关于无限时滞中立型泛函微分方程Hopf分支的研究

This proposal consists of two components. The qualitative theory on linear autonomous neutral functional differential euqation(NFDE) with infinite delays will be established in the first part. Firstly, the spectrum distribution of infinitesimal generator of its solution operators is studied under a certain phase space. Thereafter, we prove the representation theorem of the solution operators, which is later employed to obtain exponential dichotomy property in terms of semigroup theory. Secondly, formal adjoint theory for linear autonomous NFDE with infinite delays is established including such topics as formal adjoint equations, the relationship between the formal adjoint and the true adjoint and decomposing the phase space with formal adjoint equation. In the second part, the normal form theory is extended to NFDE with infinite delays, and then Hopf bifurcation for nonlinear NFDE with infinite delay is investigated combining the foregoing linear theory and normal form theory obtained, and center manifold theorem. The conditions ensuring the occurrence of Hopf bifurcaion are stated and the algorithm of computing Hopf bifurcation properties is also provided. This project will contribute to the supplement of theory of neutral functional differential equation with infinite delays.

本项目的主要内容由以下两部分组成。第一部分是对线性的自治的无限时滞中立型泛函微分方程建立起并完善有关的定性理论。首先在某个特定的相空间下研究线性方程解算子的无穷小生成元谱的分布，解算子的表示定理以及指数二分性质；然后建立线性方程的形式伴随理论，其中包括给出形式伴随方程，研究形式伴随和真实伴随之间的关系，以及如何利用形式伴随方程来分解相空间。第二部分则是将规范型理论推广到无限时滞中立型方程中，并结合上面得到的线性方程的理论和规范型理论以及中心流形定理来系统地研究非线性无限时滞中立型方程的Hopf分支问题，给出Hopf发生的条件以及分支发生时分支性质的计算方法。本项目对进一步完善无限时滞中立型泛函微分方程的相关理论有一定的意义。

中立型泛函微分方程在物理、生物等领域有着广泛的应用，但有关定性理论还并不完善。本项目主要是将泛函微分方程的有关定性理论推广到无限时滞的中立型泛函微分方程中，取得的结果主要包括：建立了线性方程的相关理论，包括谱理论和形式伴随理论；将Hassard计算泛函微分方程规范型的方法推广到了此类方程，并给出了非线性方程的Hopf分支定理。项目执行期间研究团队共发表学术论文10篇，出版教材1部，多次参加国际学术会议并作学术报告。该项目研究结果进一步完善了中立型泛函微分方程的相关理论。