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 seminal 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. Mathematically, the logistic map is written

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

 

 

 

 

()

where ${\displaystyle x_{n}}$ 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 r (sometimes also denoted μ) are those in the interval ${\displaystyle [0,4]}$. This nonlinear difference equation is intended to capture two effects:

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