Logistic map

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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, initially utilized by Edward Lorenz in the 1960s to showcase irregular solutions (e.g., Eq. 3 of ^[1]), was popularized in a 1976 paper by the biologist Robert May,^[2] in part as a discrete-time demographic model analogous to the logistic equation written down by Pierre François Verhulst.^[3] Mathematically, the logistic map is written

${\displaystyle x_{n+1}=rx_{n}(1-x_{n}),}$

(1)

where x_n 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 the environment less the current population.

The usual values of interest for the parameter r are those in the interval [0, 4], so that x_n remains bounded on [0, 1]. The r = 4 case of the logistic map is a nonlinear transformation of both the bit-shift map and the μ = 2 case of the tent map. If r > 4, this leads to negative population sizes. (This problem does not appear in the older Ricker model, which also exhibits chaotic dynamics.) One can also consider values of r in the interval [−2, 0], so that x_n remains bounded on [−0.5, 1.5].^[4]