Math 4233
Homework Set 4
1. For each of the following systems carry out the following steps.
(i) Identify the critical points.
(ii) For each critical point c, identify the corresponding linear system. Write down the general solution
of these linear systems and discuss the stability of the solutions near the critical solution x(t) = c.
(iii) Plot the direction field of the original system and discuss the evolution of the system for various
initial conditions.
(a)
dx
dt =x(1 โxโy)
dy
dt =y(1.5โyโx)
(b)
dx
dt =x(1 โ0.5y)
dy
dt =y(โ0.25 + 0.5x)
2. For each of the following systems construct a suitable Liapunov function of the form ax2+cy2where a
and care to be determined. Then show that the critical point at the origin is of the indicated type.
(a)
dx
dt =โx3+xy2
dy
dt =โ2x2yโy3,asymptotically stable
(b)
dx
dt =x3
โy3
dy
dt = 2xy2+ 4x2y+ 2y3,unstable
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