What to watch for
For ρ < 1 the only resting state is the motionless origin. Past ρ ≈ 24.06 the fixed points lose stability and the butterfly emerges: two lobes joined by a narrow bridge, the point looping one a few times before flipping: unpredictably: to the other.
The deeper insight lives in the two trajectories. They start 10⁻⁴ apart, share for a while, then peel away exponentially. That exponential sensitivity (a positive Lyapunov exponent) is the "butterfly effect" made literal, and it's the real reason weather forecasts decay over a couple of weeks. The same shape recurs far beyond meteorology: in lasers, dynamos, chemical oscillators and electronic circuits — wherever simple feedback turns nonlinear.
The knobs
- Trajectories, how many points to integrate at once (1 to 5). Two or more reveals the divergence; each starts a hair further from the attractor.
- ρ, the Rayleigh number (10 to 40). Drag it down toward 24 to watch chaos collapse into a steady orbit, or up to deform the wings.
- Speed, integration step and spin rate together (0.2 to 4). Slow it to follow a single loop; speed it up to fill the attractor fast.
- Trail — how quickly the path fades (0.02–0.3). Low values leave long ghostly trails; high values keep only the freshest arc.