Strange Attractor for a Periodically Forced Pendulum

If you periodically push on a swinging pendulum you can get some very erratic and unpredictable behavior, provided you push hard enough and if the forcing frequency is different from the natural frequency of the pendulum. Of course you would never do this when pushing a child on a swing; ordinarily you choose a forcing frequency equal to the natural frequency, where resonance maximizes the amplitude of the pendulum's swing.

The forced pendulum can be modelled by the differential equation A y'' + B y' + sin(y) = C sin(w t). The solution y(t) gives the pendulum's position as a function of time. The various constants are determined by the physical parameters describing gravity, friction, etc. By solving this equation numerically on a computer, the pendulum's motion can be simulated and analyzed. For certain parameter values the resulting motion is chaotic, and the trajectory in 3-dimensional (y,y',wt mod 2pi) phase space eventually settles down onto a "chaotic attractor". Plotting points on the attractor at some fixed value of t yields a 2-dimensional slice through the attractor in (y,y')-space, as in the images below. These images are fractal: they have a self-similar structure that reveals intricate detail at all scales, which is a consequence of the repeated "stretching and folding" evident in the animations. Animating a sequence of slices for different values of (wt mod 2pi) illustrates the evolution of the attractor in time, as shown in the animations below.

Image gallery (click on image to view larger version)


Click on an image to view a larger, animated version.

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