Stranger Things: Strange Attractors: The Lorenz Attractor

  $$\frac{dx}{dt} = y - x$$
  
  $$\frac{dy}{dt} = x - z - y$$
  
  $$\frac{dz}{dt} = xy - z$$

A beautiful example of a deterministic chaotic system, represented by three simple, yet coupled differential equations:

These equations model fluid convection, producing a complex, butterfly-shaped pattern (the "attractor") in 3D space. It's known for its extreme sensitivity to initial conditions, illustrating the butterfly effect: tiny differences in starting points lead to vastly different long-term outcomes, making long-term prediction impossible despite the system being fully determined by its equations.

This drawing was developed in Swift as a visualization tool for MacOS and other iOS devices. It would potentially be fun to run the tool on Apple Watch OS. The initial conditions were set to ⍴ = 28, σ = 10, β = 8/3.