The classical pendulum has aroused much interest in current research because of its close analogue to many physical systems that exist in nature. Of particular interest is a super conducting electronic device called the Josephson junction. This is a system for which chaotic solutions are deemed to be undesirable and need to be avoided. The pendulum is a reasonable model of the Josephson junction, and can help to predict the temporal responses of the superconducting device¹.

This project aims to investigate the dynamics of the classical pendulum using a FORTRAN program which displays its animation, phase-space, and Poincaré section and it will be shown that the pendulum behaves chaotically for certain values of damping, forcing frequency and forcing amplitude.

The equation that models the pendulum is a second order differential equation with a damping term and a forcing torque. This was solved numerically using the fourth order Runge-Kutta method, with the results being plotted after each iteration through the algorithm.

The phase-space diagrams and Poincaré sections within this report demonstrate the fascinating response of the oscillator. Although these responses often look complicated or "chaotic", they have explainable features. Oscillators with different driving parameters, when plotted can be seen to have features in common with each other. This can most easily be seen with the phase-space diagrams. These are diagrams that plot the angular displacement versus the angular velocity of the pendulum for increasing time.

Another convenient way of showing the long-term behaviour of the oscillator is by plotting the displacement versus velocity after every period of the driving force. This is called a Poincaré section and it gives a far better indication of what is happening.