I have been doing some self-study of differential equations and have finished Habermans' elementary text on linear ordinary differential equations and about half of Strogatz's nonlinear differential equations book. The thing that I am noticing is just how much these text avoid engaging the underlying differential geometry/topology of phase spaces. It also feels like the further I got in this differential equations, the more important it is to understand the underlying differential topology--for instance understanding Hamiltonian systems and symplectic manifolds, etc.
Indeed, the only text that I have seen that seems to engage the differential topology of phase spaces seems to be Arnold's 1973 book on Ordinary Differential Equations. This seems to be a really good book. The challenge is that Arnold can be a bit terse sometimes, so I was hoping to find a book to supplement Arnold's text.
I have enough background in differential topology by watching Fredric Schuller's lectures and then working through some of Renteln and John Lee's books.
Hence, I was hoping to find a book that elaborates on the differential topology side of differential equations. So all of these topics about vector fields on a manifold are fair game. Now I looked at Hirsch and Smale 1974, but this did not really get into the differential topology stuff. Perko's book was also pretty terse and did not systematically develop the topology.
If anyone has any good recommendations, that would be appreciated.
Thanks.