The Lorenz Attractor
System Analysis
The Equations
dx/dt = σ(y - x)
dy/dt = x(ρ - z) - y
dz/dt = xy - βz
Current State
Lyapunov Estimate:
λ ≈ 0.00
Understanding Chaos
Parameters
- σ (sigma): Prandtl number, ratio of momentum diffusivity to thermal diffusivity
- ρ (rho): Rayleigh number, temperature difference between top and bottom
- β (beta): Geometric factor
Key Features
- Exhibits sensitive dependence on initial conditions
- Never settles into a steady state or simple periodic orbit
- Trajectories form a strange attractor
- System is deterministic but unpredictable long-term