In synthetic differential geometry, giving a smooth dynamical system on a manifold is equivalent to giving, for each point x0, the "infinitesimally next state" s(x0), i.e a point so that if x(t) = x0, then x(t + h) = s(x_0) (where h^2 = 0).
I.e it's just a *discrete* dynamical system with the constraint that each step is infinitesimal.
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The main ingredient in proving this is the observation that the coordinate-change maps must be linear on infinitesimals, so the infinitesimal neighborhood of a point is a vector space in a canonical way, which is isomorphic to the tangent space at that point.
