Arneodo Attractor

Attractor Builder (Blender Add-on)
Equations:
ẋ = a*(x-y)
ẏ = -4*a*y+x*z+m*x**3
ż = -d*a*z+x*y+b*z**2
Parameters:
| a = 1 | b = -0.09 | d = 1.5 | m = 0.01 |
Simulation settings:
Initial state: x₀ = 1, y₀ = 0.1, z₀ = 0.1
Method: Euler
Time Step (dt): 0.01
Steps: 15000
Burn-in: 1000
Scale: 0.1

The Arneodo attractor was introduced in 1981 by Alain Arneodo, Paul Coullet, and Christian Tresser as an example of a new mechanism for the onset of turbulence. The authors studied three-dimensional differential systems that preserve the symmetry \( S: (x, y, z) \mapsto (-x, -y, z) \), for which a sequence of increasingly complex stable homoclinic orbits appears. As the control parameter approaches a critical value, the system undergoes a series of bifurcations resulting from the splitting of these homoclinic orbits, ultimately leading to the formation of a chaotic attractor. The original system, introduced in the 1981 paper (p. 220, Eq. 11), has the form:

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

\( \dot{y} = -4\alpha y + xz + \mu x^{3} \)

\( \dot{z} = -\delta \alpha z + xy + \beta z^{2} \)

In their numerical investigation, the authors fix the parameters \(\alpha = 1.8\), \(\beta = -0.07\), and \(\delta = 1.5\), and then examine the evolution of the system under changes in the control parameter \(\mu\). For \(\mu \approx 0.076\) they observe a pair of stable homoclinic orbits. For \(\mu = 0.05\) the system exhibits a stable symmetric periodic orbit, while for \(\mu = 0.034\) a pair of stable orbits appears which are mutual images under the symmetry \(S\). Further decreasing \(\mu\) produces — at \(\mu = 0.02\) — a complex geometric structure that behaves as a strange attractor.

Source:

Arneodo, A., Coullet, P., & Tresser, C. (1981). A possible new mechanism for the onset of turbulence. Physics Letters A, 81(4), 197–201. DOI: https://doi.org/10.1016/0375-9601(81)90239-5