Chaos and strange attractors

Watch deterministic equations produce unpredictable behaviour. Thousands of particles trace four strange attractors in real time, a phase portrait shows trajectories settling onto a limit cycle, and eight double pendulums released a hair apart show the butterfly effect.

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What you can explore

Lorenz attractor

The famous butterfly, traced by 1,800 particles integrated with RK4. Adjust σ, ρ and β and watch the attractor reshape, with its two wing-centre equilibria marked. Above ρ ≈ 24.74 the motion turns chaotic.

Rössler attractor

A flat spiral that periodically lifts into a spike and folds back: the stretch-and-fold mechanism of chaos in its simplest form. Raise c to follow the period-doubling route into chaos.

Thomas attractor

A cyclically symmetric tangle built from sine functions. One damping parameter, b, walks it from orderly loops into a chaotic three-dimensional lattice.

Aizawa attractor

Trajectories wrap around a sphere-like surface, then escape up a narrow tube along its axis and fall back over the top.

Phase space

The Van der Pol oscillator’s phase portrait. Thousands of trajectories stream across the plane and converge on a glowing limit cycle that turns into fast relaxation jumps as the nonlinearity μ grows.

Double pendulum

Eight double pendulums start 0.00001 radians apart. A live graph shows their separation growing exponentially until they disagree completely, with a running estimate of the Lyapunov exponent.

Key equations

  • dx/dt = σ(y − x)
  • dy/dt = x(ρ − z) − y
  • dz/dt = xy − βz
  • ẍ − μ(1 − x²)ẋ + x = 0

Curriculum

Enrichment: nonlinear dynamics and chaos.

Common questions

What is a strange attractor?

A set in phase space that trajectories approach but never repeat on. Nearby starting points separate exponentially, yet every trajectory stays on the same fractal shape.

Why is the double pendulum chaotic?

Its equations are nonlinear, so tiny differences in the starting angle grow exponentially. After a few seconds the motion is effectively unpredictable even though it is completely deterministic.

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