The effect of symmetry-breaking on the parameterically excited pendulum

The effect of the symmetry-breaking on the parameterically excited pendulum including a bias term is investigated. At first, our numerical simulations show that the area of the safe region of the unexcited pendulum (without damping and without forcing) will decrease with the increasing of the bias term. Due to the variation, the critical homoclinic bifurcation of the excited pendulum will increase, and the region where the homoclinic transversal intersection occurs between the stable and unstable manifolds in the Poincaré map will be enlarged. Second, as the bias term increases, our analysis demonstrates that the number and the type of attractors of the Poincaré map, the phase portraits, the basins of attraction, and the bifurcation diagrams will produce a considerable variation. In particular, the stability of the parameterically excited pendulum will lose once the bias term exceeds a critical value. In this case there is no longer any steady state existing. These results suggest that much attention should be paid on controlling the increasing of bias term, especially when the parameterically excited pendulum as a main device is applied to some practical systems.