4.9 Newton’s method  (Page 4/7)

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 [link] 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 $2-\text{cycles},4-\text{cycles}$ (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 [link] ), 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.

In this project we use the logistic map

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.

To visualize the long-term behavior of the iterative process associated with the logistic map, we will use a tool called a cobweb diagram . As we did with the iterative process we examined earlier in this section, we first draw a vertical line from the point $\left(x_0,0\right)$ to the point $\left(x_0,f\left(x_0\right)\right)=\left(x_0,x_1\right).$ We then draw a horizontal line from that point to the point $\left(x_1,x_1\right),$ then draw a vertical line to $\left(x_1,f\left(x_1\right)\right)=\left(x_1,x_2\right),$ and continue the process until the long-term behavior of the system becomes apparent. [link] shows the long-term behavior of the logistic map when $r=3.55$ and $x_0=0.2.$ (The first $100$ iterations are not plotted.) The long-term behavior of this iterative process is an $8$ -cycle.

1. Let $r=0.5$ and choose $x_0=0.2.$ Either by hand or by using a computer, calculate the first $10$ values in the sequence. Does the sequence appear to converge? If so, to what value? Does it result in a cycle? If so, what kind of cycle (for example, $2-\text{cycle},4-\text{cycle}.)?$
2. What happens when $r=2?$
3. For $r=3.2$ and $r=3.5,$ calculate the first $100$ sequence values. Generate a cobweb diagram for each iterative process. (Several free applets are available online that generate cobweb diagrams for the logistic map.) What is the long-term behavior in each of these cases?
4. Now let $r=4.$ Calculate the first $100$ sequence values and generate a cobweb diagram. What is the long-term behavior in this case?
5. Repeat the process for $r=4,$ but let $x_0=0.201.$ How does this behavior compare with the behavior for $x_0=0.2?$