🦋 Lorenz Equations Strange Attractor Viewer

They come from Lorenz's simplified atmospheric convection model: σ (Prandtl number) is the ratio of fluid viscosity to thermal conductivity, ρ (Rayleigh number) measures the applied temperature difference driving convection relative to the threshold needed for it to start, and β relates to the physical aspect ratio of the convecting layer. The "classic chaos" values used by default are σ=10, ρ=28, β=8/3, taken directly from Lorenz's original 1963 paper.

An attractor is a set of states a system settles into over time; it is "strange" because it has a fractal (non-integer) dimension and the trajectory on it never repeats or self-intersects, yet stays confined to a bounded region forever. In this simulation you can see the trail wind around both lobes indefinitely without the path ever exactly retracing itself, which is the defining behaviour of a strange attractor.

The simulation integrates the same three coupled ODEs every frame with RK4 from a fixed starting point (0.1, 0, 0). Because the Lorenz system has a positive Lyapunov exponent (λ₁ ≈ 0.91 at the classic parameters), two trajectories that start an infinitesimal distance apart diverge exponentially, roughly as e^(0.91t) — doubling their separation every ≈0.76 time units — even though both remain confined to the same butterfly-shaped attractor.

The Lorenz system is chaotic and highly sensitive to numerical error, so a low-order method like Euler's would accumulate error quickly and distort the trajectory's long-term shape. Fourth-order Runge–Kutta cancels error terms up to O(dt⁴) per step, so with a tiny step size of dt ≈ 0.005 the simulated attractor closely matches the true continuous solution while remaining fast enough to run in real time in the browser.

At ρ ≈ 24.74 (for σ=10, β=8/3) the system undergoes a subcritical Hopf bifurcation. Below this threshold the two symmetric equilibrium points become stable, and instead of a chaotic butterfly the trajectory spirals inward and settles onto one of them — no more chaos, no more attractor, just simple convergence. Above it, the equilibria are unstable and trajectories are repelled onto the strange attractor instead.