Chapter 7.4, Problem 31E

Let $x(t)$ and $y(t)$ be the annual defense budgets of two antagonistic nations (expressed in billions of U.S. dollars). The change of these budgets is modeled by the equations
$x(t+1)=ax(t)+by(t)$
$y(t+1)=bx(t)+ay(t)$ ,
where a is a constant slightly less than 1, expressing the fact that defense budgets tend to decline when there is no perceived threat. The constant b is a small positive number. You may assume that a exceedsb.
Suppose $x(0)=3$ and $y(0)=0.5$ . What will happen in the long term? There are three possible cases, depending on the numerical values of a andb. Sketch a trajectory for each case, and discuss the outcome in practical terms. Include the eigenspaces in all your sketches.