Langford Attractor

Attractor Builder (Blender add-on)
Equations:
ẋ = (z - b) * x - d * y
ẏ = d * x + (z - b) * y
ż = c + a * z - (z**3)/3 - (x**2 + y**2) * (1 + e * z)
Parameters:
| a = 0.95 | b = 0.7 | c = 0.6 | d = 3.5 | e = 0.25 |
Simulation settings:
Initial state: x₀ = 0.1, y₀ = 0.0, z₀ = 0.0
Method: RK4
Time Step (dt): 0.015
Steps: 15000
Burn-in: 300
Scale: 0.1

The Langford attractor is a mathematical model of a chaotic system introduced by Canadian mathematician William F. Langford in 1984 during his numerical studies of the interaction between Hopf and hysteresis bifurcations. It was one of the first smooth three-dimensional systems to demonstrate how the coupling of these bifurcations can lead to torus breakdown and the onset of chaos. Although it is sometimes confused with the Aizawa attractor, Langford’s work provided the first formal formulation of this type of torus-breakdown dynamics, while Aizawa introduced a related simplified version of the system. The original system proposed by Langford (1984, p. 287) is given by equations:

\(\dot{x} = (z - \beta)x - \omega y\)

\(\dot{y} = \omega x + (z - \beta)y\)

\(\dot{z} = \mu + \alpha z - \tfrac{1}{3}z^{3} - (x^{2} + y^{2})(\ell + p z) + \varepsilon z x^{3}\)

In this formulation, the parameters μ, α, and β govern the local growth rates and stability of the stationary states. he pair ℓ and p determine the onset and geometry of the Naimark–Sacker (secondary Hopf or torus) bifurcation, while the frequency ω controls the rotational motion in the x–y plane. The parameter ε introduces non-axisymmetric perturbations that destroy the smooth invariant torus and initiate a cascade of bifurcations — phase locking, period doubling, and eventually chaotic dynamics. As ε increases, the attractor evolves from a smooth torus to a fractal "thick torus" and finally to transient chaos.

Source: Langford, W. F. (1984). Numerical Studies of Torus Bifurcations. In: International Series of Numerical Mathematics, Vol. 70, Birkhäuser Verlag, Basel, pp. 285–295.
DOI: 10.1007/978-3-0348-6256-1_19