1550078481-Ordinary_Differential_Equations__Roberts_

440 Ordinary Differential Equations

EXERCISES 10.3

1. Consider the nonlinear Richardson's arms race model

dx

• = 5y-x^2 + 5
  dt
  (9)
  dy 2
  dt = 4x - y - 4.

a. Find the critical point in the first quadrant.

b. Write the associated linear system and determine the stability charac-

teristics at the critical point. What can you conclude about the stability

of the nonlinear system (9) at the critical point?

c. Use PORTRAIT to solve the system initial value problems consisting of

system (9) and the following five initial conditions on the interval [O, 2]:

(i) x(O) = 0, y(O) = 7; (ii) x(O) = 2, y(O) = l ; (iii) x(O) = 5, y(O) = O;

(iv) x(O) = 7, y(O) = O; (v) x(O) = 7, y(O) = 7.

Display a phase-plane portrait of y versus x on the rectangle with

0 :::; x :::; 10 and 0 :::; y :::; 10.

2. Consider the following nonlinear arms race model
   dx 2
   dt = y-x - 4
   (10)
   dy
   dt = 4x - y2 - 4.

a. Find the critical point in the first quadrant.

b. Write the associated linear system and calculate the eigenvalues. What

can you say about the stability of the associated linear system based

on the eigenvalues? What can you conclude about the stability of the

nonlinear system (10) at the critical point?

c. Use PORTRAIT to solve the system initial value problems consisting

of system (10) and the following four initial conditions on the interval

[0,4]:

(i) x(O) = 1.8, y(O) = 1.8; (ii) x(O) = 1, y(O) = 4; (iii) x(O) = 3,

y(O) = 1; (iv) x(O) = 3, y(O) = 4.

Graph the phase-plane portrait of y versus x on the rectangle with

0 :::; x :::; 5 and 0 :::; y :::; 5. What do you infer about the stability of the

nonlinear system (10) from these results?