syrinx · the voice of a nonlinear oscillator
A van der Pol oscillator — a reduced model of the bird's syrinx. Drag ε and hear the bifurcation: near zero it sings a pure whistle; past ≈ 2 it tips into relaxation and fills with harmonics — a buzz. This is the deepest math in the project, as an instrument.
The portrait on the right is the live (x, ẋ) trajectory: a smooth loop when tonal, a sharp relaxation loop when buzzy, a point at rest — the Hopf bifurcation you can hear.
stroke painter · compose song from time–frequency strokes
The analyzer decomposes real song into chirplets — oriented strokes in the time–frequency plane. Here you do the inverse: drag to draw a stroke (its slope is the chirp, its length the duration), stack a few, and play the phrase you painted.
x = time (0–2.5 s), y = frequency (0–8 kHz). Each stroke is one Gaussian chirplet A·e^(−Δ²/2σ²)·cos(2π[f₀Δ + ½cΔ²]) — the same atom the analyzer fits.