Van der Pol oscillator phase portrait
The Van der Pol oscillator was first used to model vacuum-tube circuits. It pumps energy in when the oscillation is small and drains it when it is large, so every starting point ends up on the same closed loop: a limit cycle.
Interactive, free and runs in your browser. Part of the Dynamical Systems simulations.
What you will see
A phase portrait of position against velocity. Thousands of short trails stream across the plane, coloured by speed, and all converge on a glowing loop with beads moving round it in real time. A faint curve marks where the flow is purely sideways, and the unstable centre pulses outward.
Try this
- Set μ to 0 and see circles: simple harmonic motion.
- Raise μ to 1 and watch trajectories from inside and outside meet the same loop.
- Push μ to 3 and watch the beads crawl along the slow branches and whip round the fast jumps.
Common misconception
A limit cycle is not the same as an orbit in simple harmonic motion. SHM has a whole family of loops, one per amplitude; a limit cycle is a single isolated loop that every nearby trajectory is attracted to.
Exam-style question
What does the Van der Pol phase portrait look like at μ = 0, and why?
The equation becomes ẍ + x = 0, simple harmonic motion. Energy is conserved, so each trajectory is a circle (in suitable units) whose size depends only on the starting point, and there is no limit cycle.
Key equations
ẍ − μ(1 − x²)ẋ + x = 0
Curriculum
Enrichment: nonlinear dynamics and chaos.