Butterfly Effect and Complexity

How does unpredictability emerge from deterministic equations? From Poincaré's discovery a hundred years ago, to Lorenz's butterfly, to Mandelbrot's fractals, to the chaos engineering of today.

About 30 min·6 nodes

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The path across the map

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STEP 1 · ConceptNo location

Differential Equations

Equations relating functions to their derivatives — the language of laws of change.

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STEP 2 · Mathematician~1890Paris· Main activity

Henri Poincaré

A wide-ranging mathematician who connected differential equations, celestial mechanics, topology, mathematical physics, and philosophy. Corrections to his prize memoir on the three-body problem revealed complex orbit structure that later became central to dynamical systems.

↓73 years later

STEP 3 · Concept1963Cambridge, MA· Discovery

Chaos Theory

Exact deterministic rules can amplify tiny differences in initial conditions and frustrate long-range trajectory prediction. Chaos theory studies that sensitivity together with bifurcations, attractors, and periodic windows—the structure that remains inside apparently irregular motion. Chaos is not another word for randomness or lawlessness.

↓12 years later

STEP 4 · Mathematician1975New York· Main activity

Benoît Mandelbrot

A mathematician who turned scale-dependent complexity into a common question across several fields. Born in Warsaw, Mandelbrot moved with his family to France in 1936. War severely disrupted his schooling, but he later studied at the École Polytechnique and elsewhere. His 1967 paper revisited Lewis Fry Richardson’s coastline data and asked how measured length changes with yardstick size. In an ideal fractal model with dimension above one, length can diverge as scale tends to zero, but this is not a proof that every physical coastline is literally infinite. At IBM he studied communication noise, price variation, and geometric repetition, introducing the word “fractal” in 1975. The term is a tool for comparing irregularity and scale dependence, not a claim that all natural phenomena obey one self-similar law. Around 1980 he used computer images to study the parameter set of z↦z²+c, helping make complex dynamics broadly visible.

↓Same year

STEP 5 · Concept1975New York· Discovery

Fractal

Self-similar shapes at every scale — revealing nature's jagged geometry beyond smooth curves.

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STEP 6 · ConceptNo location

Dimension

How many numbers do you need? — From 0D to ∞D and even fractional dimensions in fractals.

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