Lorenz Attractor
The canonical strange attractor: a continuous dynamical system whose trajectories never repeat yet never escape.
The System
In 1963, Edward Lorenz derived a simplified model of atmospheric convection. Three coupled differential equations:
dx/dt = σ(y − x)
dy/dt = x(ρ − z) − y
dz/dt = xy − βz
The classic parameters are σ = 10, ρ = 28, β = 8/3. These values place the system in the chaotic regime.
The phase-space axes. A trajectory starting at any point near the origin will be drawn toward the attractor — the invariant set the system orbits forever.
Sensitivity and Structure
Two nearby points diverge exponentially: this is deterministic chaos. Yet the trajectories are confined to a bounded region, tracing the same double-scroll shape indefinitely.
The attractor projected onto the XZ plane. Orbits spiral outward around one lobe until a threshold, then switch to the other lobe — the switch is unpredictable but the overall shape is invariant. The attractor has Hausdorff dimension approximately 2.06.