Logistic Map & Bifurcation Diagram
Visualize the logistic map, xn+1 = r·xn·(1 − xn), the archetypal route to chaos. For growth rates r between 0 and 4 the orbit stays in [0,1], but its long-term behaviour changes dramatically: a stable fixed point, then a cascade of period-doubling bifurcations (2→4→8→…), then deterministic chaos — interrupted by periodic windows (the period-3 window near r≈3.83). The diagram plots, for each r, the values the orbit settles on after transients die out; the ratio of successive bifurcation spacings converges to Feigenbaum's constant δ≈4.669. Click any point to inspect that r's orbit and its Lyapunov exponent. Everything runs locally in your browser.
Bifurcation diagram
x-axis: r (growth rate) · y-axis: long-term orbit values in [0,1]. Click on the diagram to open that r below.
Orbit inspector
Time series of xn for the selected r (first 120 iterations).
The logistic map xn+1 = r xn(1 − xn) is the simplest nonlinear dynamical system that exhibits chaos. For r ≤ 1 the population dies out (x → 0); for 1 < r ≤ 3 it settles to the fixed point x* = 1 − 1/r; at r = 3 the first period-doubling bifurcation splits this into a stable 2-cycle, which doubles again to 4, 8, 16… at ever-closer values of r, accumulating at r≈3.5699 (the onset of chaos). The ratios of the gaps between successive bifurcations converge to Feigenbaum's δ ≈ 4.6692, a universal constant shared by every smooth map with a quadratic maximum — a landmark discovery of 1975. Beyond the cascade the orbit is chaotic (sensitive to initial conditions, positive Lyapunov exponent ≈ ln 2 at r=4), yet periodic windows reopen: the largest is the stable period-3 window around r≈3.828–3.842. The diagram discards a transient of nTrans iterations (so initial-condition artefacts wash out) then plots nSamp samples per r. Pairs with the Mandelbrot Set, Elementary Cellular Automata, and Conway's Game of Life. Everything runs locally — nothing leaves your browser.