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Logistic cobweb and bifurcation

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

  • Vary each control and watch the rail readouts respond.
  • Compare the diagnostic plot against the live scene.