Newmark方法求解二阶常微分方程Hopf分支等价性

The equivalence of Hopf bifurcation for second-order ordinary differential equation with Newmark method

考虑一个带有参数的二阶常微分系统,对其线性化方程的特征方程根进行分析,研究系统平衡点的稳定性,得到系统产生Hopf分支的充分条件。利用Newmark方法将系统离散,分析离散系统特征方程根的分布情况,确定方法中的参数,保证对任意的步长离散系统存在Neimark-Sacker分支。证明在离散系统Neimark-Sacker分支存在的情况下,原连续系统具有Hopf分支。数值仿真验证了结果的有效性和适用性。

A second-order ordinary differential system with a parameter is considered. The stability of the equilibrium and the existence condition of Hopf bifurcation are studied by analyzing the distribution of the characteristic roots of the linearized equation. The system is discretized by Newmark method. By analyzing the distribution of the characteristic roots of the discrete system, a pair of parameters are determined such that the discrete system undergoes a Neimark-Sacker bifurcation for any time steps. On the contrary, it is proven that if there exists a Neimark-Saeker bifurcation for the discrete system, then the continuous system undergoes a Hopf bifurcation. Finally, some numerical simulations are given to show the accuracy of our theoretical analysis.