HW 5: More 1D Discrete Maps
Due:
- Friday, February 27, 9:30 AM
Introduction
In class, we started to look at maps with chaotic behavior and what the trajectories in chaotic maps look like. Here, you’ll explore two more.
Exercises
Exercise 1: Decimal Shift Map
The decimal shift map is defined as \(x_{n+1} = 10 x_n \mod 1\).
In other words: take your \(x\), multiply it by 10, and then lop off the integer part. For example, if \(x_0 = 0.145\), then \(x_1 = 0.45\).
In doing the following parts, it will be helpful to explore some values and look at patterns of what’s going on.
Part (a)
What are the fixed points of this map?
Part (b)
Find some periodic points of this map that have period \(p > 1\). In other words, look for values of \(x\) that get caught in a repeating loop of the same values. Try to find these loops for some different periods \(p\).
Part (c)
For which values of \(p\) does the map have periodic points? Explain your thinking.
Part (d)
We call a point \(x\) eventually periodic if the set \(\{f^k(x) | k \in \mathbb{Z}^+\}\) is finite. In other words, if I start at \(x\), then at some point I end up either at a fixed point or in a periodic loop. We call a point aperiodic if it is not eventually periodic.
How many aperiodic points does this map have? Describe these points, and explain your reasoning.
Challenge
What happens if we change the map to \(x_{n+1} = k x_n \mod 1\) for some other positive integer \(k > 1\)?
Exercise 2: Tent Diagram
The tent diagram is defined as \[x_{n+1} = \begin{cases} 2x_n & x_n < \frac{1}{2} \\ 2(1-x_n) & x_n \ge \frac{1}{2}. \end{cases}\]
Looking at some trajectories will be useful here again, and I highly recommend making a cobweb diagram!
Part (a)
What are the fixed points of this map?
Part (b)
Find some periodic points of this map that have period \(p > 1\). Again, try to find these loops for some different periods \(p\).
Part (c)
For which values of \(p\) does the map have periodic points? Explain your thinking.
Part (d)
How many aperiodic points does this map have? Describe these points, and explain your reasoning.
Challenge
What happens if we change the \(2x_n\) and \(2(1-x_n)\) in the function definition to \(\mu x_n\) and \(\mu (1-x_n)\) for some other real number \(\mu\) in \((0,2)\)?