Magnetic pendulum
A pendulum swings above three magnets. Where it ends up depends on where it starts — but the dependence is fractal. Move a starting point by a hair and the outcome changes completely. Drag to release the pendulum, or paint the full basin map to see chaos made visible.
Sensitive dependence on initial conditions
The magnetic pendulum is one of the simplest physical systems that produces genuinely chaotic behavior. A pendulum bob, free to swing in two dimensions, hangs over three magnets arranged symmetrically. Gravity pulls the bob toward center, friction slows it, and the magnets attract it with a force that falls off with distance. Every initial position will eventually settle on one of the three magnets — but which magnet is extraordinarily sensitive to the starting point.
Fractal basins of attraction
Color each starting position by which magnet the pendulum ultimately reaches, and a fractal pattern emerges. Near the boundaries between basins, the colors interleave at ever-finer scales. No matter how much you zoom in, you find all three colors mixed together. This is the hallmark of a fractal basin boundary — the boundary between the basins has infinite length and zero area, yet it completely determines the system’s long-term behavior.
The physics
The equations of motion combine three forces: a restoring spring force pulling the bob toward center (modeling gravity’s lateral component), magnetic attraction from each magnet proportional to 1/(r² + h²)³⁄² where r is horizontal distance and h is the pendulum height above the magnet plane, and viscous damping proportional to velocity. The interaction of these simple forces creates a potential landscape with three wells, and the path between them is where the chaos lives.
Why prediction fails
In a chaotic system, tiny uncertainties in the initial state grow exponentially over time. For the magnetic pendulum, this means that to predict which magnet the bob will reach, you need to know the starting position to arbitrary precision. No finite measurement will do near the basin boundaries. This is not a failure of our instruments — it is a fundamental property of the dynamics. Deterministic equations producing effectively unpredictable outcomes.
Chaos is not randomness
The fractal basin map is completely deterministic. Run the same initial condition twice and you get the same result. What makes it chaotic is that nearby initial conditions can produce different results, and the boundary between “nearby” and “different” has fractal structure. The pattern is not random — it is infinitely complex but precisely defined. Every swirl and filament is computed exactly from Newton’s laws. Chaos is deterministic complexity, not indeterminacy.