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The classic logistic map is widely used to illustrate chaos and bifurcation. This is especially useful in demonstrating and teaching chaos. The snapshots show the similarity in appearance of the map as you zoom in on the areas of bifurcation, sometimes known as scale invariance.

This formulation is modified from that given in the book by H. Ruskeepää, Mathematica Navigator, 3rd ed., San Diego, CA: Academic Press, 2009 p. 941 ff. This book is highly recommended for users of Mathematica 6 or 7.

A major reference is R. M. May, "Simple Mathematical Models with Very Complicated Dynamics," Nature, 261, 1976 pp. 459–467. Another widely used reference is J. Gleick, Chaos: Making a New Science, London: Viking Penguin, 1987 and 2008.