Influence of nonlinearity on transition curves in a parametric pendulum system

• In our study the hypothesis of small number in the parametric pendulum system is not necessary.
• The energy method rather than the asymptotic method which is dependent on small parameter is used to analytically calculate the transition curves and periodic solutions of the pendulum system.
• Our results based on the energy method suggest that the nonlinearity in the parametric pendulum makes the system more stable, which is very different from the results derived by using the asymptotic method in previous studies.

In this paper transition curves and periodic solutions of a parametric pendulum system are calculated analytically by employing the energy method. In previous studies this problem usually was dealt with by using the asymptotic method which is limited by small parameter. In our research, the hypothesis of small number in the pendulum system is not necessary, some different conclusions are obtained on the impacts of nonlinearity in the pendulum system on the transition curves in the parametric plane. The results based on the asymptotic method suggested that nonlinearity in the pendulum system only significantly causes decrease of the area of the stable regions in the parametric plane when the angular displacement of the pendulum is not very small. However, our analysis according to the energy method shows that nonlinearity does not significantly change the area of the stable regions in the parametric plane, but notably alter positions of the stable regions. Furthermore, position of the stable regions to a large extent is related to the amplitude of periodic vibrations of the pendulum especially when the angular displacement of the pendulum is large enough. Our results are very different from that reported in previous studies, which have been verified by numerical simulations.