1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 427

g(x, y) = g(x, y) + gx(x, y)(x - x) + gy(x, y)(y - y) + R 2 (x, y)

where Ri(x, y)/ J(x - x*)^2 + (y - y*)^2 -t 0 as (x, y) __, (x*, y*) for i = 1, 2.

Since f(x*, y*) = g(x*, y*) = 0, the nonlinear system (1) can be written as

(9)

~~ = fx(x, y)(x - x) + fy(x, y)(y -y) + R1(x, y)

~~ = gx(x, y)(x - x) + gy(x, y)(y - y) + Rz(x, y).

Since the functions Ri(x, y) are "small" when (x, y) is near (x*, y*), we antic-

ipate that the stability and type of critical point at (x*, y*) for the nonlinear

system (9) will be similar to the stability and type of critical point at (x*, y*)

for the linear system

dx

dt

dy

dt

A(x - x) + B(y - y)

C(x - x) + D(y - y)

where A = fx(x*, y*), B = fy(x*, y*), C = gx(x*, y*), and D = gy(x*, y*).

On letting X = x - x* and Y = y - y* and noting that dX / dt = dx / dt and

dY / dt = dy/dt, we anticipate that the stability characteristics of the nonli near

system (1) or (9) near (x*, y*) will be similar to the stability characteristics

at (0, 0) of the associated linear system

dX =AX BY

dt +

(10)

dY

di= ex +DY.

In fact, it turns out that the stability characteristics of the nonlinear sys-
tem (9) are the same as the stability characteristics of the associated linear
system (10) with the following two possible exceptions. (i) When the eigen-
values of the linear system (10) are purely imaginary(>.= ±i/3), the neutrally
stable critical point of the associated linear system can become asymptoti-
cally stable, unstable, or remain neutrally stable in the nonlinear system and,
correspondingly, the trajectories can become stable spirals, unstable spirals,
or remain centers. So when the eigenvalues of the associated linear system are
purely imaginary, the stability and behavior of the trajectories of the nonlin-
ear system near (x, y) must be analyzed on a case-by-case basis. (ii) When
the eigenvalues are equal , the stability properties of the nonlinear system and
the associated linear system remain the same, but the critical point might
change from a node in the linear system to a spiral point in the nonlinear
system.