1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 429

Translating the critical point (1, 1) to the origin by letting X = x - 1 and

Y = y - 1 , we are lead to consider the stability characteristics at (0, 0) of the

linear system

X'=-Y

Y'=-X.

The eigenvalues of this system, which are the eigenvalues of the matrix

A=( 0 -1)

-1 0

are >-1 = 1 > 0 and >-2 = -1 < 0. From Table 10.l the origin is an unstable

saddle point of this linear system; and, therefore, the critical point (1, 1) is an

unstable saddle point of the nonlinear system ( 11).

We used a computer software program to produce the direction field shown

in Figure 10.8a for

dy y' y-xy

dx x' x - xy

on the rectangle R = {(x, y)I - 1.5::::; x::::; 2.5 and - 1.5::::; y::::; 2.5}. From

this direction field and the stability properties of system (11) at the critical
points (0, 0) and (1, 1), we were able to sketch the phase-plane portrait shown
in Figure 10.8b for the nonlinear system (11).

2

y

0

-1

I I I I I I I \ '-......, __ ..- / / / / /

I I I I JI 1

1

I \ '---...--,.,.,,,,, / / / / /

I I I I \ '-.. ,,,.,.,,,,, / / / / /

/Jill \'-'----//////

~~

1

l

1

l

1

~=;jjj~

! l

)

\ "'.-/I I I I I I

I \ / I I I I I I I

//-,,\\\\\

\\\\\\ // ___ .._,,,,,

-~~.._,,, "'''\\\ /--//~ ---------

/

___ .,..,,_-,.,.,,,,,/I,-.---~~~----

,...,.,,,.,,,.,.,.,,,,,,.,.,,,,, / / I '-----~~~-----------­

/ / / / / / I \ -....... ---------------

//////I ,,__ ------

/////// \'- -----

/////I I \ ,--....---------------

/ / / / / / / ,,~ ~-----

-1 0 2

x

Figure 10.8a Direction Field for System (11)