THE LIMIT CYCLE

A self-winding oscillation the system returns to from every direction. The van der Pol oscillator x″ − μ(1−x²)x′ + x = 0 has nonlinear damping that pumps energy in at small amplitude and drains it at large amplitude — carving one isolated closed orbit that every nearby trajectory, inside or out, spirals onto. Rendered, not quoted.

source van der Pol, B. — On Relaxation-Oscillations, London, Edinburgh & Dublin Phil. Mag. (Ser. 7) 2, 978–992 (1926). doi:10.1080/14786442608564127

Blue Team · builds & defends
3
The Model

Write it as a plane flow with y = x′:

x′ = y
y′ = μ(1−x²)·y − x

The factor (1−x²) is the whole trick. Where |x|<1 the damping is negative (energy pumped in); where |x|>1 it is positive (energy removed). Amplitude cannot sit at zero and cannot run away — it settles on the one orbit where pumped and drained energy cancel over a cycle.

Assumed: an ideal isolated system, exact quadratic damping, no forcing or noise. A real relaxation oscillator only approximates this.

5
The Lineage

A self-sustaining clock. Van der Pol’s isolated attracting oscillation (1926) is the object that the-hopf-bifurcation gives birth to when a stable focus turns unstable, and the shape a heartbeat resembles — an amplitude the system defends against a push either way.

Unlike a conservative center (a nested family of orbits, one per energy), this cycle is isolated: perturb the amplitude and it comes back.

7
The Witness

Re-runs the live check that both a tiny-start and a large-start trajectory land on the same orbit. It confirms green, and flips red the instant the tamper (window 6) removes the nonlinear damping.

witness: booting…
The Machine
4
Data In  in ↓

Two very different initial conditions on the same plane: inside (0.5, 0) and outside (3.0, 0), plus the damping strength μ. RK4, fixed step h=0.005.

↓
0
The Panel  lit

Blue tracers start inside and spiral out; red tracers start outside and spiral in. Both wind onto the bright green orbit — the isolated attracting limit cycle, computed live.

↓
8
Data Out  out ↓
proving…

One isolated orbit; inside and outside converge to it; period set by the dynamics, not the start. Trapped in an annulus — Poincaré–Bendixson guarantees a cycle there.

Red Team · attacks & breaks
1
The Adversary

wall“An oscillation just needs a spring and small friction — a plain x″+cx′+x=0 rings and settles. Your extra term is decoration.”

Rebuttal. Linear damping has no preferred amplitude: c>0 decays to rest, c<0 blows up, c=0 gives a whole family of neutral circles that depend on the start. Only the sign-changing damping μ(1−x²) pins a single amplitude the flow returns to. That is exactly what window 6 destroys.

2
The Graveyard

“Bigger start ⇒ bigger final swing.”

→ False. Every start converges to the same amplitude ≈ 2.01; the cycle is isolated, not a nested center.

“The origin is a fine resting point.”

→ False. At small amplitude the damping is negative; the origin is an unstable spiral (Jacobian trace μ>0), so rest is never reached.

“Period grows with amplitude.”

→ False here. Both starts give the same period ≈ 6.66 — set by the dynamics, not the launch.

6
The Tamper

Replace the nonlinear damping with a constant positive one (x″+0.5x′+x=0). The origin becomes a stable spiral and the limit cycle vanishes — everything decays to 0. Window 7 catches it live.