Differential topology of the spaces of asymptotically stable vector fields and Lyapunov functions

We study the topology of the space of all smooth asymptotically stable vector fields on $\mathbb{R}^n$, as well as the space of all proper smooth Lyapunov functions for such vector fields. We prove that both spaces are path-connected and simply connected when $n\neq 4,5$ and weakly contractible when $n\leq 3$. Moreover, both spaces have the weak homotopy type of the nonlinear Grassmannian of submanifolds of $\mathbb{R}^n$ diffeomorphic to the $n$-disc. The proofs rely on Lyapunov theory and differential topology, such as the work of Smale and Perelman on the generalized Poincar\'{e} conjecture and results of Smale, Cerf, and Hatcher on the topology of diffeomorphism groups of discs. Applications include a partial answer to a question of Conley, a parametric Hartman-Grobman theorem for nonyperbolic but asymptotically stable equilibria, and a parametric Morse lemma for degenerate minima. We also study the related topics of hyperbolic equilibria, Morse minima, and relative homotopy groups of the space of asymptotically stable vector fields inside the space of those vanishing at a single point.