Chapter 4 | Applications of Derivatives
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Iterative Processes and Chaos Iterative processes can yield some very interesting behavior. In this section, we have seen several examples of iterative processes that converge to a fixed point. We also saw in Example 4.48 that the iterative process bounced back and forth between two values. We call this kind of behavior a 2 - cycle . Iterative processes can converge to cycles with various periodicities, such as 2 − cycles, 4 − cycles (where the iterative process repeats a sequence of four values), 8-cycles, and so on. Some iterative processes yield what mathematicians call chaos . In this case, the iterative process jumps from value to value in a seemingly random fashion and never converges or settles into a cycle. Although a complete exploration of chaos is beyond the scope of this text, in this project we look at one of the key properties of a chaotic iterative process: sensitive dependence on initial conditions. This property refers to the concept that small changes in initial conditions can generate drastically different behavior in the iterative process. Probably the best-known example of chaos is the Mandelbrot set (see Figure 4.83 ), named after Benoit Mandelbrot (1924–2010), who investigated its properties and helped popularize the field of chaos theory. The Mandelbrot set is usually generated by computer and shows fascinating details on enlargement, including self-replication of the set. Several colorized versions of the set have been shown in museums and can be found online and in popular books on the subject.
Figure 4.83 The Mandelbrot set is a well-known example of a set of points generated by the iterative chaotic behavior of a relatively simple function.
In this project we use the logistic map
f ( x ) = rx (1− x ), where x ∈ [0, 1] and r >0 as the function in our iterative process. The logistic map is a deceptively simple function; but, depending on the value of r , the resulting iterative process displays some very interesting behavior. It can lead to fixed points, cycles, and even chaos.
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