1550078481-Ordinary_Differential_Equations__Roberts_

418 Ordinary Differential Equations

It follows from the uniqueness of so lutions of (1) that at most one trajectory

passes through any point in R. That is, trajectories do not intersect one

another in the phase-plane.

DEFINITION Critical Point or Equilibrium Point

A critical point or equilibrium point of system (1) is a point (x*, y*)

where f(x, y) = g(x, y) = 0.

Hence, by definition, the critical points of system (1) are determined by

simultaneously solving

(2)

f(x,y) = 0

g(x, y) = 0.

DEFINITIONS Stable, Asymptotically Stable, Neutrally Stable,

and Unstable

A critical point (x*, y*) of system (1) is stable if every solution which is

"near" ( x*, y*) at time to exists for all t ;::: to and remains "near" ( x*, y*).

A stable critical point (x*, y*) of system (1) is asymptotically stable if

every solution which is "near" (x*, y*) at time t 0 exists for all t ;::: t 0 and

limt_,= x(t) = x* and limt_,= y(t) = y*.

A stable critical point which is not asymptotically stable is said to be

neutrally stable.

A critical point which is not stable is called unstable.

In order to better understand the concept of stability and to be able to

sketch trajectories of an autonomous system near a critical point, we first

need to discuss stability and phase-plane portraits for linear systems of the

form

dX

dt =AX +BY +r

(3)

dY

dt = CX+DY+ s

where A, B, C, D , r, ands are real constants. In many, but not all cases, the

type of stability at the critical point of a nonlinear system is the same as the