The Pendulum - Demo 7

Driven Damped Pendulum - the Poincare section

Further information on the chaotic dynamics is given by the Poincare section - the intersection of the phase space trajectory with a fixed plane. This is easy to calculate numerically for the driven pendulum if we take the Poincare section as a z=const plane: the Poincare section then corresponds to "strobing" the dynamics at the driving frequency. The structure shows up better if we decrease the dissipation strength e.g. to b=0.25

A "whispy" shape with intriguing structure appears. (This is more interesting than for the Lorenz model, because the rate of contraction of volumes in phase space here is smaller than in the Lorenz model with the usual parameters.) You can investigate the fine structure by enlarging a portion of the stopped plot using the mouse to outline a rectangle (with the button held down). Turning off the time display and not "allowing sleep" (if your Java implementation allows this without crashing) may increase the speed at which the plot develops. From this investigation you can probably accept that chaotic motion is on a "strange attractor" with fractal structure. Again these ideas will be studied in more detail later.

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Last modified Wednesday, November 24, 1999
Michael Cross