Above we see a handy dandy applet simulation of a pendulum which exhibits chaotic behavior. The forces on the pendulum are such that the differential equation for the angle of the pendulum, x ,measured with respect to the vertical, are given by

y = dx/dt,

and dy/dt = -cy - sin x + f cos z,

where y is the angular velocity of the pendulum, t is time, z = wt, and c, f and w are constants.

This simulation uses a Runga Kutta 2 algorithm to numerically integrate the differential equation. The time steps are kept somewhat large to speed the simulation and thus these solutions may differ from others using smaller step sizes. This simulation does exhibit the characteristics of chaos however. The applet is written in JAVA and should run on Netscape 4.5 or newer web browsers or Internet Explorer 5.5 or newer web browsers.

In this particular simulation c = 0.05, and w = 0.7. f is adjustable by using the scroll bar at the top of the applet. After initial transients die out the behavior of the pendulum is periodic for 0.80 < = f < = 0.87 and shows one attractor on the Poincare section plot. At 0.89 there is period doubling with two attractors on the Poincare section plot. Above 0.91 these is chaos.

On the left we see the pendulum swing back and forth. One the right we can graph a "phase space plot" or a "Poincare section plot" depending upon the button settings.

For initial conditions both x and y are zero at time = 0. The "Re Start" button sets x and y to the initial conditions and the simulation restarts. The "Clear" button clears the plot (phase or Poincare) on the right side.