The Logistic Map: From Order to Chaos

The logistic map is the recurrence

xn+1 = r xn (1 − xn),   0 ≤ xn ≤ 1,   0 ≤ r ≤ 4.

1. Choose the rate r and the initial value x0

Drag the sliders, or click anywhere on the bifurcation diagram in section 3 to set r.

2. Cobweb plot and time series

Left: start at x0 on the horizontal axis, go up to the parabola y = rx(1 − x), across to the diagonal y = x, and repeat. Each corner on the diagonal is one iterate. Right: the same iterates plotted against n.

Cobweb: xn ↦ xn+1

parabola y = rx(1 − x)diagonal y = xcobweb path

Time series: xn vs n

orbit of x0

3. Bifurcation diagram: long-run behaviour as r varies

current rmajor parameter-region boundaries

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