几类二维格微分方程动力学行为

Lattice differential equations, namely infinite systems of ordinary differential equations indexed by points on a spatial lattice, have been proposed as models in various fields, such as material science, image processing, chemical kinetics, biology, etc. The main goals in this thesis are to investigate the existence，uniqueness，stability and bifurcations of several different types of travelling waves including periodic moving waves、solitary waves、uniform sliding states for some representative two-dimensional lattice systems. Our methods to be used involve representation theory of Lie group, Lyapunov-Schmidt reduction, center manifold theory, nonlinear analysis and so on. By using these methods, we will construct invariant manifold of mixed-type delay differential equations in order to analyze the dynamic behavior in two-dimensional lattice systems with Hamilton and non-Hamilton, respectively. Thus, we extend some results in one-dimensional lattice systems to high-dimensional ones in the analysis of the influence of the structure of systems and travelling direction. Meanwhile, some new theorems and methods will be developed. This research will not only enrich and develop some basic theory of high-dimensional lattices, but also provide effective methods and theoretical basis for some practical problems.

格微分系统表现为由定义在具有几何结构的格上的无穷多个常微分方程耦合而成的系统，在材料科学、图像处理、化学动力学、生物学等学科领域有着广泛的实际应用背景。本项目旨在研究几类具代表性的二维格系统的不同形式的行波解，包括周期运动波解、孤立波、一致滑行状态解的存在性、唯一性、稳定性、分支等性质。综合运用李群表示论，Lyapunov-Schmidt 简约方法，中心流形，非线性泛函分析等现代数学工具，构造混合型时滞微分方程的不变流形，分析具有 Hamilton 结构的和非 Hamilton 结构的二维格系统的动力学性质，并分析系统结构和行波方向的影响，将一维格系统的相关结论推广到高维格系统中去，从而发展高维格系统的理论与方法。这些研究不仅丰富和发展了高维格系统的基本理论，而且为一些实际应用问题提供了有效的解决办法和理论依据。