Thomas Attractor

Attractor Builder (Blender Add-on)
Equations:
ẋ = - b*x + sin(y)
ẏ = - b*y + sin(z)
ż = - b*z + sin(x)
Parameters:
| b = 0.15 |
Simulation settings:
Initial state: x₀ = 0.01, y₀ = 0.02, z₀ = 0.03
Method: RK4
Time Step (dt): 0.03
Steps: 10000
Burn-in: 500
Scale: 0.7

The Thomas attractor is an example of a three-dimensional nonlinear system in which chaotic dynamics emerge from the structure of cyclic feedback between the variables. It was described by René Thomas in a study devoted to feedback circuits and their role in generating complex dynamical behaviour, including various forms of deterministic chaos. The author introduces the general form of the system (1999, p. 1896, Eq. 2):

\( \dot{x} = -b x + f(y), \)

\( \dot{y} = -b y + f(z), \)

\( \dot{z} = -b z + f(x), \)

where \(b > 0\) is a damping coefficient and the function \(f\) introduces the nonlinearity responsible for chaotic behaviour. Thomas demonstrated that many different choices of \(f\)—including \(f(u)=\sin u\)—lead to chaotic trajectories as long as the cyclic feedback structure is preserved. For values around \(b \approx 0.15\), the system produces complex, aperiodic motion in which the trajectories wrap repeatedly around symmetric regions of phase space. This model is an example of the so-called “labyrinth chaos”, a phenomenon Thomas describes as emerging from cyclic coupling combined with nonlinear response.

Source:

Thomas, R. (1999). Deterministic chaos seen in terms of feedback circuits: analysis, synthesis, “labyrinth chaos”. International Journal of Bifurcation and Chaos, 9(10), 1889–1905. https://doi.org/10.1142/S0218127499001383