时滞微分系统周期解的最小周期问题

By making use of critical point theory and equivariant degree theory and combining Nehari manifold method, this project mainly studies problem of minimal period for periodic solutions of delay differential systems. Specifically, we build the variational framework on suitable function space for delay differential systems or their couple ordinary differential systems. By making use of minimax theory, mountain pass lemma, maslov index, equivariant degree and so on, we build theorems of existence and multiplicity for critical points on the space, which the variational functional is built, its subspace or Nehari manifold defined on it. These theorems are used to study the existence and multiplicity of periodic solutions and their minimal period of delay differential systems. By studying the problem of minimal period for periodic solutions of delay differential systems, the project generalizes the results from Kaplan-Yorke type equations to nonautonomous and high dimension cases. Also, it provides new theories and methods for studying more complex delay differential equations. This project is meaningful in theory and useful in applying.

本项目主要应用临界点理论、等变度理论，结合Nehari流形方法，研究时滞微分系统周期解的最小周期问题。具体来说，对时滞微分系统或其耦合常微分系统，在适当的函数空间建立变分框架，通过应用 Minimax 理论、山路引理、Maslov 指标、等变度等工具，在变分泛函建立的空间、子空间或其上定义的Nehari流形上，探讨变分泛函临界点的存在性和多重性，进而研究时滞微分系统周期解的存在性、多重性及其最小周期。本项目通过研究时滞微分系统周期解的最小周期问题，将Kaplan-Yorke型方程相关结果推广到非自治和高维情形，为探讨更复杂的时滞微分方程提供新的理论和方法。这项研究具有重要的理论意义和广泛的应用价值。

本项目主要应用临界点理论、等变度理论，结合Nehari流形方法，研究时滞微分系统、哈密顿系统及差分系统周期解的多重性及其最小周期。具体来说，对时滞微分系统、哈密顿系统及差分系统，在适当的函数空间建立变分框架，通过应用临界点理论结合Nehari流形方法，把变分泛函限制在Nehari流形上，通过亏格理论，建立了周期解的多重性结果。此外，通过寻找限制泛函的极小值，并证明这个值就是一个临界值，它对应的周期解具有最小周期。进一步，结合等变度理论，可以证明限制泛函在Nehari流形上有多重临界值，它们对应的周期解是几何不同的，而且具有共同的最小周期。最后，作为在生物数学中的应用，本项目研究了带有时滞的离散捕食-食饵模型，得出了其行波解存在的充分条件。