1550078481-Ordinary_Differential_Equations__Roberts_

444 Ordinary Differential Equations

(3) Suppose the eigenvalues of A all have nonpositive real part and sup-

pose those eigenvalues which have zero real part are purely imaginary eigen-

values. That is, suppose the eigenvalues which have zero real part are of the

form i/3 where /3 -I 0. If every purely imaginary eigenvalue of multiplicity k

has k linearly independent eigenvectors, then the origin is a stable critical

point of system (17). Whereas, if any purely imaginary eigenvalue of multi-

plicity k has fewer thank linearly independent eigenvectors, then the origin

is an unstable critical point of system (17).

The stability characteristics of the nonhomogeneous system of n linear first-

order differential equations

(18) z' = Az + b

at the critical point z, which satisfies Az + b = 0 , is the same as the stabil-

ity characteristics of the homogeneous linear system y' = Ay at the origin,

since the linear transformation y = z - z* transforms the nonhomogeneous

system (18) into the homogeneous system (17).

One possible extension of Richardson's arms race model from a system for

two nations to a system for three nations is

(19) dy2

dt

dy3

dt = a31Y1 + a32Y2 - c3y3 + r3

where aij and Ci are nonnegative real constants and ri are any real constants.
In system ( 19) Y1, Y2, and y3 are the yearly amounts spent by nations 1,
2, and 3 for arms; the terms aiiYi represent the rate of increase in yearly
expenditures for arms by nation i due to its fear of nation j ; the terms -CiYi
represent the rate of decrease in yearly expenditures for arms by nation i due
to the resistance of its people to increased spending for arms; and ri represents
the collective underlying "goodwill" or "grievance" which nation i feels toward
the other two nations.

6. a. Suppose in system (19) c1 = c2 = c3 = 1 and aij = 2 for all appropriate

values of i and j. Is the critical point of system (19) stable or unstable?

(HINT: Write system (19) in matrix-vector form, use EIGEN or your

computer software to calculate the eigenvalues of the appropriate 3 x 3

constant matrix A, and use the stability theorem to determine the an-

swer to the question.)