We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$  we  mean 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?