Van der Pol
Flow · (ẋ, ẏ, ż) = f(x, y, z, t)
(z, −0.2y, μ · (1 − x²) · z − x)
Open in the app The dials and keys named below are the app's.
The system
The first line, ẋ = z, makes z the velocity of x. So the third line is an acceleration, and the pair is one second-order equation, ẍ − μ·(1 − x²)·ẋ + x = 0. The app is y-up, so the picture lies in the floor plane, position across and velocity in depth.
Read the middle term as friction, with coefficient −μ·(1 − x²). Inside |x| < 1 it is negative for any positive μ, so the swing is pumped rather than damped. Outside it the sign flips and energy is taken away, so a small orbit grows and a large one shrinks, with one balanced loop between them. Its amplitude in x is almost exactly 2, measured at 2.0001 for μ = 0.1, so the loop is 4 across. The slider does not move it.
What the slider changes
Where Hopf Bifurcation puts the cycle's size on the slider, this one puts its character. At μ = 0.1 the loop is nearly a circle and one turn takes 6.288 seconds at Speed 1. Raise μ and it stops being a circle: the loop grows tall in the velocity direction, reaching ±14.2 at μ = 10. There the fastest speed on the loop is over a thousand times the median.
That is a relaxation oscillation. The seed creeps along one slow branch, snaps across to the other, and creeps back. It arrives well below the top of the slider: at μ = 2 the fastest point is already twelve times the median, and the loop still fits the opening plot to about μ = 3. The creeping sets the pace, so the period grows with μ, 19.1 seconds at μ = 10 against the asymptotic (3 − 2·ln 2)·μ ≈ 16.1. Below zero the friction changes sign everywhere and the seeds wind into the origin.
History
Balthasar van der Pol met this equation at Philips in the 1920s, in vacuum tube circuits. A tube feeding current back into its own tuned circuit is just such a friction, changing sign with the amplitude. He named the large-μ behaviour in a paper of 1926, On relaxation-oscillations.
Try
- Drag μ to 0.1: a circle 4 across, turning once every 6.288 seconds, against 2π ≈ 6.283.
- Drag μ to 3: the creep and the snap are plain, the fastest point 38 times the median, and the loop still fits the plot.
- Drag μ to 10: one turn takes 19.1 seconds. The loop reaches ±14.2 in the velocity direction, so raise Span of x, y, z to 16 to see all of it.
- Drag μ to −1: nothing is pumped and every seed winds into the origin.
- Divide the square by 4, μ(1 − x²/4)z − x: the pumped band is twice as wide, so the cycle settles at an amplitude of 4 instead of 2.