# 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:

$b = 0.19$

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