Logistic map

The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May,[1] in part as a discrete-time demographic model analogous to the logistic equation written down by Pierre François Verhulst.[2] Mathematically, the logistic map is written

 

 

 

 

(1)

where xn is a number between zero and one, that represents the ratio of existing population to the maximum possible population. The values of interest for the parameter[3] r (sometimes also denoted μ) are those in the interval [−2, 4], so that xn remains bounded on [−0.5, 1.5]. 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.

However, as a demographic model the logistic map has the pathological problem that some initial conditions and parameter values (for example, if r > 4) lead to negative population sizes. This problem does not appear in the older Ricker model, which also exhibits chaotic dynamics.

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.


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