Burke–Shaw Attractor

Attractor Builder (Blender add-on)
Equations:
ẋ = -a * (x + y)
ẏ = -y - a * x * z
ż = a * x * y + b
Parameters:
| a = 10 | b = 13 |
Simulation settings:
Initial state: x₀ = 0.1, y₀ = 0.0, z₀ = 0.0  
Method: RK4  
Time step (dt): 0.01  
Steps: 15000  
Burn-in: 500  
Scale: 0.3

The Burke–Shaw attractor was proposed by Robert Shaw in 1981 during his studies on nonlinear chaotic flows conducted jointly with physicist Bill Burke. It represents an example of an unbounded strange attractor — a system without fixed points, whose trajectories can extend infinitely along one axis. The original system of equations (1981, p. 104) is given by:

\(\dot{x} = -10x - 10y\)

\(\dot{y} = -10xz - y,\) \(\quad V = 13\)

\(\dot{z} = 10xy + V\)

In this system, the coefficient a defines the strength of nonlinear coupling between x and y, while the parameter b acts as a constant offset in the equation for z, causing a slow drift of trajectories along the z-axis. Although the flow is not bounded, its solutions form a locally organized structure exhibiting chaotic features, known as the Burke–Shaw attractor.

Source: Shaw, R. (1981). Strange Attractors, Chaotic Behavior and Information Flow. Zeitschrift für Naturforschung A, 36(1), 80–112.
Online access: https://www.degruyterbrill.com/document/doi/10.1515/zna-1981-0115/html?lang=en