Chaos Theory Lab

A system is <strong>chaotic</strong> when it's deterministic (no randomness — the same input always gives the same output) yet shows <strong>sensitive dependence on initial conditions</strong>: two starts that differ by a billionth eventually diverge completely. That's the butterfly effect. Chaos also tends to be bounded and to settle onto a <strong>strange attractor</strong> — an intricate fractal shape the trajectory never exactly repeats.

The logistic map is the deceptively simple equation <strong>x_n+1 = r·x_n·(1 − x_n)</strong>, a toy model of population growth. As you raise <strong>r</strong>, the long-term behaviour doubles from a steady value to a 2-cycle, 4-cycle, 8-cycle… and at r ≈ 3.5699 it becomes chaotic. The <strong>bifurcation diagram</strong> plots those final values against r, revealing the famous period-doubling tree and the Feigenbaum constant (≈ 4.669).

It's the butterfly-shaped solution of Edward Lorenz's 1963 weather model — three coupled equations (dx/dt = σ(y−x), dy/dt = x(ρ−z)−y, dz/dt = xy−βz). With σ=10, ρ=28, β=8/3 the trajectory loops forever around two wings without ever crossing itself or repeating. It was the discovery that launched chaos theory, and it's why long-range weather forecasting is fundamentally limited.

It measures how fast nearby trajectories pull apart. A <strong>positive</strong> Lyapunov exponent is the signature of chaos. For the logistic map at r = 4 it equals exactly ln 2 ≈ 0.693, meaning the distance between two nearby points roughly doubles each step — so even tiny measurement error makes long-term prediction impossible. The Butterfly-effect mode estimates it live.

Both come from iterating <strong>z → z² + c</strong> in the complex plane. The <strong>Mandelbrot set</strong> is the map of which c values stay bounded; each point of it corresponds to a whole <strong>Julia set</strong>. Their fractal boundaries are where order meets chaos — and the real axis of the Mandelbrot set is literally the logistic map's bifurcation diagram in disguise. For deep zooming there's also the dedicated <a href="https://utilitiesbunker.com/tools/fractal-explorer">Fractal Explorer</a>.

No. Every iteration, integration and pixel is computed in your browser with plain JavaScript and the canvas API. It works offline and nothing you do is uploaded.

Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.

Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.

Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.