# Van der Pol

Flow · (ẋ, ẏ, ż) = f(x, y, z, t)

`(z,  −0.2y,  μ · (1 − x²) · z − x)`

[Open in the app](https://www.wavelace.com/app#p=148) · [This page](https://www.wavelace.com/presets/van-der-pol)

### 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.

### Read more

- [Van der Pol oscillator](https://en.wikipedia.org/wiki/Van_der_Pol_oscillator)
- [Balthasar van der Pol](https://en.wikipedia.org/wiki/Balthasar_van_der_Pol)
- [Relaxation oscillator](https://en.wikipedia.org/wiki/Relaxation_oscillator)
- [Limit cycle](https://en.wikipedia.org/wiki/Limit_cycle)