一类周期系数力学系统的Hopf-Flip分岔

Hopf-Flip bifurcations of a two-degree-of-freedom mechanical system with periodic coefficients

研究了一类周期系数力学系统因周期运动失稳而产生Hopf-Flip分岔的问题。首先根据拉格朗日方程给出了该力学系统的运动微分方程,并确定其周期运动的具有周期系数的扰动运动微分方程,再根据周期系数系统的稳定性理论建立了其给定周期运动的Poincaré映射,进一步根据该系统的特征矩阵的特征值穿越单位圆情况分析判断该Poincaré映射不动点失稳后将发生Hopf-Flip分岔,并用数值计算加以验证。结果表明,非共振条件下,系统的周期运动可通过Hopf-Flip分岔,进而演变成次谐运动,而三阶强共振条件下系统周期运动失稳后形成不稳定的次谐运动。

In this paper, the Hopf-Flip bifurcation problem of a mechanical system with periodic coefficients is investigated while the periodic motion losing its stability. Its differential motion equations are established. The perturbed differential equations with periodic coefficients of its periodic motions are derived. Then the Poineare map of its period motion is established by utilizing the stability theory of system with periodic coefficients. Further more, the probability of the Hopf-Flip bifurcation oeeurenee is analyzed according to the eigenvalue of its eigen-matrix cross the unit circle and is demonstrated by numerical simulation results. It reveals that through Hopf-Flip bifurcations of non-resonance, periodic motions may lead to subharmonie motion. But in the ease of 3 order resonance, periodic motions may lead to unstable subharmonie motion after loss of stability.