Lorenz Attractor — The Butterfly Effect

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Lorenz System (1963)

Edward Lorenz discovered that a simple 3-equation weather model produced wildly unpredictable behavior — the "butterfly effect." The attractor never repeats the same path yet stays bounded forever.
dx/dt = σ(y−x) | dy/dt = x(ρ−z)−y | dz/dt = xy−βz

Mandelbrot Set

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The Mandelbrot Set

A point c is in the set if z→z²+c never escapes |z|>2. The boundary is infinitely complex — zooming reveals endless self-similar detail. Colors represent how quickly each point escapes.
zₙ₊₁ = zₙ² + c

Julia Set

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Julia Sets

Each value of c produces a different Julia set — a fractal cousin of the Mandelbrot. When c is inside the Mandelbrot set, the Julia set is connected. Outside: it's "dust." As c changes, the Julia set continuously morphs.
zₙ₊₁ = zₙ² + c, z₀ = pixel position

Double Pendulum — Chaos in Motion

Sensitivity to Initial Conditions

Two pendulums (red and cyan) start at nearly identical angles — separated by just 0.001°. Their paths diverge rapidly, demonstrating chaos: tiny differences compound into completely different outcomes. This is the defining feature of a chaotic system.