Saturation on compact manifolds via Stone-Weierstrass type theorem, and forced nonlinear PDEs

We consider polynomial nonlinear PDEs on compact manifolds with finite-dimensional forcing terms. We provide necessary and sufficient conditions under which the forcing, through the nonlinear dynamics, attains a dense subspace on manifolds with and without boundary. Our key argument is Nachbin's theorem, which is an analogue of the well-known Stone-Weierstrass theorem in spaces of differentiable functions. As applications, we prove small-time approximate controllability by additive finite-dimensional controls, topological irreducibility, and almost sure unboundedness of Sobolev norms.