STATEMENT OF PROBLEM

The Logistic Map and Iterations.

Graphical Analysis

The Feigenbaum Diagram

Divergence Diagrams (Extensions of Feigenbaum Diagrams for viewing Cantor Sets.)

The Problem.
Part I. A study of the logistic map f(x) = a x ( 1 - x ).

Do all computations with the parameter a as an arbitrary number between 0 and 4 inclusive unless indicated otherwise and concentrate on values of x in [0, 1].

The Problem.
Part II. A study of the quadratic map Q(x) = x^2 + b.

Do all computations with the parameter b as an arbitrary number between -2 and 1/4 inclusive unless indicated otherwise and concentrate on values of x in the interval [-(1 + Sqrt[1 - 4 b])/2, (1 + Sqrt[1 - 4 b])/2].

The Problem.
Part III. A study of the Feigenbaum diagrams for the logistic family.

Do all computations with parameter and state variable ranges as described in part I.

The Problem.
Part IV. A study of the Feigenbaum diagrams for the quadratic family.

Do all computations with parameter and state variable ranges as described in part II.

The Problem.
Part V. Divergence Diagrams (extensions of Feigenbaum diagrams.)

The divergence diagram for the logistic map provides a way to see periodic points that still remain when a > 4. Each shade (color) corresponds to the number of iterates before x > 1 (after which the orbit will clearly diverge to negative infinity.) Refer to the Divergence Diagram given in the background materials preceding the problem sets.