Oscilloscopes, phase plots and beyond

Similarly to how oscilloscopes trace out two signals x(t) and y(t), one might plot any related variables against each other.

The above illustration shows x(t) against y(t), both of which seem to be periodic, albeit quite complicated waveforms.

Phase plots of, say, position versus momentum are another common way to display orbits of differential equations. Any recorded signal may be plotted with its numerically estimated derivative on the y-axis. Hint: use a higher order differentiating filter, not just a forward or backward difference.

Graphs of the amplitude spectrum of one signal against that of another signal are harder to interpret. The axes then represent amplitudes, and each point is a unique frequency whose coordinates is the amplitude in signal x versus its amplitude in signal y. If the points gather near a thin line along the main diagonal, the two spectra are highly correlated.

Another novel kind of plot is the signed amplitude spectrum. It combines phase information with the amplitude spectrum and distinguishes positive and negative amplitues depending on the sign of the phase. Below, the red line is the signed amplitude spectrum of one variable and the blue line is that of a related variable in the same system.

All the illustrations come from a 3D ODE that is currently under investigation, suspected guilty of exhibiting interesting behaviour.