Rössler attractor simulation

Otto Rössler designed his system in 1976 to be the simplest possible continuous system that is chaotic. Only one of its three equations is nonlinear, yet it produces a folded band that never repeats.

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

What you will see

Particles spiral outward in a flat disc, then occasionally lift up in a spike and fold back into the centre. The stretch-and-fold motion is the basic mechanism of chaos, shown here more clearly than in almost any other system.

Try this

  1. Lower c towards 3 and watch the band collapse into a simple loop.
  2. Raise c slowly and watch the loop double, then double again, before chaos.
  3. Follow one particle through a spike and see where it lands.

Common misconception

A strange attractor is not just a messy tangle. It has a precise fractal structure, and every trajectory stays on it forever, even though no two trajectories ever coincide.

Exam-style question

With a = b = 0.2, what happens to the Rössler attractor as c increases from about 2.5 to 5.7?

It follows a period-doubling route to chaos: a single loop (period 1) splits into a double loop (period 2), then period 4 and so on, with the doublings coming faster until the motion becomes chaotic, as at the classic value c = 5.7.

Key equations

  • dx/dt = −y − z
  • dy/dt = x + ay
  • dz/dt = b + z(x − c)

Curriculum

Enrichment: nonlinear dynamics and chaos.

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