Chapter 12: Problem 21
Here is an iterated map that is easily studied with the help of your calculator: Let \(x_{t+1}=f\left(x_{t}\right)\) where \(f(x)=\cos (x) .\) If you choose any value for \(x_{0},\) you can find \(x_{1}, x_{2}, x_{3}, \cdots\) by simply pressing the cosine button on your calculator over and over again. (Be sure the calculator is in radians mode.) (a) Try this for several different choices of \(x_{0}\), finding the first 30 or so values of \(x_{t}\). Describe what happens. (b) You should have found that there seems to be a single fixed attractor. What is it? Explain it, by examining (graphically, for instance) the equation for a fixed point \(f\left(x^{*}\right)=x^{*}\) and applying our test for stability [namely, that a fixed point \(\left.x^{*} \text { is stable if }\left|f^{\prime}\left(x^{*}\right)\right|<1\right].\)
Short Answer
Step by step solution
Key Concepts
These are the key concepts you need to understand to accurately answer the question.