Aizawa attractor simulation

The Aizawa attractor is one of the most striking shapes in chaos theory: trajectories spiral around a roughly spherical surface, then shoot up through a narrow tube along its axis and fall back over the top.

Interactive, free and runs in your browser. Part of the Dynamical Systems simulations.

What you will see

Thousands of particles trace the sphere and its central tube, coloured by height. Change a and b to reshape the attractor and watch how small parameter changes alter its whole geometry.

Try this

  1. Watch a single particle travel up the tube and back down over the surface.
  2. Change a and see how the sphere flattens or stretches.
  3. Compare the density of particles near the tube with the rest of the surface.

Common misconception

The particles are not following a fixed path around the shape. Each one moves on its own unpredictable trajectory; the shape appears only because all trajectories stay on the same attractor.

Exam-style question

How can a deterministic system like this one be unpredictable?

Nearby starting points separate exponentially fast (a positive Lyapunov exponent). Any tiny error in the initial state grows until it is as large as the attractor itself, so long-term prediction fails even though each step is completely determined.

Key equations

  • dx/dt = (z − b)x − dy
  • dy/dt = dx + (z − b)y
  • dz/dt = c + az − z³/3 − (x² + y²)(1 + ez) + fzx³

Curriculum

Enrichment: nonlinear dynamics and chaos.

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