x'=-y-x^3

y'=x

Using linearization, I see that the only fixed pt, (0,0) is a center. But the real parts of the eigenvalues are 0, so maybe linearization hasn't worked. I tried changing to polar to analyze, and I now have

theta' = 1+rcos^3(theta)

r'=-r^3cos^4(theta)

What do these tell me about the vector field? Have I made a mistake?