Abstract

We describe the topology of the space of all smooth asymptotically stable vector fields on Rⁿ, as well as the space of all proper smooth Lyapunov functions for such vector fields. In particular, both spaces are path-connected and simply connected when n is not equal to 4, 5 and weakly contractible when n is less than 3. 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 nonhyperbolic but asymptotically stable equilibria, and a parametric Morse lemma for degenerate minima. (Note that Hatcher's aforementioned result is a solution to the "SMALE conjecture".)