The Pendulum - Demo 6

Driven Damped Pendulum - chaos

Increasing the driving strength in tiny steps would lead to more and more subharmonic bifurcations, i.e. frequencies of 1/8, 1/16 etc. These are hard to see, but you can try to do so. However if we increase the driving to a=1.09 the character of the motion changes significantly:

The orbit does not seem to repeat itself, and the power spectrum ...

now shows peaks that (except maybe for the fundamentals and harmonics) are broadened, and there is a broad band background. (You should run the demonstration long enough for the spectrum to reach a steady form.) These are characteristics of chaos, and the simplest of all dynamical systems has undergone a "period doubling route to chaos". There are many quantitative predictions that can be made about this particular route to chaos, and we will study it in detail later.

Increasing the driving to 1.15 give stronger chaos:

but perhaps at first sight surprisingly, increasing the driving further, e.g. to 1.35 gives periodic motion again (now a "running" solution):

The driven, damped pendulum shows a complicated series of transitions to and from chaos as the driving is increased. Intuitively we might try to understand chaos as arising from the delicacy in the motion near the unstable fixed point, so that when the driving and internal dynamics conspire to bring the orbit here, chaos might occur. This can happen after oscillations, or after one complete orbit, or two etc., so a succession of chaotic and nonchaotic regions might be expected.

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Last modified Wednesday, November 24, 1999
Michael Cross