Showing dynamical systems are not topologically conjugate

Question

Consider the two dynamical systems $$x'=y(x^2+y^2+1)$$
$$y'=-x(x^2+y^2+1)$$ and $$x'=y$$ $$y'=-x.$$ The second system is just the linearization of the first one around $(0,0)$.
I would like to prove that the two systems are not topologically conjugate. Now, the matrix corresponding to the linearization is not hyperbolic, so the linearization theorem cannot be used and in any case the theorem says that if the matrix is hyperbolic, then the system is topologically conjugate to its linearization.
I know that for instance the second system has solutions with period precisely $2 \pi$, but I am not sure how to use this in order to show the two are not topologically conjugate.