Two Van der Pol oscillators — nonlinear systems obeying ẍ − μ(1−x²)ẋ + x = 0 — are wired together with a coupling term κ(xⱼ−xᵢ). Each oscillator alone settles into its own stable limit cycle, but once κ rises above zero their rhythms pull toward each other, a phenomenon called spontaneous synchronization. It is the same mechanism thought to keep pacemaker cells in the heart, neurons, and circadian clocks beating in step.
An RK4 integrator solves the coupled ODEs in real time. The Time series view plots x₁(t) and x₂(t) stacked on top of each other; the Phase portrait view plots (x, ẋ) trajectories side by side so you can watch two limit cycles drift into lock-step.
Drag Damping μ to change how sharply each oscillator relaxes back to its cycle, Coupling κ to strengthen the link between them, and Phase offset Δφ to set how far apart they start. Switch between Time series and Phase portrait, or hit Reset to restart from the chosen offset.
The order parameter r = cos(Δφ) and the |Δφ| readout are exactly how researchers quantify synchronization in real biological networks — the same statistic underlies the Kuramoto model used to study crowd clapping and firefly flashing.
It's a nonlinear second-order ODE, ẍ − μ(1−x²)ẋ + x = 0, originally used to model vacuum-tube circuits. Unlike a simple pendulum, it has a self-sustaining limit cycle: no matter the starting point, x(t) settles into the same periodic orbit, with μ controlling how strongly nonlinear damping shapes that cycle.
When κ > 0, each oscillator feels a pull toward the other's current state via the term κ(xⱼ−xᵢ). If that pull is strong enough relative to the phase difference, it nudges both oscillators' rhythms until they lock in-phase or anti-phase — the diffusive coupling term is doing all the work.
κ sets how much one oscillator's state influences the other's acceleration. At κ=0 they evolve independently and drift out of phase; as κ increases past a critical threshold, the |Δφ| readout collapses toward zero (or 180°) and the sync badge switches from "Not synced" to "In-phase" or "Anti-phase".
In-phase locking (|Δφ| < 15°) means both oscillators rise and fall together, like two hearts beating in unison. Anti-phase locking (|Δφ| > 165°) means they alternate, peaking exactly when the other troughs — both are stable synchronized states depending on initial phase offset and coupling.
Van der Pol dynamics are a standard simplified model for cardiac pacemaker cells, circadian pacemaker neurons, and bursting neurons. Coupling many such oscillators together is how biologists explain how millions of independent cells produce one coherent heartbeat or a single 24-hour circadian rhythm.