Driven Damped Pendulum II: Diverging Trajectories and Chaos



This applet compares two close together solutions of the damped driven pendulum equation of motion. In the chaotic regime, we can set initial positions 10-4 radians apart, and see that the paths are apparently identical for several cycles—actually because we cannot see the exponentially increasing difference until it is a radian or so on our scale. Eventually the paths completely and randomly separate. In nonchaotic systems, the paths usually converge, but there are exceptions. Taylor gives the example gamma = 1.077 where different starting angles give different final cyclic behavior. There is a switch over at some angle. In fact, for 1.077, check phi_2 = -0.028, -0.0285, -0.029, and do it for different dt. Here we need speed and smallest dt to be sure.
We allow the possibility of same initial conditions, different gamma. Around gamma = 1.073, changes of 0.0001 dramatically alter the initial wandering before the cycle settles down.

Try gamma = 1.105, reproduces Taylor fig 12.10, but a slight difference gives a 3-cycle (possibly also chaotic?).
Gamma = 1.4 is rolling motion: here it is natural to plot angular velocity (which is periodic) rather than angle, which quickly leaves the graph. To reproduce Taylor's figure 12.19, set dt = 0.0005, speed = 2.

The next applet, Damped Driven Pendulum III, plots the logarithm of the difference between the two solutions, making the exponential behavior evldent: exponential increase of separation, on average, in the chaotic regime, an exponentially falling difference in the nonchaotic case.


Here's the relevant lecture, which includes more details.

Program by: Carter Hedinger