We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here we use $H^{s}$ for the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$.

We consider the following polynomial vector field on $\mathbb{R}^{2}$ $$ \begin{cases} \dot x=ax+by+cx^{2}+dxy+ey^{2}\\\dot y=a'x+b'y+c'x^{2}+d'xy+e'y^{2} \end{cases}$$

Using Poincare compactification, the above vector field can be considered as an analytic vector field on $S^{2}$. These vector fields define derivations on the plane and sphere, repectively. In both cases this operator is denoted by $D:H^{2} \to H^{1}$.

My question is about the spectrum of $\phi \circ D$ as a bounded operator on $H^{2}$:

Can one compute this spectrum in terms of the coefficient of the above vector field? Is it true to say that the number of connected components of the spectrum is finite? Are there any relation between the dynamics of the vector field and the topology-geometric properties of the spectrum? Does the number of limit cycles of the above quadratic polynomial system effect on the shape of the spectrum? More generally, are there some results about some relations between the dynamical behaviour of an arbitrary vector field $X$ and the spectrum of the above operator on Sobolev space?