We explore connections between stability of equilibrium points of cubic differential equations and feasibility of basic semialgebraic sets. This allows for (i) a methodology for providing certificates of infeasibility of such sets presented as Lyapunov functions and obtained by semidefinite programming, and (ii) a methodology for (``often") finding feasible solutions to a broad class of constraint satisfaction problems by simulating the solution of a differential equation. We present preliminary work in this direction and demonstrate applications to (intractable) problems such as that of finding Nash equilibria in games, or Euclidean embeddings given pairwise distances.