The Logistic Map, Repeating One Equation

Definition The logistic map is the rule

where the parameter $r$ is a fixed number with $0 \le r \le 4$ and the input $x$ lies in the interval $[0, 1]$. Iterating it means choosing a starting value $x_0$ in $[0, 1]$ and computing $x_1 = f(x_0)$, then $x_2 = f(x_1)$, and so on, so that

The values never leave $[0, 1]$. For $x$ in $[0, 1]$ the product $x(1 - x)$ is never negative and is at most $1/4$, which it reaches at $x = 1/2$, so $f(x)$ lies between $0$ and $r/4$, and $r/4 \le 1$. The surprise is that this one rule, depending on $r$, can produce orbits that settle at a single value, orbits that fall into a repeating cycle, and orbits that never repeat at all. For $0 \le r \le 1$ every orbit tends to $0$. For $1 < r \le 3$ every start strictly between $0$ and $1$ tends to the value $1 - 1/r$. For $3 < r < 1 + \sqrt{6} \approx 3.4495$ every start strictly between $0$ and $1$ approaches a pattern that alternates between two values, except special starts whose orbits land exactly on $1 - 1/r$. Nothing random is involved, since once $r$ and $x_0$ are fixed every later value is determined. The rule is often read as a population model, with $x$ the population as a fraction of some maximum and the factor $1 - x$ standing for crowding, and the ecologist Robert May made it famous in 1976 as a simple model with very complicated behavior. It is a simplified model, not a forecast for any real population.

Definition A discrete dynamical system consists of a set of possible states and a rule $f$ that sends each state to a state in the same set. Time moves in whole steps, and the state after one more step is $x_{n+1} = f(x_n)$. The sequence $x_0, x_1, x_2, \ldots$ is called the orbit of the starting state $x_0$. The system is deterministic, meaning that the current state alone fixes the next one, with no random choice. The early part of an orbit, before it comes close to its long-term pattern, is called the transient. It has no sharp end, and how long it lasts depends on how close to the pattern one asks the orbit to be. A statement about long-term behavior always concerns one rule, including the value of any parameter such as $r$, and one starting state, and it describes what happens as $n$ grows without bound, which no finite list of steps can show completely.

Definition A cobweb diagram draws the orbit of a rule $f$ on an interval. Draw the graph of $y = f(x)$ and the diagonal line $y = x$ on the same axes, and start at the point $(x_0, x_0)$ on the diagonal. Move vertically to the graph, which reaches the point $(x_0, x_1)$ because $x_1 = f(x_0)$. Then move horizontally to the diagonal, which reaches $(x_1, x_1)$. Repeating the two moves visits $(x_1, x_2)$, $(x_2, x_2)$, $(x_2, x_3)$ and so on, and the corners on the diagonal have the orbit $x_0, x_1, x_2, \ldots$ as their first coordinates. The points where the graph meets the diagonal are exactly the fixed points, because there $f(x) = x$. When the path returns exactly to a corner it has visited before, the orbit has become a cycle.

Definition A fixed point of a rule $f$ is a state $x^*$ with $f(x^*) = x^*$, so the orbit that starts there never moves. The fixed point is stable, also called attracting, when there is an interval around $x^*$ such that every orbit starting in that interval stays in it and converges to $x^*$. Suppose $f$ has a slope $f'(x)$ that changes continuously near $x^*$. If $\lvert f'(x^*) \rvert < 1$, then $x^*$ is stable. If $\lvert f'(x^*) \rvert > 1$, then $x^*$ is unstable, and every start close to $x^*$ but different from it is pushed farther from $x^*$ at the next step. If $\lvert f'(x^*) \rvert = 1$, this test decides nothing. The reason is that near $x^*$ each step multiplies the gap $x - x^*$ by approximately $f'(x^*)$. For the logistic map the slope is $f'(x) = r(1 - 2x)$. The fixed point $0$ has slope $r$, so it is stable for $r < 1$ and unstable for $r > 1$. For $r \ge 1$ the second fixed point $1 - 1/r$ has slope $2 - r$, so it is stable for $1 < r < 3$ and unstable for $r > 3$.

Definition A two-cycle, or cycle of period two, of a rule $f$ is a pair of different states $p$ and $q$ with $f(p) = q$ and $f(q) = p$. An orbit that starts at $p$ alternates $p, q, p, q, \ldots$ forever. Each of $p$ and $q$ is a fixed point of the rule applied twice, $f(f(x))$, but not of $f$ itself. The two-cycle is stable when there is an interval around each of its points such that every orbit starting in one of these intervals approaches the cycle. When $f$ has a continuously changing slope, the chain rule shows that the slope of $f(f(x))$ at $p$ and at $q$ is the product $f'(p)\,f'(q)$, called the multiplier of the cycle. The cycle is stable if the multiplier lies strictly between $-1$ and $1$, and unstable if it is larger than $1$ in size. Within $0 \le r \le 4$, the logistic map has a two-cycle exactly when $r > 3$, given by

and its multiplier is $4 + 2r - r^2$. That multiplier lies between $-1$ and $1$ exactly when $3 < r < 1 + \sqrt{6}$.