2D Van der Pol Oscillator — Multi-Trajectory Limit Cycle

Van der Pol parameter

μ (nonlinear damping) 1.0

Integration speed 1.0×

Trail length 1500

μ presets

Convergence to the limit cycle

Spread across 5 starts

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Cycle amplitude (x)

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Time elapsed

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Five trajectories start at very different points in the (x, ẋ) plane — inside, outside and off-axis from the eventual loop. All five are integrated with the same real RK4 solver and the same μ, and every one of them spirals onto the identical closed curve. That single attracting curve is the limit cycle.

Equation & what changes with μ

The main canvas integrates the real Van der Pol equation with 4th-order Runge–Kutta, rewritten as a first-order system:

ẍ − μ(1 − x²)ẋ + x = 0
⇒  ẋ = y
   ẏ = μ(1 − x²)y − x

At small μ the nonlinear term is weak and the limit cycle is close to a circle — the time series looks almost sinusoidal. As μ grows, energy is pumped in hard near x = 0 and bled off hard near |x| = 2, producing a relaxation oscillation: long slow drifts broken by fast jumps, visible directly in the x(t) panel below as the waveform sharpens from a smooth wave into a near-square, sawtooth-edged shape.

Trajectory 1 (primary, drawn in time series) Trajectories 2–5 (other starts)

Time series — x(t), trajectory 1t = 0.0