于建立分段连续型微分系统分支相容性问题的理论框架具有重要意义。本研究将丰富延迟微分系统数值分析的内涵，具有重要的理论价值和广泛的应用前景。

中文关键词： 分段连续项；非线性延迟微分方程；数值解；稳定性；振动性

英文摘要： The bifurcational consistency of delay differential systems is a hot issue in numerical analysis in recent years. As a part of delay differential systems, differential equations with piecewise continuous arguments (EPCA) have been widely applied in control science, biology and mechanics, etc. Hence there is no doubt that the bifurcational consistency of numerical methods for EPCA is of great importance. This project treats the nonlinear EPCA with one [t], the nonlinear EPCA of mixed type and the nonlinear EPCA of alternately advanced and retarded type as the research objects. In this project, the normal form theory, the center manifold theorem and the improved discrete Hopf bifurcation theory are all adopted. We not only investigate the bifurcation behavior of EPCA concluding of the stability of equilibrium point and the existence of Hopf bifurcation point, but also study the conditions under which the numerical methods preserve bifurcation structure, such as the value and numerical approximation of Neimark-Sacker bifurcation parameter and the direction of the Neimark-Sacker bifurcation and their stability. The project will reveal the inner mechanism in which the integer function influences the bifurcation behavior of EPCA. And the achievements of this research are of great significance for building the the

英文关键词： piecewise continuous arguments；nonlinear delay differential equations；numerical solution；stability；oscillation