Morse–Smale system

In dynamical systems theory, an area of pure mathematics, a Morse–Smale system is a smooth dynamical system whose non-wandering set consists of finitely many hyperbolic equilibrium points and hyperbolic periodic orbits and satisfying a transversality condition on the stable and unstable manifolds.

Morse–Smale systems are structurally stable and form one of the simplest and best studied classes of smooth dynamical systems.

They are named after Marston Morse, the creator of the Morse theory, and Stephen Smale, who emphasized their importance for smooth dynamics and algebraic topology.

Consider a smooth and complete vector field X defined on a compact differentiable manifold M with dimension n. The flow defined by this vector field is a Morse-Smale system if

This mathematical analysis–related article is a stub.

Flow lines on an upright torus: the stable and unstable manifolds of the saddle points do not intersect transversely, so the height function does not satisfy the Morse-Smale condition.
Flow lines on a tilted torus: the height function satisfies the Morse-Smale condition.