HW 4: 1D Discrete Maps

Important

Due:

  • Friday, February 20, 9:30 AM

Introduction

In class, we briefly looked at discrete one-dimensional maps written in terms of the difference in \(x\), like \(\Delta x = x\), where \(\Delta x\) told us what to add to \(x_n\) to get to \(x_{n+1}\). For example, with this map, when we started with \(x_0 = 1\), we found

\(\begin{align*} x_1 &= x_0 + \Delta x = 1 + 1 = 2, \\ x_2 &= x_1 + \Delta x = 2 + 2 = 4, \\ x_3 &= x_2 + \Delta x = 4 + 4 = 8, \end{align*}\)

and so forth.

The more typical way to write these discrete 1D maps is in terms of \(x_{n+1}\). So the map above, for instance, would be \(x_{n+1} = 2x_n\).

We can still look for fixed points: values of \(x_n\) such that \(x_{n+1} = x_n\). We can also still look at their stability – if we’re just a little bit away from that fixed point value \(x^*\), do we head towards \(x^*\) (stable) or away (unstable)?

But compared to the continuous systems we’ve looked at previously, some more interesting behavior can arise. We can get periodic points, basically getting caught in a loop. For example, the map \(x_{n+1} = -x_n^3\) has a pair of points of period 2: -1 and 1 keep flip-flopping.

Exercises

Exercise 1: Exploring 1D maps

Part (a)

Choose three of the discrete 1D maps below. Find their equilibrium points and classify their stability.

  • \(x_{n+1} = \sqrt{x_n}\)
  • \(x_{n+1} = x_n^3\)
  • \(x_{n+1} = x_n^2\)
  • \(x_{n+1} = \sin x\)
  • \(x_{n+1} = 0.5(3x_n - x_n^3)\)
  • \(x_{n+1} = \ln x\)
  • \(x_{n+1} = \frac{2x_n}{1+x_n}\)
  • \(x_{n+1} = 3x_n - x_n^3\)

Part (b)

For notation, much like in the continuous case, let \(f(x_n)\) be the right side of the equation. In other words, \(x_{n+1} = f(x_n)\).

For each equilibrium point \(x^*\) that you found in Exercise 1, find the derivative \(f'(x^*)\). What do you notice about this value and stability of \(x^*\)? Try to explain what’s going on here.

Exercise 2: Trajectories and Cobweb Diagrams

Part (a)

You’ll keep working with the three maps you chose in Exercise 1 here. For each one, choose a few initial values \(x_0\) (that are not a fixed point you found above) and calculate the next \(x_n\) values from each of those initial points, until you feel like someone could see what’s going on. What kinds of things do you notice about your map?

Part (b)

Now for each of your maps, do the following:

  • Draw a coordinate system with \(x_n\) on the horizontal axis and \(x_{n+1}\) on the vertical axis.
  • Draw the diagonal line \(x_{n+1} = x_n\).
  • Draw the curve corresponding to that map. (For example, \(x_{n+1} = 2x_n\) would be a straight line through the origin with slope 2.) I’ll call this the graph of \(f\) (where \(x_{n+1} = f(x_n)\).)
  • For each of your initial values \(x_0\), go to that point on the horizontal axis. Draw a line straight up/down until you hit the graph of \(f\). Then move horizontally until you hit the \(x_{n+1} = x_n\) line. Then move vertically to the graph of \(f\) again, then horizontally to the diagonal line again, etc.

This graph is called a cobweb diagram. How does it show what’s happening in your maps?

Exercise 3: Periodic Points

Look for points of period 2 in your chosen maps. These should be points such that \(x_{n+2} = x_n\) that are not your fixed points. If you don’t think there are any, explain why.