482
Chapter 4 | Applications of Derivatives
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 ⎛ ⎝ x 0 , 0 ⎞ ⎠ to the point ⎛ ⎝ x 0 , f ( x 0 ) ⎞ ⎠ = ( x 0 , x 1 ). We then draw a horizontal line from that point to the point ( x 1 , x 1 ), then draw a vertical line to ⎛ ⎝ x 1 , f ( x 1 ) ⎞ ⎠ = ( x 1 , x 2 ), and continue the process until the long-term behavior of the system becomes apparent. Figure 4.84 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.
Figure 4.84 A cobweb diagram for f ( x ) =3.55 x (1− x ) is presented here. The sequence of values results in 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 − cycle, 4 − 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?
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online