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The logistic map is one of the "simplest" maps that gives rise to chaotic behavior. It is defined as the series: \[x_{n+1}=rx_n(1-x_n) \] For \( 1<r<3 \), \(x\) converges to \( \frac{r-1}{r} \), but for higher \( r \), \(x\) begins to oscillate between values, and eventually its behavior becomes chaotic. This is best visualized with a bifurcation diagram (as shown in the visualization).
This applet applies the logistic map to a random value repeatedly, and translates the result into an audio waveform (using a sample rate of 4096Hz). Only values of \(r\) between 3 and 4 are allowed (lower values lead to waveforms that are just DC offsets). An FFT of the signal is shown in the bottom pane. The applet was made using AudioWorkletNode and p5.js.