Collection: Logistic Map

The logistic map is one of the classic examples of chaos theory. It can be summarised as follows: great complexity may arise from very simple rules. The equation can be used to model many natural processes, for example how a population of animals grows and shrinks over time. How the population behaves turns out to be enormously sensitive to the value of r, in counterintuitive ways. If r is between 0 and 1 the population will always die, but if it is between 1 and 3 the population will approach a fixed value – and if it is above 3.56995 the population becomes wildly unpredictable.

These behaviours are described as “chaotic” by mathematicians and they are not what we would instinctively expect. But they all emerge from an equation that is mathematically quite simple. As we marvel at the diversity and complexity of the natural world, the Universe or the small constituents of matter we should keep remembering that at some fundamental level all of these things share some common simplicity.

The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behavior can arise from very simple non-linear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst. The logistic map, pictured here, was a seminal discovery by May that demonstrated how even a simple equation could result in chaos.