8.1. Logistic map Consider the map xn+1 = fr(xn) = r xn(1−xn).

Introduced by Verhulst in 1845, the map models the growth of a population in a region of a finite area: the population at time n+1, expressed by xn+1, is proportional to the population xn at time n, but at the same time is also proportional to the remaining area.

This quantity is decreased proportional to xn, namely (1−xn). Despite the innocent aspect, this map has a remarkable mathematical structure.

a. Prove that for r < 1 the map has a unique (stable) fixed point at the origin.

b. Prove that for 1 < r < 3, there exist two fixed points, at x = 0 and x = 1−1/r, where the first is unstable and the second stable.

c. Prove that the second fixed point becomes unstable when r > 3. Show that for 3 < r < r1 there are two stable fixed points x1 and x2 of the map fr[fr[x]] ≡ f (2)

r (x), i.e. they are solutions of the equations x2 = fr(x1) and x1 = fr(x2).

d. Show that there is a value r2 > r1 when the two previous fixed points x1 and x2 become unstable. More generally, prove that there exists a sequence of values rn such that for rn−1 < r < rn there is a set of 2n−1 points characterized by the conditions fr(x∗

i ) = x∗

i+1, f (2n−1 r (x∗

i ) = x∗

i .

e. Define the family of functions gi(x) by the limit gi(x) = lim n→∞

(−α)n f (2n)

rn+i x

(−α)n

.

Show that they satisfy the functional equation gi−1(x) = (−α)gi gi −x

α

 ≡ T gi(x).

Study the features of the function g(x), defined as the ‘fixed point’ of the transformation law T g(x) = T g(x) = −α g g −x

α

.

Prove, in particular, that α is a universal parameter.