# The Pendulum - Demo 3

## Driven Damped Pendulum - small amplitude oscillations

If small driving is turned on, the long time solution becomes a stable limit cycle. First look at how the transients decay from various initial conditions - the dynamics can be quite complicated with the driven motion at frequency 0.667 "beating" with the free decay at frequency near 1. Eventually however all the trajectories will converge onto the same periodic orbit.

To study the periodic orbit further it is useful to eliminate the transient before applying our diagnostics.

The limit cycle at this small amplitude is seen to be close to circular.

The "power spectrum" shows the motion to be nearly simple harmonic:

Note that the vertical axis is the log (to base ten) of the power, so the first peak at the driving frequency 0.667 is about six orders of magnitude higher than the third harmonic at frequency 2! The peaks are also quite narrow - again they look broad because of the log scale. They are not perfectly sharp, as might be expected for a periodic signal, because necessarily they are calculated over a finite time interval. You can investigate how the peak characteristics are affected by adding further samples (iterate in time), changing the number of points contributing to each sample ("Points"), and the windowing function ("WinNum" from 0 to 3). We will study more about the details of power spectra in chapter 6.

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