具时滞和分层结构的耦合振子模型的动力学研究

The collective behavior of coupled oscillators has very wide applications in many scientific and technological fields. In the research area of delay differential equations, there has been a flurry of research activity on dynamics of the models of coupled oscillators. The purpose of this project is to research the dynamics of coupled oscillator models with delays and hierarchical structure, by the stability and bifurcation theory of delay differential equations, the theory of equivariant degree, Lie group, and the approach of normal form. Our research contents of this project mainly include three aspects as follows: (1) the global asymptotic stability of the equilibrium; (2) the stability and the global continuation of the equivariant bifurcated periodic solutions; (3) the mode interaction of equivariant pitchfork bifurcation and equivariant Hopf bifurcation. In addition, we would like to point out that the combined effects of the delays and the hierarchical structure on the dynamics of the models of coupled oscillators should be investigated urgently. The research results of this project will not only give an impetus to the development in the theory of symmetric delay differential equations, but also be a convenience for applying the models to some areas of engineering and technology.

耦合振子模型的集体行为在科学工程领域有着广阔的应用前景。耦合振子模型的动力学分析是目前时滞微分方程研究领域的一个热点。本项目旨在运用时滞微分方程稳定性和分岔理论，并借助 Lie 群和等变度理论以及规范型的计算方法，研究具时滞和分层结构的耦合振子模型的动力学。具体研究内容主要包括三个方面：(1) 平衡点的全局渐近稳定性；(2) 等变分岔周期解的稳定性和全局持续性；(3) 等变pitchfork分岔和等变Hopf分岔模式的相互作用。值得指出的是，研究时滞和分层结构对耦合振子模型动力学的共同影响是亟待解决而又尚未研究的全新课题。本项目的研究成果不仅可推动对称性时滞微分方程理论的发展，而且对相关应用领域也将有重要的指导意义。

耦合振子模型的动力学是近年来微分方程及其应用领域一个非常重要的研究课题。本项目基于前期研究工作，建立了两类由时滞微分方程描述的具分层结构的耦合振子模型。我们运用泛函微分方程的稳定性理论、分岔理论，并结合李群表示论和等变度理论等工具，深入研究了各模型丰富的动力学性质。.本项目关于两类具时滞和分层结构的耦合振子模型的动力学研究成果主要集中在：平衡点的稳定性，等变分岔周期解的时空模式、稳定性和全局存在性。部分结果已投稿到国际杂志 Neurocomputing。此外，我们还得到了一个关于右端不连续型时滞微分方程的指数稳定性的结果，并尝试应用到某些具不连续信号传输函数和分层对称结构的神经网络模型的稳定性研究上。上述研究成果不但丰富了有关对称性泛函微分方程理论，而且对技术应用领域也有重要的指导意义。后续研究已获得国家自然科学基金青年基金项目（编号：11401051）资助，目前相关研究工作正在开展。