Double Pendulum Chaos

Exploring Deterministic Chaos and Sensitivity to Initial Conditions

Single Pendulum
Double Pendulum
Sensitivity Demo
Phase Space

Simple Harmonic Motion

A single pendulum exhibits predictable, periodic motion. The equation of motion for small angles is:

d²θ/dt² + (g/L)sin(θ) = 0

This serves as a baseline to understand the dramatic difference when we add a second pendulum.

45.0°
Angle (θ)
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Angular Velocity (ω)
100%
Total Energy

Chaotic Motion

The double pendulum is a classic example of deterministic chaos. Despite having simple, deterministic equations of motion, its behavior is:

  • Unpredictable: Long-term behavior cannot be predicted
  • Sensitive: Tiny changes in initial conditions lead to vastly different outcomes
  • Non-periodic: Motion never exactly repeats
  • Deterministic: No randomness - same initial conditions always give same result
Kinetic
Potential
Total
90.0°
θ1
90.0°
θ2
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ω1
0.00
ω2

Butterfly Effect

Watch 8 double pendulums with nearly identical initial conditions diverge dramatically over time. Each pendulum starts with θ1 differing by only 0.0001 radians (0.0057°) - smaller than a human can perceive.

This demonstrates the Lyapunov exponent - a measure of how quickly nearby trajectories diverge. For a chaotic system, this exponent is positive, meaning small differences grow exponentially.

This is why weather prediction is limited to ~10 days despite deterministic physics!

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Max Divergence (pixels)

Phase Space Portrait

Phase space plots position (θ) vs velocity (ω) for each pendulum. Each point represents the system's state at one moment.

  • Simple pendulum: Closed elliptical orbits (periodic)
  • Double pendulum: Complex, never-repeating trajectories (chaotic)
  • Poincaré section: Intersection points reveal hidden structure

Chaotic systems fill the available phase space densely but never exactly repeat.

Pendulum 1 (θ1 vs ω1)

Pendulum 2 (θ2 vs ω2)