1550078481-Ordinary_Differential_Equations__Roberts_

434 Ordinary Differential Equations

We analyzed the linear a rms race model (1) earlier. Suppose that the term
representing t he resistance of the first nation's people to increased spending
for arms, -Cx, is changed to -Cx^2 and, likewise, the resistance of the second
nation 's people to increased spending for arms, -Dy, is changed to -Dy^2.
Then the nonlinear arms race to be analyzed b ecomes

(2)

dx

• = Ay-Cx^2 +r
  dt

dy 2

- =Bx-Dy +s.

dt

The critical points (x*, y*) of (2) simultaneously satisfy

(3a) Ay-Cx^2 +r = 0

and

(3b) Bx - Dy^2 + s = 0.

For A # 0, the graph of equation (3a) is a parabola. The vertex of the

parabola is at (0, -r/A), the parabola's axis of symmetry is the y-axis, and

the parabola opens upward. For B # 0, the graph of equation (3b) is also a

parabola. The vertex of t his parabola is at (-s/ B, 0), its axis of symmetry is

the x-axis, and the parabola opens to t he right. Visualizing these parabolas
to be free to slide along the x-axis and y-axis and free to open and close, we
see that it is possible for (2) to have 0, 1, 2, 3 or 4 critical points and we see
that at most two critical points can li e in the first quadrant.

Solving equation (3a) for y yields

(4)

Cx^2 - r

y=---

A

Substituting this expression into equation (3b) and rearranging terms, we find
x* satisfies the fourth degree equation

(5) C^2 D x^4 - 2CDrx^2 - A^2 Bx+ Dr^2 -A^2 s = 0.

The real, nonnegative crit ical points of system (2) can be found by using com-
puter software to solve (5) and then substituting the real, nonnegative roots,
x, into equation (4). Next, the stability characteristics of the nonlinear sys-
tem (2) at a critical point (x, y) can often be determined from the stability
characteristics of the associated linear system at (x, y*).