This demonstration illustrates a simple phase plane plot. The particular system plotted in this example is
x'=y
y'=-sin(x) -0.1 y
which is equivalent to the standard equation of motion
x"+0.1x'+sin(x) = 0
for the damped nonlinear pendulum.
In this application, the variable x is really the angle theta between the
pendulum arm and vertical and y is x', which is proportional to the speed
of the pendulum bob.
To plot the trajectory that starts at any specific point, just click the mouse there. The program stops plotting a trajectory when it reaches the edge of the screen or when the user clicks the "Stop" button. To restart a stopped trajectory click "Continue". The user may clear the screen by clicking the "Clear" button. By default, trajectories are plotted forwards in time. The user may also chose to have trajectories plotted backwards in time. Just click the "Forwards" button. It toggles between "Forwards" and "Backwards".
The demonstration below is an applet. Google Chrome, Firefox, Safari and Microsoft Edge no longer execute applets natively because of security issues with NPAPI plugins. However applets can be played in the Google Chrome browser by using the CheerpJ Applet Runner extension.