高维微分系统和中立型小时滞系统的不变流形约化

高维微分系统和小时滞系统的约化是微分方程与动力系统领域重要的研究课题，它们可以分别通过寻求系统的法向双曲不变流形和惯性流形来实现。本项目将围绕这两类约化问题开展研究：(1)研究含参数高维微分系统的法向双曲不变流形约化。我们将克服稳定和不稳定法向共存带来的困难，将前人关于稳定慢流形约化的结果推广到鞍型慢流形约化的情况，给出确定约化所需小参数条件的方法。进一步地，我们将利用分叉理论研究法向双曲不变流形低阶逼近的影响。(2)研究中立型小时滞系统的惯性流形约化。我们将推广已有结果得到更弱条件下惯性流形的存在性和光滑性。进一步地，我们将利用奇异摄动常微分系统的慢流形理论给出惯性流形的逼近。这些结果将有利于理论研究和实际应用中高维微分系统和中立型小时滞系统的降维和化简，为系统的定性分析提供方便。

Reductions for high-dimensional differential systems and small-delay systems are important research topics in the field of differential equations and dynamical systems, they can be obtained by the normally hyperbolic invariant manifolds theory and the inertial manifold theory, respectively. Concerning these two reduction problems, we will study the followings: (1) studying the normally hyperbolic invariant manifold reduction for high-dimensional systems with parameters. We will overcome the difficulties caused by the coexistence of stable and unstable normal directions, and further study how to determine the small parameter conditions for the slow manifold of saddle type reduction, which will extend previous works on the slow manifold of stable type reduction. Furthermore, we will study the effects of the low-order approximations for normally hyperbolic invariant manifolds by applying bifurcation theory. (2) studying the inertial manifold reduction for small-delay systems of neutral type. We will study the existence and smoothness of inertial manifolds under weaker conditions than those in the previous works. Furthermore, we will give the approximation of inertial manifolds by applying the normally hyperbolic invariant manifolds theory for singularly perturbed systems of ordinary differential equations. These results will be applied to the dimension reductions and simplifications of high-dimensional differential systems and small-delay systems, which arise from theoretical researches and practical applications. So they are helpful for the qualitative analysis of these two kinds of differential systems.