Chaos Lab

Deterministic rules, unpredictable outcomes. Nudge the starting point by a hair and watch the futures part ways.

Ensemble

Readout (lead pendulum)

Kinetic–Potential– Total energy–Energy drift– Lyapunov λ–Poincaré points0
What am I seeing?
  • Drag either bob to set a new start. The ensemble copies it with a tiny offset ε in the lower angle.
  • Bottom left: log of the phase-space distance between two neighbours. A straight rising line is exponential divergence; its slope is the Lyapunov exponent.
  • Bottom right: Poincaré section. A dot is stored each time the upper arm passes straight down going right. Regular motion draws closed curves, chaos fills an area.

Parameters

Two trajectories

Readout

Time0Separation–
What am I seeing?
  • Blue and orange start a tiny distance apart. At first they travel together, then one lobe choice differs and they decouple.
  • Drag to rotate, scroll to zoom. For ρ < 1 the origin is stable; above about 24.74 the two fixed points (rings) lose stability and motion is chaotic.

Readout

r–Lyapunov λ–Behaviour–View–
What am I seeing?
  • x → r·x·(1−x). Each column of the diagram shows where the orbit settles for one value of r.
  • Drag a box to zoom, scroll to zoom at the cursor, click to pick r, double-click to reset. The diagram is self-similar: each window contains a copy of the whole.
  • Bottom left: cobweb. Bounce between the parabola and the diagonal. Bottom right: the iterates in time.
  • λ > 0 means chaos; λ < 0 means a stable cycle.

θ″ = −θ′/Q − sin θ + g·cos(Ω t)

Readout

Drive periods0Poincaré points0θ (wrapped)–ω–
What am I seeing?
  • A damped pendulum shaken sinusoidally. Strobing the state once every drive period gives the Poincaré map (large panel).
  • Periodic motion lands on a few dots. Chaos traces a folded, stretched strange attractor with fractal layers.
  • Bottom left: phase portrait of the live motion. Bottom right: angle against time.

Magnets & pendulum

Readout

Computed0%Pendulum–
What am I seeing?
  • Each pixel is a pendulum released from rest there. Its colour is the magnet it finally settles on; brighter means it settled sooner.
  • Near the magnets the regions are solid. Farther out, the pendulum swings over several magnets and the boundaries braid into a fractal.
  • Click anywhere to release a real pendulum from that spot and follow its path.

All motion is integrated with fourth-order Runge–Kutta in your browser. Nothing is pre-rendered.