Learning Outcomes
- Apply iterative processes to various situations
As mentioned earlier, Newton’s method is a type of iterative process. We now look at an example of a different type of iterative process.
Consider a function [latex]F[/latex] and an initial number [latex]x_0[/latex]. Define the subsequent numbers [latex]x_n[/latex] by the formula [latex]x_n=F(x_{n-1})[/latex]. This process is an iterative process that creates a list of numbers [latex]x_0,x_1,x_2, \cdots ,x_n, \cdots[/latex]. This list of numbers may approach a finite number [latex]x^{*}[/latex] as [latex]n[/latex] gets larger, or it may not. In the next example, we see an example of a function [latex]F[/latex] and an initial guess [latex]x_0[/latex] such that the resulting list of numbers approaches a finite value.
example: Finding a Limit for an Iterative Process
Let [latex]F(x)=\frac{1}{2}x+4[/latex] and let [latex]x_0=0[/latex]. For all [latex]n \ge 1[/latex], let [latex]x_n=F(x_{n-1})[/latex]. Find the values [latex]x_1,x_2,x_3,x_4,x_5[/latex]. Make a conjecture about what happens to this list of numbers [latex]x_1,x_2,x_3, \cdots,x_n, \cdots[/latex] as [latex]n\to \infty[/latex]. If the list of numbers [latex]x_1,x_2,x_3, \cdots[/latex] approaches a finite number [latex]x^*[/latex], then [latex]x^*[/latex] satisfies [latex]x^*=F(x^*)[/latex], and [latex]x^*[/latex] is called a fixed point of [latex]F[/latex].
Watch the following video to see the worked solution to Example: Finding a Limit for an Iterative Process.
Try It
Consider the function [latex]F(x)=\frac{1}{3}x+6[/latex]. Let [latex]x_0=0[/latex] and let [latex]x_n=F(x_{n-1})[/latex] for [latex]n \ge 2[/latex]. Find [latex]x_1,x_2,x_3,x_4,x_5[/latex]. Make a conjecture about what happens to the list of numbers [latex]x_1,x_2,x_3, \cdots, x_n, \cdots[/latex] as [latex]n\to \infty[/latex].
Try It
Activity: 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 the example where Newton’s method fails 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 7), 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 7. 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
as the function in our iterative process. The logistic map is a deceptively simple function; but, depending on the value of [latex]r[/latex], 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 [latex](x_0,0)[/latex] to the point [latex](x_0,f(x_0))=(x_0,x_1)[/latex]. We then draw a horizontal line from that point to the point [latex](x_1,x_1)[/latex], then draw a vertical line to [latex](x_1,f(x_1))=(x_1,x_2)[/latex], and continue the process until the long-term behavior of the system becomes apparent. Figure 8 shows the long-term behavior of the logistic map when [latex]r=3.55[/latex] and [latex]x_0=0.2[/latex]. (The first 100 iterations are not plotted.) The long-term behavior of this iterative process is an 8-cycle.
Figure 8. A cobweb diagram for [latex]f(x)=3.55x(1-x)[/latex] is presented here. The sequence of values results in an 8-cycle.
- Let [latex]r=0.5[/latex] and choose [latex]x_0=0.2[/latex]. 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-cycle, 4-cycle)?
- What happens when [latex]r=2[/latex]?
- For [latex]r=3.2[/latex] and [latex]r=3.5[/latex], 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?
- Now let [latex]r=4[/latex]. Calculate the first 100 sequence values and generate a cobweb diagram. What is the long-term behavior in this case?
- Repeat the process for [latex]r=4[/latex], but let [latex]x_0=0.201[/latex]. How does this behavior compare with the behavior for [latex]x_0=0.2[/latex]?
