The cobweb on the left is a single trajectory of x → x² − r: bounce off the curve, across to the diagonal, repeat. Watch it spiral into a fixed point, lock onto a cycle or fill the whole interval. The Lyapunov exponent on the right turns that into one number: negative = order, positive = chaos.
Lyapunov spectrum λ(r) across the whole map
In the ordered regime, change x₀ to anything and the cobweb still ends on the same cycle. The start is forgotten.
The attractorThe Lyapunov curve just kisses zero at each period-doubling, then dives back down: order lost and regained.
Why it touches zeroNudge r by 0.001 at the edge of a window and a clean 3-cycle dissolves into a full chaotic smear.
Sensitive in r too