几类非线性动力系统的局部结构分支相关问题研究

本项目主要研究几类非线性动力系统中局部结构的分支动力学行为, 尤其是反转或非反转、保守或非保守，Z_2对称或非Z_2对称等具有不同性质的动力系统的扰动分支问题。研究静态或时间周期局部斑点、螺旋波和环等局部结构的重要意义在于局部结构是物理、化学、生物等过程发生复杂动力学行为的重要标志, 有着极为重要的理论价值和应用价值。本项目我们将主要采用几何退化扰动技巧和动力系统分支理论对Turing分支和Hamiltonian Hopf分支附近的一维或二维局部结构展开进一步深入的研究, 运用解析和数值方法研究局部结构的存在性、非线性稳定性及参数依赖性等特征, 发现产生局部结构的条件、分类及决定和影响空间结构的稳定性的因素, 对局部结构的蛇形分支、孪形分支等分支行为及其形成机理进行深入的研究和探讨, 力求在该方面取得令国内外同行关注的一系列具有突破性和原创性的研究结果。

This project will focus on investigating related bifurcations of localized structures in several nonlinear dynamical systems, including the reversible or non-reversible, conservative or non-conservative systems, Z_2 symmetric or non-Z_2 symmetric systems etc. Localized structures (such as spots, spiral waves, rings, etc.) organize the dynamics of many biological, chemical and physical processes and have many important applications in all kind of subjects. We will aim at studying the existence, nonlinear stability and parameter dependence of localized structures. The geometric singular perturbation theory and homoclinic bifurcation theory will be exploited in the process of studying the one-dimensional and two-dimensional localized structures near Turing bifurcation and Hamiltonian Hopf bifurcation. By using analytical and numerical techniques, we will determine the condition of occurence, nonlinear stability and classification of localized structures, including the snaking region. Finally, we hope to obtain some particular concern, ground-breaking and original achievements.