时滞耦合系统共振双Hopf分岔分析

The delayed coupled dynamical systems will be detailedly researched in this programme. An analytical method based on the method of multiple scales will be proposed to investigate the double Hopf bifurcation occurring in delayed coupled dynamical systems. By using the method, the normal form equations of resonant double Hopf bifurcation are obtained. Then by analyzing the normal form equations, the conditions which can determine whether a resonant double Hopf bifurcation is strong resonance or not are obtained. The mathematical relationship between these conditions and the physical parameters of original system is established. We focus on the effects of time delay, coupling strengthen and nonlinear terms on the resonance. The mechanism of the codimension three singularity of strong resonance are studied. Furthermore, some important theoritical problems and analytical methods of some application fields will be solved. At the same time, the base of the application of time delay will be established and the theory of the delayed dynamics be developed and even consummated.

本项目定位基础性研究项目，将以一般的时滞耦合动力系统为研究对象，通过对多尺度方法进行改进和推广，发展一种适合研究时滞耦合系统共振双Hopf分岔的新方法，并利用该方法得到共振双Hopf分岔的规范型方程。通过对其规范型方程的研究，找到决定共振强弱的条件，并建立这些条件与系统物理参数之间的数学关系。重点关注时滞、耦合强度和非线性与共振强弱之间的关系，更加深入认识强共振余维三奇异性的产生机理，解决某些应用研究领域内的一些重要的理论问题和分析方法，同时建立利用时滞的理论依据和应用基础，推动时滞动力学理论的完善和发展。

本项目以时滞耦合系统中发生的共振双Hopf分岔为研究对象，主要取得两个方面的成果。第一，发展出一套基于多尺度方法的研究时滞耦合系统中发生的共振双Hopf分岔的新方法，利用该方法可以得到时滞系统共振双Hopf分岔的规范型方程。第二，通过分析1:1、1:2、1:3共振的规范型方程，首次明确给出了决定强弱共振的条件，建立了共振强弱与系统物理参数之间的数学关系。此外，本项目的结果也纠正了以往只根据线性系统频率的比值区分共振强弱的错误做法，明确提出低阶共振也有可能是弱共振。