1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 417

DEFINITION Autonomous System

An autonomous system of differential equations has the form

dx

dt = f(x , y)

(1)

dy

dt = g(x, y).

System (1) is call ed autonomous because the functions f and g depend

explicitly only on the dependent variables x and y and not upon the inde-

pendent variable t (which often represents time).

In performing a qualitative analysis of an autonomous system of the form (1 ),
we are interested in answering the following questions:
"Are there any equilibrium points?" That is, "Are there any constant
solutions?"

"Is a particular equilibrium point stable or unstable?"
In chapter 7, we stated that if f, g , fx, fy, gx, and gy are continuous
functions of x and y in some rectangle R in the xy-plane and if (xo, Yo) E
R, then there is a unique solution to system (1) which satisfies the initial

conditions x(to) = xo, y(to) =Yo· Furthermore, the solution can be extended

in a unique manner until the boundary of R is reached. In the discussions
which follow, we will assume that f(x, y) and g(x, y) and their first partial
derivatives are all continuous functions in some rectangle R.

DEFINITIONS Phase-Plane, Trajectory, and

Phase-Plane Portrait

The xy-plane is called the phase-plane.

A solution (x(t), y(t)) of (1) traces out a curve in the phase-plane. This

curve is called a trajectory or orbit.

A phase-plane portrait is a sketch of a few trajectories of (1) together

with arrows indicating the direction a particle will fl.ow along the trajectory

as t (time) increases.