dx/dt = σ(y−x)
dy/dt = x(ρ−z)−y
dz/dt = xy−βz
The Lorenz equations are a simplified model of atmospheric convection — three coupled nonlinear ODEs solved here with Euler integration. For classic parameters (σ=10, ρ=28, β=8/3) the system is chaotic: it never repeats and is exquisitely sensitive to initial conditions.
- σ, ρ, β — the three physical parameters of the system; changing ρ below ~24 collapses the chaotic butterfly into stable fixed points.
- Integration Step dt — numerical step size; too large a step visibly destabilises the trajectory, illustrating ODE solver accuracy trade-offs.
- Spawn Nearby Trajectory — starts a second point a tiny distance away and tracks how fast the two diverge (the hallmark of chaos).
Real-world use: this exact model launched modern chaos theory and explains why weather forecasts become unreliable beyond about two weeks — tiny measurement errors amplify exponentially.