时滞系统的强共振双Hopf分支及相关研究

There are numerous phenomena involved in time delay, such as the feedback control in the field of engineering technology, the mature period and incubation period of biological populations. In fact, most scholars currently focus on the Hopf bifurcation, weak resonance and non resonance double Hopf bifurcation in time-delay systems. However, the study on the strong resonance double Hopf bifurcation and the existing results on this study are few. This topic studies strong resonance double Hopf bifurcation of time-delay systems and the related problems, by mainly using the theory of functional differential equation, reaction diffusion equation theory, the method of center manifold and Normal form theory. First of all, we have a analysis on strong resonance double Hopf bifurcation caused by the unique time delay in differential systems after analyzing its stability; Secondly, a study is given, which is about delay-induced resonance double Hopf bifurcation due to the existence of one or more time delays; Finally, based on the acquired outcomes and in light of the theory of delay reaction diffusion equations, we make an investigation on the resonance and non-resonance double Hopf bifurcation of reaction diffusion system with time-delay. The research results will indicate the fundamental change of dynamical behavior resulting from the time delay. This can provide certain theories to solve practical problems and further enrich the bifurcation theory in a time-delay system.

时滞现象在自然界中是普遍存在的, 比如工程技术领域的反馈控制、生物种群的成熟期和孕育期等。目前，大部分学者的工作基本都集中在时滞系统的Hopf分支、弱共振和非共振双Hopf分支等方面，对强共振双Hopf分支及相关研究成果甚少。本项目主要利用泛函微分方程理论、反应扩散方程理论、中心流形方法和规范型理论，研究时滞系统的强共振双Hopf分支及相关问题。首先，对具有单个时滞的微分系统进行稳定性分析，研究由时滞诱发的强共振双Hopf分支问题；其次，探讨由两个(或多个)时滞导致的共振双Hopf分支问题；最后，综合前期得到的成果，利用时滞反应扩散方程理论，研究时滞反应扩散系统中时滞诱发的共振和非共振双Hopf分支问题。本项目的研究成果将充分展示时滞量的变化导致系统的动力学行为发生本质影响，丰富了时滞系统的分支理论，还为解决实际问题提供一定的理论依据。

本课题我们主要利用中心流形方法和规范型理论研究了时滞微分系统的高余维分支问题，以及时滞反应扩散系统高余维分支问题。首先，我们考虑了时滞微分系统的弱共振和非共振双Hopf分支问题；接着，探讨了时滞微分系统的强共振双Hopf分支问题；最后，研究了时滞反应扩散系统的Turing-Hopf分支、Turing-Turing分支、双Hopf分支问题。在高余维分支点附近，我们给出了详细的动力学分类，并利用数学软件模拟分支点附近的动力学行为。