Rikitake Attractor

Attractor Builder (Blender add-on)
Equations:
ẋ = -b * x + z * y
ẏ = -b * y + (z - a) * x
ż = 1 - x * y
Parameters:
| a = 5 | b = 2 |
Simulation settings:
Initial state: x₀ = 1, y₀ = 1, z₀ = 1  
Method: RK4  
Time step (dt): 0.01  
Steps: 20000  
Burn-in: 500  
Scale: 0.4

The Rikitake attractor originates from the two-disk dynamo model proposed by Tsuneji Rikitake in 1957 to explain irregular reversals of the Earth's magnetic field. The model describes two coupled electrical circuits driving and influencing each other through magnetic induction. In its original physical formulation (1957, pp. 90–91), the system was written as follows:

\( L_1 \frac{dI_1}{dt} + R_1 I_1 = 2\pi M \Omega_1 I_2, \)

\( L_2 \frac{dI_2}{dt} + R_2 I_2 = 2\pi N \Omega_2 I_1, \)

\( G_1 \frac{d\Omega_1}{dt} = G_1 - 2\pi M I_1 I_2, \quad G_2 \frac{d\Omega_2}{dt} = G_2 - 2\pi N I_1 I_2. \)

Rikitake showed that for certain parameter values, the coupled equations for the electric currents (I₁, I₂) and angular velocities (Ω₁, Ω₂) produce irregular—chaotic—oscillations that could model geomagnetic polarity reversals. After simplifying and nondimensionalizing the system, the model reduces to the classical three-variable form now known as the Rikitake attractor (McMillen, 1999, pp. 1):

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

\(\dot{y} = -\mu y + (z - a)x\)

\(\dot{z} = 1 - x y\)

In the Blender add-on implementation, parameters a and b correspond to the coupling strength and damping coefficient (the analog of μ). For a = 5 and b = 2, the system generates the well-known double-wing chaotic attractor, illustrating spontaneous polarity switches—an analogy to the Earth's geomagnetic field reversals.

Sources:
Rikitake, T. (1957). Oscillations of a System of Disk Dynamos. Proceedings of the Cambridge Philosophical Society, 54(1), 89–105. DOI: 10.1017/S0305004100033223
McMillen, T. (1999). The Shape and Dynamics of the Rikitake Attractor. The Nonlinear Journal, 1, 1–10.