Hopf Bifurcation
Flow · (ẋ, ẏ, ż) = f(x, y, z, t)
(μ · x − z − x · (x² + z²), −0.2y, x + μ · z − z · (x² + z²))
Open in the app The dials and keys named below are the app's.
The system
The app is y-up, so the motion worth watching happens in the floor plane, spanned by x and z. The height has a line to itself, ẏ = −0.2·y, an e-fold every 5 seconds at Speed 1, so the seeds settle onto that plane in the first few seconds and stay there.
Write r² = x² + z² and the two plane lines collapse into two short ones. The angle advances at exactly one radian per second, a turn every 2π ≈ 6.28 seconds. The radius obeys ṙ = r·(μ − r²), which is zero at the origin and at r = √μ. So the slider decides how many rest states there are. For μ ≤ 0 there is only the origin, and every seed winds into it. For μ > 0 the origin pushes seeds away and the circle of radius √μ catches them all: radius 1 at μ = 1, radius 2 at μ = 4.
Where the cycle is born
Near the origin the system is a rotation with a growth rate: the eigenvalues are μ ± i. As μ rises through 0 that pair crosses the imaginary axis, and the origin turns from a sink into a source. What is left behind matters more. The orbits do not simply run away, they pile onto a closed loop that appears out of the point itself and widens as the slider rises. This is the supercritical Hopf bifurcation, and these equations are its normal form.
The growth is a square root, so the loop opens fast and then slows. At μ = 0 exactly neither behaviour holds. The radius then obeys ṙ = −r³ alone, so the origin still wins but only algebraically, r = r₀/√(1 + 2·t), which from radius 1 is still at 0.079 after 80 seconds. That single value of the slider is the preset Cubic Spiral Sink.
History
Poincaré met this bifurcation around 1892, in celestial mechanics. Aleksandr Andronov proved a theorem for it in the plane in 1929, and Eberhard Hopf carried the result to any number of dimensions in 1942. The full name is the Poincaré–Andronov–Hopf bifurcation.
Try
- Drag μ down to 0: the ring shuts and the threads spiral in without ever arriving, which is the preset Cubic Spiral Sink.
- Drag μ to 4: the ring settles at radius 2. Four times the slider buys only twice the loop.
- Drag μ to −1: the origin takes every seed, and quickly, the decay now being exponential.
- Put a 4 in front of both cubic terms, μ · x − z − 4x · (x² + z²) and x + μ · z − 4z · (x² + z²). The radius now obeys ṙ = r·(μ − 4r²), and the ring halves to 0.5 at μ = 1.
- Top for the plane portrait, with the height projected away.