Concept# Logistic map

Summary

The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple nonlinear 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 written down by Pierre François Verhulst.
Mathematically, the logistic map is written
where xn is a number between zero and one, which represents the ratio of existing population to the maximum possible population.
This nonlinear difference equation is intended to capture two effects:

- reproduction, where the population will increase at a rate proportional to the current population when the population size is small,
- starvation (density-dependent mortality), where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of

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