1550078481-Ordinary_Differential_Equations__Roberts_

412 Ordinary Differential Equations

so y( t) is decreasing in this region and the particle tends to move downward
toward the line L 2. That is, the second nation in the arms race tries, at all
times, to adjust its expenditures by moving them vertically toward the line
L2.
We wish to examine the limiting b ehavior of the expenditures for arms in
Richardson's model. We divide the limiting behavior into three categories and
classify the types of arms races as follows:
l. If x --) 0 and y --) 0 as t --) oo, then we say the arms race results in

mutual disarmament.

2. If x--) x and y --) y as t--) oo, then we say the arms race is a stable
   arms race.


3. If x --) oo and y --) oo as t --) oo, then we say there is an unstable

arms race or a runaway arms race.

We will now consider two cases under the assumptions that A, B, C, and
D are nonzero. Solving the equations of Li and L 2 for y (see equations (3)),
we find
Cx r
y=- --
A A

Bx s

y=-+-

D D

(for Li)

Thus, the slope of line Li is C /A and the slope of L 2 is B / D.

Case 1. Supposer = s = 0 and Li and L 2 are not parallel. Since r = s = 0 ,

the equilibrium point is the origin, (0, 0) - see equations (4). Since Li and L 2
are not parallel, they intersect at the origin and either (a.) the slope of Li is
larger than the slope of L 2 or (b.) the slope of L 2 is larger than the slope of Li.
Representative graphs for case l a. and case lb. are shown in Figures 10.2a.
and 10.2b., respectively. We numbered the regions in the first quadrant, I, II,
and III, as shown in the figures.

y y

III III

x

a. b.

Figure 10 .2 Graphs of Lines Li and L 2 for r = s = 0 and

a. C/A > B/D and b. C/A < B/D

x