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By Karl-Heinz Becker

This learn of chaos, fractals and intricate dynamics is meant for someone conversant in desktops. whereas preserving the math to an easy point with few formulation, the reader is brought to a space of present clinical examine that was once scarcely attainable until eventually the provision of pcs. The booklet is split into major components; the 1st presents the main fascinating difficulties, each one with an answer in a working laptop or computer application layout. quite a few routines let the reader to behavior his or her personal experimental paintings. the second one half offers pattern courses for particular computing device and working structures; info confer with IBM-PC with MS-DOS and Turbo-Pascal, UNIX 42BSD with Berkeley Pascal and C. different implementations of the images exercises are given for the Apple Macintosh, Apple IIE and IIGS and Atari ST.

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Try it out! In the above print-out it goes without saying that other factors such as -1 and 3 and other summands are possible. The equations under consideration no longer have anything to do with the original ‘measles’ problem. They are not entirely unknown to mathematicians: they appear in a similar form in approximation methods for the solution of differential equations. 2-l In summary we might say that we always obtain a Feigenbaum diagram if the recursion equation is nonlinear. In other words, the underlying graph must be curved.

5); . 57 (Chaos). Under certain conditions, moreover, we cannot predict what will happen at all. Insignificant differences in the initial value lead to totally different behaviour, giving virtually unpredictable results. 57. This is the ‘point of no return’ , dividing the region of order from that of chaos. To avoid misunderstandings, we must again emphasise that the above remarks refer to a completely deterministic system. But - from a practical point of view - the chaos effect produces the bitter aftertaste of indeterminacy.

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. We should henceforth make tiny changes to the program l to make the point being worked on flash, and l to avoid choosing a fixed iteration number at the start. It is easy to make a point flash by changing its colour repeatedly from black to white and back again. We can change the iteration number by using a different loop construction.

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