By Karl-Heinz Becker
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Extra info for Dynamical Systems and Fractals: Computer Graphics Experiments with Pascal
99, a limit point, with screen dialogue. For each given k-value we get distinct pictures. 2-1). In all other cases the horizontal and vertical lines tend towards segments of the original curve, which correspond to limiting p-values. 2-2 represent two different p-values. 35 Between Order and Chaos: Feigenbaum Diagrams a new initial condition. After many such feedbacks (iterations) we establish that the results run through certain fixed values. By graphical feedback we refer to the picture of the function in an x,y-coordinate system (x stands for p, y for f(x) or f(p)).
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. 2 *) *) c* (* 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 .
This is the initial value for the next stage of the feedback. The value must be carried across to the x-axis. For this purpose we use the diagonal, with equation y = x. From the point on the parabola (with coordinates (x, f(x)) we draw a line horizontally to the right (or left), until we encounter the diagonal (at the coordinates (f(x), f(x))). Then we draw another vertical to meet the parabola, a horizontal to meet the diagonal, and so on. This procedure will be explained further in the program and the pictures that follow it.