Given any two continuous dynamical systems on Euclidean space such that the origin is globally asymptotically stable and assume that both systems come equipped with-possibly different-convex smooth Lyapunov functions asserting the origin is indeed globally asymptotically stable. We show that this implies those two dynamical systems are homotopic through qualitatively equivalent dynamical systems. It turns out that relaxing the assumption on the origin to any compact convex set or relaxing the convexity assumption to geodesic convexity does not alter the conclusion. Imposing the same convexity assumptions on control Lyapunov functions leads to a Hautus-like stabilizability test. These results ought to find applications in optimal control and reinforcement learning.