Why is the magnetic pendulum chaotic?

The pendulum is governed by nonlinear ODEs where three competing magnetic attractors create a phase space with exponential sensitivity to initial conditions. Infinitesimally nearby starting positions can end at different magnets, giving a positive Lyapunov exponent and the defining hallmark of chaos.

What are basins of attraction?

Each magnet's basin of attraction is the set of all starting positions from which the pendulum eventually comes to rest at that magnet. In the magnetic pendulum these basins have fractal boundaries — they are nowhere smooth, so no matter how precisely you specify the starting position near a boundary, you cannot predict which magnet wins.

What do the colours in basin mode show?

Red, green, and blue regions correspond to the three magnets. Each pixel is coloured by simulating a pendulum released from rest at that world-space coordinate and recording which magnet it settles near. The fractal boundary between colours is a visual proof of chaos: nearby initial conditions can produce wildly different outcomes.

How does damping affect the chaos?

Lower damping means the pendulum oscillates longer before settling, greatly increasing trajectory complexity and fractal boundary intricacy. Very high damping causes the pendulum to settle quickly and predictably near the nearest magnet, reducing chaos. The transition between behaviours is itself a rich area of nonlinear dynamics research.

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🧪 Magnetic Pendulum — Chaos & Fractal Basins of Attraction

Simulate a magnetic pendulum above 3 magnets — watch chaotic motion and reveal fractal basins of attraction showing which magnet wins from each starting point. RK4 integration, 6 presets.

Chaos & Dynamics2DModerate60 FPS

⚙ Under the hood

This simulation demonstrates a magnetic pendulum, showcasing chaotic behavior and fractal basins of attraction. The pendulum's motion is influenced by both gravity and the magnetic force, leading to complex oscillations that exhibit sensitive dependence on initial conditions.

Magnetic PendulumChaosBasins

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install