Halvorsen Attractor

Attractor Builder (Blender add-on)
Equations:
ẋ = -a*x - 4*y - 4*z - y**2
ẏ = -a*y - 4*z - 4*x - z**2
ż = -a*z - 4*x - 4*y - x**2
Parameters:
| a = 1.4 |
Simulation settings:
Initial state: x₀ = 0.01, y₀ = 0, z₀ = 0.01
Method: DP5
Tolerance: 0.0001
Min step: 0.000001
Max step: 0.1
Steps: 10000
Burn-in: 300
Scale: 0.2

The Halvorsen attractor is a symmetric three-dimensional chaotic system originally proposed by Arne Dehli Halvorsen in the 1990s on the Usenet discussion group sci.fractals. It later gained attention when Julien C. Sprott analyzed it formally and published his results in 1997 in the note A Symmetric Chaotic Flow (University of Wisconsin–Madison). The attractor is notable for its cyclic symmetry and elegant, three-lobed geometry, making it both mathematically interesting and visually appealing. The original system of equations (Sprott, 1997, Eq. 1) is given as:

\(\dot{x} = -a x - 4y - 4z - y^{2}\)

\(\dot{y} = -a y - 4z - 4x - z^{2}\)

\(\dot{z} = -a z - 4x - 4y - x^{2}\)

This system is characterized by its complete cyclic symmetry: the equations remain unchanged under the transformation \(x \rightarrow y \rightarrow z \rightarrow x\). The single parameter a controls the level of damping and determines the transition from periodic to chaotic behavior. Sprott demonstrated that for values around a ≈ 1.27, the system follows a classical period-doubling route to chaos, producing a complex and aesthetically symmetric strange attractor.

Source:
Sprott, J. C. (1997, revised 2004). A Symmetric Chaotic Flow. Department of Physics, University of Wisconsin–Madison. Online: https://sprott.physics.wisc.edu/chaos/symmetry.htm