Coupled Oscillator Chaos H = ½(px² + py²) + ½(x² + ω²y²) + x²y²

One mass on springs that can swing two ways, x and y, which stiffen each other as they move: two oscillators with a quartic coupling, struck once and left to ring down. A gentle strike rings regularly; a strong one goes chaotic until damping takes the energy back below the threshold, and then it rings on with whatever split of energy the chaos left behind. Strikes A and B differ by a tiny change in direction. Listen to both.

Physical setupOne mass, free to slide in a plane, held by springs whose anchors roll along the walls, so each spring pushes only along its own axis.

The mass (m = 1) sits at rest in the middle until the strike: a sharp blow that gives it energy E in the chosen direction. It then oscillates along x at frequency 1 and along y at frequency ω.

What couples the two motions is the stiffness. The springs along x get stiffer the farther the mass is from the x axis, and the springs along y get stiffer the farther it is from the y axis:

kx = 1 + 2y²    ky = ω² + 2x²

That is the x²y² term in the energy. A plate behaves the same way: bending it one way stretches its midplane, which stiffens it for every other way of bending. A light friction on the mass sets the ring time.

Motion in the bowlThe same mass seen as a ball rolling in a bowl, V(x, y) = ½x² + ½ω²y² + x²y². Bands are heights of 10%, 25%, 50% and 100% of the strike energy; the ball cannot climb past the outer one. The line is the path of the selected strike (A unless you choose another below).
Phase space at the strike energyPoincaré section of the undamped system (y = 0, py > 0). Closed curves are regular motion, scattered dots chaos. The ring marks where the strike starts.
Who gets the energyShare of the energy in the x oscillator for 12 strikes, each differing from the next by the A/B difference. The shaded span is the chaotic phase of strike A.
Strike AStrike B10 more strikes
How fast nearby strikes separateLocal Lyapunov exponent of strike A (per second, 20-period windows). Above zero, a difference between two strikes grows exponentially. The dashed line is the energy, in dB.
What you hearSpectrogram of the selected strike, as heard by the listener. Broadband smear means chaos; clean lines mean regular ringing. Click a strike, or use the ← and → keys, to flip between the 12; the color scale is shared, so differences in level are real. The selected strike is also drawn bold in the energy-share plot and as the path in the bowl.