Index theoretical interpretation for the number of attractors of a dynamical system

Question

In this  MO post  I  search for  some  index  interpretation for  the  number  of  attractors of  a  dynamical  system.

https://mathoverflow.net/questions/182415/elliptic-operators-corresponds-to-non-vanishing-vector-fields

The  initial  operator we  associate  to  a  vector  field  is  the  derivation operator.  Theoretically, the  codimension of the  range of this derivation operator  is  an  upper  bound  for the  number of  limit cycles of the vector  field. But there  some  some practical difficulties. The  main difficulty is that this  codimension is  almost always infinity.  So in the  above post  I  search  for  some remedies   and  resolutions  for  such  difficulties.

I  would  appreciate if you give  comment on the  above  MO  question.

Thank  you