A chaotic balance

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.

Butterflies, hurricanes and... pools!

Chaos is nothing more than order seen from the opposite side.

This defintion by Fethry Duck in the italian story Il mobile caotico (The chaotic furniture) can be considered very centered on the heart of chaos. And the mathematical tool that we used to study it is the theory of chaos.

Flapping the wings

What best identifies chaos theory is the butterfly effect, which identifies in a simple and effective way the strong dependence of chaotic systems on initial conditions. The name was first used by Edward Lorentz, who published the first article on this effect in 1963⁽¹⁾.
The popular version of the butterfly effect goes something like this: The flapping of a butterfly's wings in Brazil causes a hurricane in New York and the use of the butterfly was probably suggested to Lorentz from Ray Bradbury's 1952 short story A sound of thunder in which an unwary time traveler, stepping out of the path set by the travel agency and thus stepping on a butterfly, even manages to change the result of the last US presidential elections, allowing a fascist to become the most powerful man on the planet!
From a scientific point of view, one of the most typically chaotic problems is that of weather forecasts, because of the large amount of variables that are present. The appearance of chaotic behaviors, however, would not be so scientifically interesting if it were not for one of their particular characteristics: the fundamental laws that govern, for example, time are deterministic and individually easily solved, but by combining together a large number of such equations, not only the resolution of the system is more complicated, so much so that it is necessary to use electronic calculators, but also the solution shows a chaotic behavior graphically well identified by the Lorentz attractors: