Birdsong

The normal form's two muscles: the labial tension across, the air sac pressure up (α downward). Above the line the labia swing; below it they rest. Right of the corner the line is a Hopf bifurcation (solid), where a swing grows from nothing into a pure tone; left of it a saddle-node (dashed), where the swing starts at full size and zero frequency, a click train that runs up into a buzz. The dot is the newest note now, with its gesture's path behind it; the ring is where its breath is headed.

The labia's phase portrait: their opening x across, their speed up. The loop is the limit cycle the model settles on at the note's full breath; the dots are the last swing the engine played. Near the Hopf line the loop is a small ellipse, a sine; near the saddle-node it is lopsided, crawling past the ghost of the vanished fixed points (the crosses) and snapping back, a pulse rich in harmonics.

The trachea's quarter-wave resonances (c/4L, 3c/4L …) times the mouth's (the OEC's) single resonance, and the note's harmonics as they leave the beak: the labia's cycle (faint) filtered by the tract (bright). A mouth tuned to the note sits on the fundamental and leaves a whistle; a fixed one picks out whichever harmonics fall under it.