Thomas attractor simulation

René Thomas’s system is beautifully symmetric: each variable is driven by the sine of the next, minus a little damping. A single parameter, b, controls how dissipative it is, and sweeping it walks the system from simple order into chaos.

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

What you will see

Particles weave a cyclically symmetric tangle of loops through 3D space. With strong damping the motion settles down; reduce it and the loops multiply into a chaotic lattice that keeps the same three-fold symmetry.

Try this

  1. Raise b and watch the motion settle into simple orbits.
  2. Lower b slowly and look for the point where chaos appears.
  3. Orbit the camera to see the symmetry between the x, y and z axes.

Common misconception

More damping does not always mean less interesting. Damping is what makes trajectories collapse onto an attractor at all; with zero damping the system has no attractor and simply wanders.

Exam-style question

What role does the parameter b play in the Thomas system?

b is a damping term. For larger b (above about 0.21) trajectories settle onto fixed points or periodic orbits; as b decreases below that the system becomes chaotic, and at b = 0 it no longer dissipates at all and trajectories wander through space.

Key equations

  • dx/dt = sin y − bx
  • dy/dt = sin z − by
  • dz/dt = sin x − bz

Curriculum

Enrichment: nonlinear dynamics and chaos.

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