Liu Chungen, Wang Qi
School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006 People's Republic of China.
School of Mathematics and Statistics, Henan University, Kaifeng, 475000 People's Republic of China.
J Dyn Differ Equ. 2022 Nov 16:1-19. doi: 10.1007/s10884-022-10226-2.
In this paper, we consider a kind of second-order delay differential system. By taking some transforms, the property of delay is reflected in the boundary condition. The wonder is that the corrseponding first-order system is exactly the so-called -boundary value problem of Hamiltonian system which has been studied deeply by many mathematicians, including the authors of this paper. Firstly, we define the relative Morse index for the delay system and give the relationship with the -index of Hamiltonian system. Secondly, by this index, topology degree and saddle point reduction, the existence of periodic solutions is established for this kind of delay differential system.
在本文中，我们考虑一类二阶时滞微分系统。通过进行一些变换，时滞的性质反映在边界条件中。奇妙的是，相应的一阶系统恰好是所谓的哈密顿系统的 - 边值问题，许多数学家，包括本文作者，都对其进行了深入研究。首先，我们为该时滞系统定义相对莫尔斯指标，并给出它与哈密顿系统的 - 指标的关系。其次，借助这个指标、拓扑度和鞍点约化，建立了这类时滞微分系统周期解的存在性。
