I was just experimenting with the “classic” feedback FM and stumbled on a related technique that seems quite interesting to me, especially because of its chaotic behavior. It’s actually very simple: The output of an oscillator is delayed and, scaled by some factor, fed back into that same oscillator as its frequency. So there is no external input, no carrier frequency (or a zero carrier frequency if you will), which means that it is a symmetric “through-zero” FM.

By guessing I figured out how the delay time determines the frequency of the oscillator, but I still don’t understand a few things: Why is there a factor of 4 involved so that the delay is one quarter of a cycle? And why is there still some error? I found two ways to fix it, first by oversampling and second by reducing the delay time by the duration of one half sample. To be honest I don’t have a clue why this works ... Any ideas?

What I find particularly interesting is the path from sinusoidal tones at modulation index 1 (which seems to be a critical value) to different modes of periodic oscillation and finally to chaos when the modulation index is increased: First odd harmonics are added, then even harmonics, then a period doubling or “bifurcation” occurs, and shortly after that we get into the chaotic region. This seems to be not too different from the logistic map or related chaotic maps. So I’m wondering if it’s possible to analyze the oscillator in terms of chaos theory to better understand and control its behavior.

Apart from being mathematically interesting, I think this could also be a usable noise generator with spectral control. Other effects can be achieved by adding an external input ... but for now, I’m more interested in understanding that simpler case.