1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 419

type of stability of a corresponding linear system, and near the critical point

the trajectories of the nonlinear system resemble the trajectories of the linear

system.

Suppose (x*, y*) is a critical point of the linear system (3). Hence, (x*, y*)

simultaneously satisfies

Ax* + By* + r = 0

Cx* + Dy* + s = 0.

In order to locate the origin of a new xy-coordinate system at (x*, y*) with

x-axis parallel to the X-axis and y-axis parallel to the Y-axis, we make the

changes of variables x = X - x* and y = Y - y*. Differentiating and substi-

tuting into (3), we find

and

Next, letting

dx dX

dt = dt = A(x + x*) + B(y + y*) + r

=Ax + By+ (Ax*+ By*+ r)

= Ax+By

dy dY

dt = dt = C(x + x*) + D(y + y*) + s

= Cx +Dy+ (Cx +Dy+ s)

=Cx+Dy.

z = G) and A = ( ~ ~)

we see that the stability of the nonhomogeneous linear system (3) at the
critical point (x, y) and the behavior of the trajectories of (3) nea r (x, y)
is the same as the stability and behavior of the linear homogeneous system

(4) dz= Az

dt

at (x, y) = (0, 0). Hence, we need to study only the various types of stability
of system ( 4) at the origin and the associated behavior of the trajectories near
the origin to understand the stability and behavior of systems of the form (3)
at (x, y).

In chapter 8, we saw how to write the general solution of ( 4) in terms of
the eigenvalues and eigenvectors of the matrix A. The equation Az = 0 has

the unique solution z = 0 if and only if det A -=f. 0. That is, the origin is the

unique critical point of system ( 4) if and only if det A -=f. 0. If det A= 0, there

is a line of critical points which passes through the origin. Throughout, we

will assume det A -=f. 0. Let us now consider several examples.