非线性分段连续型微分系统数值方法的分支相容性研究

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 theoretical framework of bifurcation problem of EPCA. The study of this project will enrich the connotation of the numerical analysis of delay differential systems, and hence has important theoretical value and broad application prospect.

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。

分段连续型微分系统在控制、生物、信息和生理学等领域有着广泛的应用，故对其进行数值求解具有重要的理论意义和应用价值。本课题针对两种类型的延迟微分系统，分别研究了数值解的相关性质，获得了一些重要的结论。首先，对于几类分段连续型微分系统（包括向前型分段连续微分系统，带一个延迟项的分段连续型微分系统，混合型分段连续微分系统，分段连续型偏微分系统和分段连续型随机微分系统），运用相应的数值方法（包括Runge-Kutta方法，Euler-Maclaurin方法，θ-方法和Euler-Maruyama方法）进行离散，对于前几类系统，给出了收敛阶，获得了数值解稳定区域包含解析解稳定区域的充要条件，由解析解振动的条件获得了数值方法保持原系统振动和非振动性的结论，进一步揭示了稳定性和振动性之间的关系；对于偏微分系统，获得了解析解和数值解渐近稳定的充分条件；对于随机系统，得到了数值解均方稳定和T-稳定时步长所满足的条件；其次，对于非线性和中立型延迟微分系统，应用θ-方法进行求解，分析了数值解的振动性和非振动数值解的渐近行为。以上理论结果都通过了数值实验的验证，取得了理想的数值结果。综上所述，本课题所获成果将丰富延迟微分方程数值计算理论，在控制科学和生物信息学等领域具有广泛的应用前景。