Our mathematical history begins in a discipline that, apparently, has very little to do with mathematics: biology. In 1975 on the journal Nature Robert May, an australian ecologist, publishes a review article with a rather indicative title: Simple mathematical models with very complicated dynamics(1). The heart of the paper is the following equation:
\[x_{t+1} = a x_t (1 - x_t)\]
The equation, or logistic map, this is its name, describes the rate of change of a population in function of the parameter \(t\) (the time), that varies in a discrete rather than continuous way, while \(a\) is a constant that identifies the growth rate of a population. Insteed \(x_t\) is the ratio between the existing population and the maximum possible population at time \(t\).
The model thus described is deterministic, i.e. the population at instant 0 determines the population at subsequent instants. The equation predicts the existence of a stationary state, i.e. a situation in which the population at time \(t + 1\) is equal to the population at time \(t\). This state is stable, that is, it is maintained for a sufficiently long time, but only for \(a\) lower than or equal to 3. However when the growth rate exceeds this value, the size of the population begins to oscillate between 0 and 1, apparently in a random way. But if we observe carefully, we notice small more or less periodic recurrences, which show how the behavior of the equation is actually chaotic.
