410 Ordinary Differential Equations
However , instead of finding the general solution to (2'), we will use tech-
niques from elementa ry calculus t o answer important questions concerning
the nature of the solutions without first finding the solutions. The first and
most important question to ask is "Is there an equilibrium point?" By an
equilibrium p oint, we mean a point (x, y) which is a constant solution of
(1). At a n equilibrium point, the amount spent on arms per year by each
nation remains constant for all time-the first nation always sp ends x* and
the second nation always spends y. If there is a n equilibrium point, we then
want to a nswer the question: "Is the equilibrium point stable or unsta ble?"
That is, we want to know if an initial condition (xo, Yo) is sufficiently close to
the equilibrium point, does the associated solution (x(t), y(t)) remain close to
(x, y*) for all t ime-in which case, we say the equilibrium point is stable-or
does t he associated solution move away from the equilibrium p oint- in which
case, we say the equilibrium point is unstable.
For there to b e an equilibrium point (constant solution) for Richardson's
arms race model, the rate of ch ange of the amount spent for arms p er year
by both nations must simultaneously b e zero. Thus, we must have dx/dt =
dy/dt = 0. Setting dx/dt = 0 and dy/dt = 0 in system (2), we see that an
equilibrium point (x, y) must simultaneously satisfy
L 1 : -Cx + Ay + r = 0
(3)
L2 : Bx - Dy + s = 0.If E =CD - AB-/= 0, then there is a unique solution to (3)- namely,
(4) x * = rD E + sA ' y * = Cs+ Br
EIn discussing Richardso n 's arms race model (2), you might think we should
sketch graphs of x and y as functions of time and draw co nclusions from
these graphs. However, it is often more informative to look a t a "phase-
plane" diagram in which one dependent varia ble is graphed versus the other
dependent variable. In this instance, we could plot y versus x or x versus y.
The graph in the xy-plane of both L 1 and L 2 a re lines- see equations (3).
Let us assume in the discussion which follows that the parameters A, B ,
C, and D are all nonzero. A typical sketch of the graph of L 1 for r > 0
is shown in Figure 10 .1. The line L 1 divides the x y-plane into three sets-
(i) the line L1, itself, on which dx/dt = -Cx + Ay + r = 0, (ii) the half-
plane where dx/dt = -Cx + Ay + r > 0, and (iii) the h alf-plane where
dx/dt = -Cx + Ay + r < 0.
If we view Rich ardson's arms race model (2) as the equations of motion for
a particle in the xy-plane, the first equation in (2), dx/dt = -Cx + Ay + r,
tells us the horizontal component of the velocity and the second equation in
(2), dy/dt =Bx - Dy+ s, tells us the vertical component of the velocity. The