Unlike a simple pendulum that slows down, the Van der Pol oscillator sustains its own oscillation: it draws energy from a source at small amplitude and dissipates it at large amplitude, settling on a unique limit cycle independent of starting conditions.
The equation ẍ − μ(1−x²)ẋ + x = 0 has a nonlinear damping term. For μ = 0 it is a harmonic oscillator. As μ grows, the limit cycle becomes increasingly non-sinusoidal — the 'relaxation oscillation' typical of cardiac pacemakers and electronic circuits.
Watch the phase portrait — position vs velocity — spiral onto the limit cycle from any initial condition. Increase μ to strengthen nonlinearity and observe the waveform becoming sawtooth-like. With external forcing enabled, find the Arnold tongues of frequency-locking.
Balthasar van der Pol discovered this oscillator studying vacuum-tube radio circuits in the 1920s. He also noticed that under certain forcing conditions the circuit made 'irregular noise' — one of the first experimental observations of deterministic chaos, decades before the word existed.
This simulation integrates the Van der Pol equation ẍ − μ(1−x²)ẋ + x = 0, rewritten as the first-order system ẋ = y, ẏ = μ(1−x²)y − x. It advances the state with a fourth-order Runge–Kutta scheme at a fixed step of 0.02. The left panel plots the phase portrait (x against ẋ), while the right panel shows the time series x(t), revealing how any trajectory settles onto a single closed limit cycle.
The μ slider (0 to 5) sets the strength of the nonlinear damping; x₀ chooses the starting position; Speed controls how many integration steps run per frame; and Pause, Reset and Trail manage playback and the fading path history. This self-sustaining oscillation models real systems such as vacuum-tube radio circuits, cardiac pacemaker rhythms and neuronal firing, where energy is pumped in at small amplitude and bled off at large amplitude.
What is the Van der Pol oscillator?
It is a nonlinear oscillator described by ẍ − μ(1−x²)ẋ + x = 0, originally derived by Balthasar van der Pol to model vacuum-tube electronic circuits. Its defining feature is a damping term that depends on amplitude, which lets the system sustain a stable, self-generated oscillation rather than dying away or growing without bound.
What is a limit cycle?
A limit cycle is an isolated closed loop in the phase plane that nearby trajectories spiral towards over time. In this simulation, no matter where you start, the trajectory converges onto the same loop, so the long-term amplitude and waveform are set by the equation rather than by the initial conditions.
What does the μ (damping) slider do?
The μ parameter scales the nonlinear damping term μ(1−x²)ẋ. Near μ = 0 the motion is almost a smooth sine wave like a harmonic oscillator. As μ rises towards 5, the limit cycle distorts into a sharp, sawtooth-like relaxation oscillation with quick jumps separated by slow drifts.
The damping coefficient μ(1−x²) changes sign with amplitude. When |x| is small the term is negative, adding energy and pushing the system outward; when |x| is large it becomes positive, removing energy. This balance produces a stable, self-sustaining oscillation instead of a steady decay to rest.
The left panel is the phase portrait, plotting position x on the horizontal axis against velocity ẋ on the vertical axis, so a periodic motion traces a closed loop. The right panel is the time series x(t), showing the same motion as a waveform unfolding in time so you can watch the shape change.
The page uses the classical fourth-order Runge–Kutta (RK4) method with a fixed time step of 0.02. RK4 evaluates the slope at four points within each step and combines them in a weighted average, giving high accuracy and good stability for this smooth, deterministic system.
The x₀ slider sets only the starting position (velocity begins at zero). Because the limit cycle is globally attracting, different starting points converge to the very same loop. You can see this by resetting with various x₀ values and watching each trajectory spiral onto the identical cycle.
At large μ the motion splits into two distinct timescales: long, slow phases where x drifts gently, punctuated by rapid jumps. This stop-and-go pattern is called a relaxation oscillation, and the resulting waveform looks far more like a sawtooth than a sine wave.
Yes, within the limits of numerical integration. The RK4 solver reproduces the true Van der Pol dynamics closely, including the correct limit-cycle amplitude and the shift towards relaxation behaviour as μ grows. Tiny rounding from the fixed 0.02 step has no visible effect on the qualitative behaviour shown.
It describes self-oscillating systems across many fields: triode and vacuum-tube radio circuits where it first arose, the rhythmic firing of cardiac pacemaker cells, neuronal spiking models, and certain mechanical and acoustic systems. It is a standard textbook example of self-sustained oscillation in dynamical systems theory.