Morse–Smale system

Remarks

For a flow $\varphi_t$ generated by a vector field on $M$, the non-wandering set must be the union of finitely many equilibria and closed orbits. All of them must be hyperbolic, with transverse stable and unstable manifolds. Example: consider the flow on the round sphere $S^2$ that moves every point along its meridian from the north pole $N$ to the south pole $S$. Let $f$ be its time-one map. Then $\Omega(f) = \set{N, S}$, the point $N$ is a hyperbolic source and $S$ a hyperbolic sink. Here $W^s(S) = S^2 \setminus \set{N}$ is open, so it meets $W^u(N)$ transversally, and the other intersections are single points with full-dimensional manifolds or empty. Non-example: an irrational rotation of the circle has no periodic points, and its non-wandering set is the whole circle, so clause 1 fails. By the stable manifold theorem for hyperbolic periodic points, $W^s(p)$ and $W^u(p)$ are injectively immersed submanifolds of $M$. Smale showed that the gradient flow of a Morse function satisfies clause 3 after a small change of the metric.