The Analysis of Rotated Vector Field for the Pendulum
  The pendulum, in the presence of linear dissipation and a constant torque, is a non-integrable, nonlinear differential equation. In this paper, using the idea of rotated vector fields, derives the relation between the applied force $\beta$ and the periodic solution, and a conclusion that the critical value of $\beta$ is a fixed one in the over damping situation. These results are of practical significance in the study of charge-density waves in physics. 
Lian-Gang Li