Math 308 - Differential Equations Fall 2002
Homework Assignment 11
Due Friday, December 13
1. Consider the nonlinear system
dx
dt =x(1 โxโy)
dy
dt =y(3 โ2xโy)
(a) Find all the equilibrium points.
(b) Find the linearization at each equilibrium, classify the equilibrium point, and sketch the phase
portrait for the linearized system.
(c) Sketch the phase portrait for the nonlinear system. Use the results of (b) to determine the phase
portrait near the equilibrium points. Use nullclines to help determine what happens elsewhere.
Sketch and label the separatrices associated with any saddle points. (You may use Maple or the
textโs CD to check your answer, but you do not have to hand in a computer generated phase
portrait.)
(d) Consider the solution (x(t), y(t)) for which x(0) = 1/2 and y(0) = 1/2. Use your phase portrait
in (c) to determine each of the following: lim
tโโ
x(t), lim
tโโโ
x(t), lim
tโโ
y(t), lim
tโโโ
y(t).
2. Repeat parts (a)-(d) of the previous question for the system
dx
dt = (x3โ1)(1 โy)
dy
dt =y(2xโ1)
3. Consider the linear system
dx
dt =ax,
dy
dt =by,
where aand bare constants.
(a) Show that the function g(x, y) = xโbyais a conserved quantity.
(b) What condition on aand bwill make this a Hamiltonian system?
Continued on the back...
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