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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
(1)
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 the environment less the current population.
The usual values of interest for the parameter r are those in the interval [0, 4], so that xn 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 xn remains bounded on [−0.5, 1.5].[4]
^Lorenz, Edward N. (1964-02-01). "The problem of deducing the climate from the governing equations". Tellus. 16 (1): 1–11. Bibcode:1964Tell...16....1L. doi:10.1111/j.2153-3490.1964.tb00136.x. ISSN 0040-2826.
^May, Robert M. (1976). "Simple mathematical models with very complicated dynamics". Nature. 261 (5560): 459–467. Bibcode:1976Natur.261..459M. doi:10.1038/261459a0. hdl:10338.dmlcz/104555. PMID 934280. S2CID 2243371.
^Weisstein, Eric W. "Logistic Equation". MathWorld.
^Cite error: The named reference Takashi Tsuchiya, Daisuke Yamagishi, 1997 was invoked but never defined (see the help page).
The logisticmap is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic...
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qualitative behaviour of one-dimensional iterated functions, such as the logisticmap. The technique was introduced in the 1890s by E.-M. Lémeray. Using a...
these points is called a periodic point. This is illustrated by the logisticmap, which depending on its specific parameter value can have an attractor...
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medicine. Dynamical systems are a fundamental part of chaos theory, logisticmap dynamics, bifurcation theory, the self-assembly and self-organization...
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measured in this study is now known as the first Feigenbaum constant. The logisticmap is a prominent example of the mappings that Feigenbaum studied in his...
interval to the next between every period-doubling bifurcation. The logisticmap is a polynomial mapping, often cited as an archetypal example of how...
conditions remain on a [ 0 , 1 ] {\displaystyle [0,1]} boundary in the logisticmap x n + 1 = r x n ( 1 − x n ) {\displaystyle x_{n+1}=rx_{n}(1-x_{n})} form...
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initial conditions is provided by a particular parametrization of the logisticmap: x n + 1 = 4 x n ( 1 − x n ) , 0 ≤ x 0 ≤ 1 , {\displaystyle x_{n+1}=4x_{n}(1-x_{n})...
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