1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations

EXAMPLE 9 A Nonlinear Modified Richardson's Arms Race

Model

Consider the modified arms race model

dx dt = 3y - 2x (^2) - 1 = f ( x, y )
(6)
dy
dt = 8x- y^2 - 7 = g(x,y).
a. Find the critical points in the first quadrant.
435
b. Write the associated linear systems and determine the stability charac-
teristics at each critical point. What are the stability characteristics of
the nonlinear system (6) at each critical point?
SOLUTION
a. System (6) is a special case of system (2) in which A= 3, B = 8, C = 2,
D = 1, r = -1, ands= -7. Substituting these values into equation (5) and
then into ( 4), we see that the x -coordinate of the critical points of system ( 6)
must satisfy 4x^4 + 4x^2 - 72x + 64 = 0 and the y-coordinates must satisfy
y = (2x^2 +1)/3. Using POLYRTS to calculate the roots of the first equation,

we find x = 1, 2, -l. 5 ±2. 39 792i. To be in the first quadrant x and y must both

. be real and nonnegative. Substituting x = 1 into the equation y = (2x^2 +1)/3,
we get y = l. So (1, 1) is a critical point in the first quadrant. Substituting
x = 2 into the equation y = (2x^2 +1)/3, we get y = 3. Thus, (2, 3) is a second
critical point in the first quadrant.
b. Calculating first partial derivatives, we see f x = -4x, fy = 3, gx = 8

and gy = -2y.

At (1, 1) the associated linear system is

x' = fx(l, l)(x - 1) + fy(l, l)(y - 1) = -4(x - 1) + 3(y - 1)

y' = gx(l, l)(x - 1) + gy(l, l)(y - 1) = 8(x - 1) - 2(y - 1).

The coefficient matrix of this linear system,

A= (!x( l , 1) fy(l, 1)) = (- 4 3)

gx(l, 1) gy(l , 1) 8 -2

has eigenvalues 2 and -8. (Verify this fact by using EI GEN or your computer

software to compute the eigenvalues of A.) Since one eigenvalue is positive

and the other is negative, the critical point (1, 1) is a saddle point of both the

associated linear system and the given nonlinear system (6).