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Chaos from the Logistic Map. Part 1: “Periodic Points.”
The aim with this story is to present one of the most famous theorems in the research of chaotic maps. Before getting shrug-off too much, let me note early, this theorem is mathematically not more involved from what one learns in a first course of university level mathematics.
Throughout this account we will assume any considered map f to be a continuous real-valued function defined on some compact interval I. Moreover, we will assume f(I) \subset I. Therefore, we are able to build iterates like f^n(x), that denote n-times application of f onto the point x.
My intend is to make the text as readable as possible, but due to this environment’s lack of support for latex symbols, please apologize for making the following abbreviations:
f^k(x): denotes k-times composition of f, that is: f^3(x) f(f(f(x)))A \intersect B: means the intersection of the sets A and BA \subset B: means A is a subset of BA \superset B: means A is a superset of B
