I have updated the Lissajous figure so that it will come out of the FFT spectrum in the middle of my sketch.
The FFT spectrum used to look like this:

To incorporate the Lissajous figure, I found where the parametric equations gave a line that was near the FFT spectrum and used an if statement to “activate” the Lissajous figure. It now looks like:

Initially, six lines will be in the Lissajous figure (and six lines taken out of the FFT spectrum). After one full cycle of the figure, three lines will be reincorporated in to the FFT spectrum. After another full cycle, all lines will be incorporated in to the spectrum.
The next step for this is to figure out how to ensure this happens continuously and not hard coded in.