小世界延时网络的稳定性与霍普夫分岔

针对带有延时的一维小世界网络模型，通过分析其线性化系统对应的超越特征方程，来研究其平衡点的局部稳定性，把延时看作分岔参数。发现当延时穿过某一临界值时，系统会产生霍普夫分岔。从平衡点分岔出一类周期轨道，利用标准型理论和中心流形定理。得到判断分岔周期解的方向、稳定性以及其他特性的精确计算公式，最后通过数值仿真进行了验证．

Stability and Hopf bifurcation of delayed small-world network

The one dimensional small-world network model with time delay was considered in this pa per. The local stability of the equilibrium in this network model was investigated by analyzing its corresponding transcendental characteristic equation of the linearization system. Regarding the time delay as a bifurcation parameter, it was found that Hopf bifurcation occurs in the system when the time delay passed through a critical value and a branch of periodic solutions was bifurcated from the equilibrium. The explicit formulas that determined the direction, the stability and other properties of bifurcating periodic solutions were derived by using the normal form theory and the center manifold theorem. The theoretical analysis was verified by a numerical simulation.