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The science
The map and its long-run behaviour
The logistic map x(n+1) = r x(n) (1 - x(n)) turns today's value into tomorrow's with one parameter r. For r < 1 everything dies out. Above that the value settles on the fixed point x* = 1 - 1/r, which stays stable while its slope |f'(x*)| = |2 - r| is below 1, that is until r = 3. There the fixed point loses stability and a 2-cycle appears, x = (r + 1 -/+ sqrt((r+1)(r-3))) / 2r. Each diagram column is one value of r: the page throws away a transient and then plots where the orbit keeps landing, so a single dot means a fixed point, two dots a 2-cycle, and a smear means chaos.
Period doubling and the Feigenbaum constants
The 2-cycle becomes unstable and splits into a 4-cycle, then 8, 16 and so on, each doubling arriving faster. Let R_n be the superstable parameter where the critical point x = 1/2 lies on a cycle of length 2^n, i.e. F^(2^n)(1/2) = 1/2. The page solves this by Newton iteration. The gaps shrink geometrically, (R_(n-1) - R_(n-2)) / (R_n - R_(n-1)) tends to delta = 4.669201609..., and the width of the cycle element nearest the critical point shrinks by alpha = -2.502907875... per doubling. Extrapolating the geometric series gives the accumulation point r_inf = R_n + (R_n - R_(n-1)) / (delta - 1) = 3.5699456718..., where chaos begins. The same delta and alpha appear for the sine map r sin(pi x) and for any smooth map with a single quadratic maximum: this is universality, which is why the constant was also seen in experiments with fluids and electronic circuits.
Lyapunov exponent
The Lyapunov exponent lambda = mean(ln |f'(x(n))|) measures how fast two nearby orbits separate, e^(lambda n). It is negative on every periodic stretch, touches zero at each period-doubling point, and is positive in the chaotic bands (red in the strip under the diagram). At r = 4 the exact value is ln 2 = 0.6931. Inside the chaotic region the diagram contains periodic windows; the widest is period 3, opening at r = 1 + sqrt 8 = 3.8284, and each window repeats the whole doubling cascade at a smaller scale, which you can see by zooming in. A positive exponent gives a prediction horizon of about ln(tolerance ratio) / lambda steps, shown in the readouts.
What is simplified
The diagram is computed in ordinary double precision from one starting value, so very deep zooms run into rounding limits and need long transients near the accumulation point; the transient length grows automatically as you zoom. The Feigenbaum constants are measured from the first nine doublings, which fixes them to about four digits. The page treats the maps as pure mathematics, not as a model of any particular population.