Cubic Spiral Sink

The system

The app is y-up, so the height is y and the interesting motion happens in the floor plane, spanned by x and z, the plane a textbook writes as x, y. The height is along for the ride: ẏ = −0.2·y alone, an e-fold every 5 seconds, so the seeds drop onto the floor plane early and what is left is the plane picture.

Write r² = x² + z² and the two remaining lines say something short: ṙ = −r³, and the angle advances at one radian per second, a turn every 2π ≈ 6.28 seconds at Speed 1. That gives r = r₀/√(1 + 2·r₀²·t). From radius 1 a seed is at 0.577 after a second, and at 0.5 after 1.5 seconds. After 20 seconds it is only at 0.156: the approach is algebraic, not exponential, and the threads pile up near the centre without ever arriving.

A centre that is not one

Drop the cubic terms and the plane system is ẋ = −z, ż = x, whose eigenvalues are ±i: circles, every orbit closed, nothing winding in or out. That is the linearisation at the origin, and it is exactly as wrong as a linearisation can be. The cubic term is too small near the origin for the eigenvalues to notice it, yet it decides the answer everywhere. ṙ = −r³ is negative for any nonzero radius, so every orbit spirals in and the origin is a sink.

This is the standard warning about linearising a flow. The Hartman–Grobman theorem says the linearisation gets the picture right near a rest point, but only when no eigenvalue sits on the imaginary axis. Here both do, and the theorem has nothing to say. A centre in the linearisation may be a sink, a source, or a genuine centre, and only the higher terms know which.

Try

• Subtract 1 inside both brackets, −z − x·(x² + z² − 1) and x − z·(x² + z² − 1), so that ṙ = r − r³. Seeds inside and outside now both end on the circle of radius 1: the limit cycle of the Hopf normal form.
• Flip the sign of both cubic terms instead. Then ṙ = +r³ and a seed at radius r₀ runs off to infinity at t = 1/(2·r₀²), two seconds from radius 0.5, and the threads leave the box.
• Raise Seed spread to its top: the outer seeds cover their first turn much faster than the inner ones, since the cubic bites hardest far out, and the ring of threads bunches up.
• Top in the deck for the plane portrait, the spiral with the height projected away.
• Front instead, to watch the heights collapse in the first few seconds.

Read more

• Hartman–Grobman theorem
• Hopf bifurcation
• Limit cycle
• Lyapunov stability