Approximate stability of the parametric inverted pendulum

In this paper, we study the stability border of the damped parametric inverted pendulum by variational method. The stability relation of the parametric inverted pendulum between parameters is obtained in the case of the large pendulum length, but the forcing amplitude is not confined, especially, the forcing amplitude can be bigger than 1, which makes up for the small amplitude case. The comparison between the stability and instability domains is given by the numerical simulation for the nonlinear system. The numerical results show that the border is reliable.