On a representation of the Verhulst logistic map

One of the simplest polynomial recursions exhibiting chaotic behavior is the logistic map xn+1=axn(1−xn) with xn,a∈Q:xn∈[0,1]∀n∈N and a∈(0,4], the discrete-time model of the differential growth introduced by Verhulst almost two centuries ago (Verhulst, 1838)  [12]. Despite the importance of this dis...

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Bibliographic Details
Published in:Discrete mathematics 2014-06, Vol.324, p.19-27
Main Authors: Rudolph-Lilith, Michelle, Muller, Lyle E.
Format: Article
Language:English
Subjects:
Carleman linearization
Chaos theory
Chaotic dynamics
Cognitive Sciences
Discrete map
Dynamical systems
Dynamics
Life Sciences
Logistic map
Logistics
Mathematical analysis
Mathematical models
Neurobiology
Neurons and Cognition
Operator formalism
Polynomials
Psychology and behavior
Recursive map
Representations
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Summary:One of the simplest polynomial recursions exhibiting chaotic behavior is the logistic map xn+1=axn(1−xn) with xn,a∈Q:xn∈[0,1]∀n∈N and a∈(0,4], the discrete-time model of the differential growth introduced by Verhulst almost two centuries ago (Verhulst, 1838)  [12]. Despite the importance of this discrete map for the field of nonlinear science, explicit solutions are known only for the special cases a=2 and a=4. In this article, we propose a representation of the Verhulst logistic map in terms of a finite power series in the map’s growth parameter a and initial value x0 whose coefficients are given by the solution of a system of linear equations. Although the proposed representation cannot be viewed as a closed-form solution of the logistic map, it may help to reveal the sensitivity of the map on its initial value and, thus, could provide insights into the mathematical description of chaotic dynamics.
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2014.01.018