1550078481-Ordinary_Differential_Equations__Roberts_

430 Ordinary Differential Equations

2

1

y

0

-1

-1 0 1 2

x

Figure 10.8b Phase-Plane Portrait for System (11)

EXAMPLE 8 Determining Critical Points and Type of Stability

for a Nonlinear System

Find all critical points of the nonli near system

x' = -y-x^3 = f(x,y)

(13)

y' = x - y^3 = g(x, y).

For each critical point determine the type of stability and type of critical
point.

SOLUTION

Solving -y - x^3 = 0 for y and substituting the res ult into x - y^3 = 0, we

see that the x coordinate of the critical points must satisfy

x - (- x^3 )^3 = x + x^9 = x(l + x^8 ) = 0.

Since x = 0 is the only real solution of x(l + x^8 ) = 0, the only critical point
of system (13) is (0, 0).