442 Ordinary Differential Equations
c. (i) Find the critical points in the first quadrant of the system
dxdt = 3y-x -1
(13)
dy 2
dt = x -y -1.(ii) Write the associated linear system and determine the stability
characteristics at each critical point. What can you conclude about
the stability of the nonlinear system (13) at each critical point?(iii) Use PORTRAIT to solve system (13) and the following six initial
conditions on the interval [O, 2]:(i) x(O) = 1, y(O) = 1; (ii) x(O) = 0, y(O) = 4; (iii) x(O) = 3, y(O) = O;
(iv) x(O) = 4, y(O) = O; (v) x(O) = 7, y(O) = O; (vi) x(O) = 3, y(O) = 5.
Display the phase-plane graph of y versus x on the rectangle with
0 :::; x :::; 10 and 0 :::; y :::; 5.- Consider the following modification of Richardson's arms race model
dx- = Ay^2 - Cx+r
(14)
dtdy- = Bx^2 -Dy+s
dt
where A , B, C, and D are positive real constants and r and s are real
constants. Here the fear of each nation for the other is modelled as being
proportional to the square of the other nations yearly expenditures for
arms.
a. What is the graph of Ay^2 - Cx + r = O? What is the graph of
Bx^2 - Dy+ s = O? How many critical points can system (14) have?
What is the maximum number of critical points system (14) can have
in the first quadrant?b. Write a single equation which the x-coordinate of the critical point must
satisfy.
c. (i) Find the critical point in the first quadrant of the system
dx- = y^2 -4x +4
dt
(15)
dy 2
dt = x - 5y-5.
(ii) Write the associated linear system and determine the stability
characteristics at the critical point. What can you conclude about the
stability of the nonlinear system (15) at the critical point?