Global Behavior in Functional Iteration Problems

Functional iteration, the process of forming a sequence x0, x1 = f(x0), x2 = f(x1) = f(f(x0)) = f2(x0), . . ., by repeated application of a function f, is fundamental to the approximation of solutions to nonlinear equations and plays an important role in the study of nonlinear dynamics. In this note...

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Bibliographic Details
Published inJournal of the Kentucky Academy of Science Vol. 80; no. 1; pp. 47 - 59
Main Author Robinson, Mark P
Format Journal Article
LanguageEnglish
Published Journal of the Kentucky Academy of Science 19.02.2020
Subjects
convergence
fixed point
iteration
REGULAR ARTICLES
sequences
visualization
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Summary:Functional iteration, the process of forming a sequence x0, x1 = f(x0), x2 = f(x1) = f(f(x0)) = f2(x0), . . ., by repeated application of a function f, is fundamental to the approximation of solutions to nonlinear equations and plays an important role in the study of nonlinear dynamics. In this note we examine another aspect of functional iteration: the global behavior of the sequence f, f2, f3, . . ., fn, . . . of composite functions as n → ∞. This topic can be presented to students at many levels using computer graphics to stimulate conjectures that can then be investigated theoretically.
ISSN:1098-7096
1938-2960
DOI:10.3101/1098-7096-80.1.47