Some systems don’t settle to rest — they settle into a LOOP. The Van der Pol oscillator pumps energy in when the swing is small and bleeds it out when the swing is large, so from ANY start the trajectory spirals onto one closed orbit and circles it forever. The calm would be the lie. Slide the nonlinearity and watch every path fall onto the same ring.
The Van der Pol oscillator x″ − μ(1−x²)x′ + x = 0 has an UNSTABLE fixed point at the origin (for μ>0) and a single stable LIMIT CYCLE: trajectories from inside spiral out and from outside spiral in, both converging to one closed orbit of fixed amplitude (~2 for small μ). The attractor is a loop, not a point — the system converges to sustained oscillation. A fail-loud self-check throws unless two different initial conditions reach the same nonzero amplitude and keep oscillating. ◆ real nonlinear dynamics, node-verified.
Forward-Euler integration and a fixed μ are the illustration; the existence of an isolated stable limit cycle (Liénard’s theorem) and origin instability are exact. ‘Settles into motion’ is literal here, not a metaphor.