Following is the problem that I solved the first part:

Consider the system x' = f(x), where f: R^2 into R^2, is defined by:

f(x) = [ (x1)^3 + (x1) (x2)^2 ]
[ (x1)^2 (x2) + (x2)^3 ]

a. Find all equilibrium points of the system.

b. Use an appropriate Liapunov function to determine the stability of the equilibrium points. If an equilibrium point is stable, is is asymptoticall stable?

Solution:

- For question a: II found that (xo) = (0,0)^T is the only equilibrium point with (x1)=0 and (x2)= 0., and since the eigenvalues are 0 implies that it is a nonhyperbolic point

- For question b: I tried the lyapunov function V(x) = a(x1)^2+ b(x2)^2, and I got 1/2 V'(x) = a {(x1)^4 + (x1)^3 (x2)} + b {(x2)^4 + (x1) (x2)^3}.