变分框架下时滞微分方程周期解的研究进展

Research Progress on Periodic Solutionsof Delay Differential Equations under Variational Framework

摘要: 时滞微分方程是生物数学、控制工程、非线性物理等领域刻画含记忆效应动态系统的核心工具，其周期解存在性与多解性分析是系统稳定性评估的关键问题。受时滞项带来的解空间拓扑复杂、泛函自伴性缺失等限制，传统常微分方程分析方法难以直接迁移应用，临界点理论与变分方法为该类问题提供了新的求解框架，通过将微分方程的解转化为对应能量泛函极值点或鞍点，实现解的存在性与多解性判定。本文梳理了该理论从常微分系统到时滞微分方程周期解分析领域的技术演化脉络，总结了核心研究成果。

Abstract: Delay differential equations serve as core tools for characterizing dynamic systems with memory effects in fields such as biomathematics, control engineering, and nonlinear physics, where the analysis of the existence and multiplicity of periodic solutions is a critical issue for system stability assessment. Restricted by the complex topology of the solution space and the loss of functional self-adjointness caused by delay terms, traditional analysis methods for ordinary differential equations are diffcult to directly migrate and apply. Critical point theory and variational methods provide a new solution framework for such problems: by transforming solutions of differential equations into extreme points or saddle points of the corresponding energy functionals, the existence and multiplicity of solutions can be determined. This paper sorts out the technical evolution context of this theory from ordinary differential systems to periodic solution analysis of delay differential equations, and summarizes the core research achievements.