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By Karl-Heinz Becker

This learn of chaos, fractals and complicated dynamics is meant for somebody accustomed to desktops. whereas holding the math to an easy point with few formulation, the reader is brought to a space of present medical learn that used to be scarcely attainable until eventually the provision of desktops. The ebook is split into major elements; the 1st presents the main fascinating difficulties, every one with an answer in a working laptop or computer software layout. various routines let the reader to behavior his or her personal experimental paintings. the second one half presents pattern courses for particular laptop and working platforms; information consult with IBM-PC with MS-DOS and Turbo-Pascal, UNIX 42BSD with Berkeley Pascal and C. different implementations of the pictures workouts are given for the Apple Macintosh, Apple IIE and IIGS and Atari ST.

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Extra resources for Dynamical systems and fractals: computer graphics experiments in Pascal

Sample text

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. 2--l Situation after the onset of oscillations, iteration number = 50.

Between Order and Chaos: Feigenbaum Diagrams I I 51 What form does the attractor have? Where are the boundaries of the basins of attraction? 2-3 We obtain a further extension of the regions to be examined, and hence extra questions to be answered, if we work with a different equation from that of Verhulst. One possibility is that we simplify the previous formula for f(p) to f(p) = k*p*(p-I). This is just the term that describes the change from one generation to the next in the Verhulst equation.

2-l as follows: Invisible := 0; Visible := 10; As regards the working of the program, the following remarks should be made: Input data are read from the keyboard and assigned to the corresponding variables intheprocedure Initialise. It is then easy to set up arbitary values from the keyboard, However, the program must then be initialised on each run. The type of input ZTmnsJafor’s note: It is also Geman for ’ fig-tree’ , hence the section title. 3; FOR i := 0 to Invisible DO BEGIN population := f(population, Feedback); END FOR i := 0 TO Visible DO BEGIN SetUniversalPoint (Feedback, population); population := f(population, Feedback); END; END; END; (* END: Problem-specific procedures *) (* -----------------------------------------------------*) PROCEDURE Initialise; BEGIN ReadReal ('Left ReadReal ('Right ReadReal ('Bottom ReadReal ('Top ReadInteger ('Invisible ReadInteger ('Visible END; >'r >'I >'r >'r Left); Right); Bottom); Top) ; >'r Invisible); >'I, Visible); PROCEDURE ComputeAndDisplay; BEGIN EnterGraphics; FeigenbaumIteration; ExitGraphics; END; The new or modified parts of the program are shown in bold type.

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