Problem: Analysis of Difference Equations

We are really concerned with the long term behavior of this physical system, as predicted by the mathematical model generated. As we already indicated, you may step as many times as you want to and see what happens. But there is a more subtle approach which you are to consider in this problem. Perform the following exercises, and write a brief report on the result. You may think of this as a mathematical experiment.

(1) Let {x0,v0} represent an initial state. Find the particular {x0,v0} which maximizes m({x1,v1}). Find the particular {x0,v0} which minimizes m({x1,v1}). Also find the amplification factors L1 = m({x1,v1}) in each of the two cases. (Since the initial magnitude is 1, then L1 will be the magnitude of the solution at the end of 1 step.)

The next two parts are open-ended and invite you to explore the behavior of the system.

(2) Use this knowledge about the difference system to predict an upper bound on m({x100,v100}). How good is the bound?

(3) Suppose {x0,v0} was the vector that minimizes m({x1,v1}). Predict a lower bound on m({x100,v100}). Finally, try this out by calculating m({x100,v100}) with a computer program or computer algebra system. Write a brief paragraph in which you compare your bound to the actual predicted value. How good was the bound?
Write comments describing what you have learned.