The Analysis of Rotated Vector Field for the Pendulum

Abstract

The pendulum, in the presence of linear dissipation and a constant torque, is a nonintegrable, nonlinear differential equation. In this paper, using the idea of rotated vector fields, derives the relation between the applied force β and the periodic solution, and a conclusion that the critical value of β is a fixed one in the over damping situation. These results are of practical significance in the study of charge-density waves in physics.

The analysis on the single particle model of CDW

“…Delicately, the CDW and the Josephson junction, revealing the chaotic behaviors of conducting [22,23], are both well described by the driven damped pendulum model [24]. By a rigorous mathematical analysis are derived these conclusions of the pendulum equation applied in [15], which provide us a more direct relation between the pendulum model and the behavior of the CDW. Such a simple model of pendulums indicates us an interesting relations of these physical subjects.…”

“…For which, let Γ increase from zero to a positive, by the analysis of rotated vector field, it is proved that trajectory L 0 will roll down and become trajectory R as shown in Figure 3. While Γ > 0, let β increase from zero to a positive value, it is proved that trajectory R will stretch up and singularity A 1 moves toward left as well, such that trajectory R enters singularity A 1 again at a certain positive value of β which is no greater than 1 [15]. Since point A 1 is the successive periodic point of A 0 , trajectory R connecting these two saddles will become a cycle on the cylindrical surface, a critical periodic solution, as shown in Figure 2.…”