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Logistic Map Cobweb and Bifurcation Diagram

The logistic map xn+1 = r xn(1 − xn) is the canonical scalar dynamical system showing a period-doubling cascade and chaos. Drag the vertical line on the bifurcation diagram (right) to set the parameter r; the cobweb on the left redraws to show how iterates converge, oscillate, or wander.

Figure 1. Cobweb and bifurcation diagram of the logistic map xn+1 = r xn(1 − xn). Method: scalar iteration in IEEE 754 double precision; Feigenbaum δ is computed by bisection on the superstable cascade f2n(1/2; Rn) = 1/2.
Keys: ← / → nudge r (Shift for fine).

WHAT TO TRY

  • Drag r up through the bifurcation diagram: the single fixed point splits to a 2-cycle, then 4, 8, and on into chaos, the period-doubling cascade traced live in the cobweb.
  • Watch the Lyapunov exponent: it stays negative on stable cycles and turns positive once the map is chaotic, the quantitative onset of sensitive dependence on initial conditions.
  • The ratio of successive bifurcation spacings converges to the Feigenbaum constant 4.669, a universal number shared by every smooth single-humped map.