Applications of Systems of Equations
- Consider the nonlinear arms race model
(11)
dx 2
dt = 9y-x - 9
dy 2
- = 4x-y - 8.
dt
a. Find the critical points in the first quadrant.
441
b. Write the associated linear systems and determine the stability char-
acteristics of each critical point. What can you conclude about the
stability of the nonlinear system (11) at each critical point?
c. Use PORTRAIT to solve the system initial value problems consisting
of (11) and the following seven initial conditions on the interval [O, 2]:
(i) x(O) = 1, y(O) = 1; (ii) x(O) = 2, y(O) = 3; (iii) x(O) = 4, y(O) = O;
(iv) x(O) = 4, y(O) = 3; (v) x(O) = 4, y(O) = 5; (vi) x(O) = 6, y(O) = 3;
(vii) x(O) = 6, y(O) = 6.
Display the phase-plane graph of y versus x on the rectangle with 0 :::;
x :::; 10 and 0 :::; y :::; 10.
- Consider the following modification of Richardson's arms race model
dx
- =Ay-Cx+r
dt
(12)
dy
- = Bx-Dy^2 +s
dt
where A, B, C , and D are positive real constants and r and s are real
constants. Here each nation's fear of the other nation is modelled as
being proportional to the amount the other nation spends annually for
arms. The resistance of the first nation's people to increased spending for
arms is modelled as being proportional to the amount spent yearly for
arms, while the resistance of the second nation's people to increased spend-
ing for arms is modelled as being proportional to the square of the amount
spent yearly for arms.
a. What is the graph of Ay - Cx + r = O? What is the graph of
Bx - Dy^2 + s = O? How many critical points can system (12) have?
What is the maximum number of critical points system (12) can have
in the first quadrant?
b. Write a single equation which the x-coordinate of a critical point must
satisfy.
