Before the program could be used to perform proper experiments, it had to be tested to make sure that the results it produced were reliable. It was stated in section 2.1 that the solution to the linear form of equation [1] is an ellipse. Thus, the program was tested by using the linearised form of equation [1]. The equation was linearised by putting the forcing term equal to zero (f=0), the damping term equal to zero (q = 99999999) and setting the forcing frequency equal to the natural frequency (w_D = w = 1). It must be noted here that the forcing frequency cannot be set to zero. This would cause the program to crash because w_D is used to calculate the step increment. This being the case, the required ellipse was produced. (Fig.4.1.1)

As a further test, some damping was added to the equation by setting q = 5. This produced a spiral into the center of the graph, in good agreement with section 2.1. (Fig.4.1.2). The quality factor q, controls the damping and approximates to the ratio of the energy stored in the pendulum to the energy dissipated in friction during a single natural oscillation⁵. The spiral illustrates how the pendulum's energy oscillates between kinetic energy (which is proportional to the square of the velocity), and the potential energy (proportional to the drive torque) and is eventually lost completely to friction.

Experiments involving the variation of the forcing amplitude whilst keeping the other parameters constant has already been extensively investigated ^6-7 and will not be discussed any further here except to say that they were used as a basis for this project.

For this project the quality factor q was varied and all the other parameters kept constant. The values used were; 0.6666..., f ( or g in the case of the display screen.) = 0.90 and the values used for q were; 2.1, 2.51, 2.6, 2.7, 2.8, 3.5. The number of steps per driving period was 200 and the number of iterations varied from plot to plot and are given on the diagrams (see appendix C for the plots). All of the initial conditions for the parameters (x, v, t ) were identical for all of these experiments and each experiment was repeated several times to test the consistency of the program. The results were identical for each value of the quality factor.

The first value of q was 2.1 and the results can be seen in figs. 4.3.1 (a) and 4.3.1 (b). After the first 1000 points, the trajectory settled down to the closed curve in fig.4.3.1 (a). The transient points that precede the stable trajectory illustrate the asymptotic behavior of the oscillator. Appropriately, then, the curve is called a periodic attractor.

The corresponding Poincaré section in fig. 4.3.1 (b) consists of a single dot (enlarged for clarity). Thus, once the transient points have been eliminated a steady state has been achieved, indicating that the same point is plotted over and over again with every drive cycle. For obvious reasons this is called a period-1 solution.

The next value of q is 2.51, figs. 4.3.2 (a) and 4.3.2 (b). This time, two distinct curves can be observed in the phase-space diagram. This is an example of a phenomenon known as period doubling. The driving torque does not change from one cycle to the next. The pendulum, however, displays different motions on alternate cycles. As a result, x and v assume different values at the beginning of consecutive drive cycles. Hence, the Poincaré section alternates between two points - after the usual transients have been exhausted. This is the bifurcation that was described in section 2.4, where one solution gives way two, two gives way to four, four to sixteen...etc. The period doubling observed is the first in an infinite series of bifurcations that occur as q is increased and all the other parameters held fixed. Each bifurcation doubles the period of a given attractor and, consequently, the number of points in its Poincaré section.

For q = 2.6, fig. 4.3.3(a) and fig. 4.3.3(b), the system is clearly chaotic. This is seen more clearly with the corresponding Poincaré section. The animated pendulum clearly lacks regularity. The system for which q is 2.7 (figs. 4.3.4 (a) and 4.3.4 (b)) is also chaotic. For q = 2.8 (figs. 4.3.5 (a) and 4.3.5 (b)) the system is periodic again, confirmation coming from the fact that its Poincaré section contains three distinct points. In this case the pendulum performs periodic motion over a period of three drive cycles. During these three drive cycles the pendulum performs one anti-clockwise rotation, one clockwise rotation and two normal oscillations. For q = 3.5 (figs. 4.3.6 (a) and 4.3.6 (b)) the system returns to chaotic motion.