By Bernard Friedman

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**Extra resources for Principles and techniques of applied mathematics**

**Example text**

Compute this natural constant as accurately as possible: kIl-k”-l 6=lim IHcm k n+l -k II ’ In order to work out 6, the values kj must be calculated as accurately as possible. 2-2 you can look at the interesting intervals of k and p, and pursue the branching of the lines. By repeatedly magnifying windows taken from the diagram you can compute the k-values. Near the branch-points, convergence is very bad. It can happen that even after 100 iterations we cannot decide whether branching has taken place.

87 . . . . . . . . . . . . . 8 . . . . . . . . . . . 8 . . . . . . . %8 . .. . .. . . . .. I.?!! . . . . . . . . . . . . . . :A@. . . . . . . . . . . “AS”C? . . . . . . . D? 1-8 List of formulas. Now represent your calculations graphically. You have six individual calculations to deal with. Each diagram, in a suitable coordinate system, contains a number of points generated by feedback. 5. 0. 5. 2 *) : Useful subroutines *) c* END (* BEGIN : Procedures of main program *) PROCEDURE Hello; BEGIN InfoOutput ('Calculation of Measles-Values'); Infooutput ('---------------------------I) ; Newlines CarryOn Newlines E-NJ; (2); ('Start : '); (2); PROCEDURE Application; BEGIN ReadReal ('Initial Population p (0 to 1) >I, Population); ReadReal ('Feedback Parameter k (0 to 3) >I, Between Order and Chaos: Feigenbaum Diagrams 23 I P 4 : I .

2-2 Devise some other functions and apply graphical iteration to them. 1-l). Until now we have computed with small values, in order not to occupy too much of the computer’s time. But now we will make our first survey of the entire range. 1-2. 2-l. The name ‘Feigenbaum’ is in honour of the physicist Mitchell Feigenbaum, 2 who carried out the pioneering research described in this chapter. 2-l a Feigenbaumdiagram. In the program fragment we introduce two new variables I nv i s i b 1 e and Visible, which in the example are given the value 50.