Sprott Attractors

Attractor Builder (Blender add-on)
Equations:
ẋ = -y
ẏ = x + z
ż = x*z + a*y**2 
Parameters:
| a = 3 |  
Simulation settings:
Initial state: x₀ = 0.1, y₀ = 0, z₀ = 0  
Method: RK4  
Time step (dt): 0.01  
Number of steps: 30000  
Burn-in phase: 500  
Scale: 0.7

The Sprott attractors form a family of simple chaotic systems introduced by Julien C. Sprott in 1994. The author analyzed the general form of three-dimensional autonomous differential equations with quadratic nonlinearities and identified nineteen distinct examples exhibiting deterministic chaos. These systems — labeled from A to S — are remarkably simple: each contains only five or six terms, with one or two nonlinearities. Below are all systems (A–S) in their original form (1994, p. 649, Table I):

A: \(\dot{x}=y,\ \dot{y}=-x+y\,z,\ \dot{z}=1-y^{2}\)

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

C: \(\dot{x}=y\,z,\ \dot{y}=x-y,\ \dot{z}=1-x^{2}\)

D: \(\dot{x}=-y,\ \dot{y}=x+z,\ \dot{z}=x\,z+3\,y^{2}\)

E: \(\dot{x}=y\,z,\ \dot{y}=x^{2}-y,\ \dot{z}=1-4\,x\)

F: \(\dot{x}=y+z,\ \dot{y}=-x+0.5\,y,\ \dot{z}=x^{2}-z\)

G: \(\dot{x}=0.4\,x+z,\ \dot{y}=x\,z-y,\ \dot{z}=-x+y\)

H: \(\dot{x}=-y+z^{2},\ \dot{y}=x+0.5\,y,\ \dot{z}=x-z\)

I: \(\dot{x}=-0.2\,y,\ \dot{y}=x+z,\ \dot{z}=x+y^{2}-z\)

J: \(\dot{x}=2\,z,\ \dot{y}=-2\,y+z,\ \dot{z}=-x+y+y^{2}\)

K: \(\dot{x}=x\,y-z,\ \dot{y}=x-y,\ \dot{z}=x+0.3\,z\)

L: \(\dot{x}=y+3.9\,z,\ \dot{y}=0.9\,x^{2}-y,\ \dot{z}=1-x\)

M: \(\dot{x}=-z,\ \dot{y}=-x^{2}-y,\ \dot{z}=1.7+1.7\,x+y\)

N: \(\dot{x}=-2\,y,\ \dot{y}=x+z^{2},\ \dot{z}=1+y-2\,z\)

O: \(\dot{x}=y,\ \dot{y}=x-z,\ \dot{z}=x+x\,z+2.7\,y\)

P: \(\dot{x}=2.7\,y+z,\ \dot{y}=-x+y^{2},\ \dot{z}=x+y\)

Q: \(\dot{x}=-z,\ \dot{y}=x-y,\ \dot{z}=3.1\,x+y^{2}+0.5\,z\)

R: \(\dot{x}=0.9-y,\ \dot{y}=0.4+z,\ \dot{z}=x\,y-z\)

S: \(\dot{x}=-x-4\,y,\ \dot{y}=x+z^{2},\ \dot{z}=1+x\)

Cases A–E contain five terms and two nonlinearities, while F–S have six terms and one. They are three-dimensional dissipative systems with strange attractors, often exhibiting a spiral structure with a single fold, similar to the Rössler attractor. In the Blender add-on, the system corresponding to attractor D is provided, which belongs to the group of systems with two nonlinearities. It features a complex spiral motion with a single focus point and a folded-band structure reminiscent of the Rössler attractor. This is one of the classical examples where a simple set of equations produces rich, irregular trajectories in phase space. These simple systems generating deterministic chaos were later described in detail by Sprott in his book Elegant Chaos: Algebraically Simple Chaotic Flows (2010), which provides a comprehensive overview of algebraically simple three-dimensional chaotic flows.

Sources:

Sprott, J. C. (1994). Some simple chaotic flows. Physical Review E, 50(2), 647–650. DOI: https://doi.org/10.1103/PhysRevE.50.R647

Sprott, J. C. (2010). Elegant Chaos: Algebraically Simple Chaotic Flows. World Scientific Publishing, Singapore. DOI: https://doi.org/10.1142/7183