Thomas Attractor

Originally proposed by the biologist René Thomas (1928–2017), this set of differential equations is cyclically symmetric.1 The constant \(b\) is a bifurcation parameter that describes how dissipative the system is. Dissipation is a property from thermodynamics that can be thought of as a kind of dampening which is the result of a loss of energy, i.e. in form of heat. Depending on the value of \(b\), the attractor changes its shape. Here is a list of what happens:

  • \(b \ge 1\): origin is a stable point
  • \(1 > b \gtrsim 0.329\): two stable points
  • \(0.329 > b \gtrsim 0.208\): two stable oscillations
  • \(0.208 > b > 0\): mostly chaos
  • \(b = 0\): no dissipation, brownian walk into infinity

There are several particularly interesting values for which more than just one attractor stabilise seperately:

  • \(b = 0.22\): two cycles
  • \(b = 0.19\): single cycle
  • \(b = 0.17\): three cycles
  • \(b = 0.13\): three cycles

Renders

Differential system:

\[\dot{x} = \sin(y) - b\, x\] \[\dot{y} = \sin(z) - b\, y\] \[\dot{z} = \sin(x) - b\, z\]

Constants:

\[b = 0.19\]

Thomas

Thomas


  1. R. Thomas, 1999. "Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, ‘labyrinth chaos’". Int. J. Bifurc. Chaos. 09(10). doi:10.1142/s0218127499001383