Consider a smooth and
complete vector fieldX defined on a compact differentiable manifold M with dimension n. The flow defined by this vector field is a Morse-Smale system if
X has only a finite number of singular points (i.e. equilibrium points of the flow), and all of them are
hyperbolic equilibrium points.
The
limit sets of all orbits of X tends to a singular point or a periodic orbit.
The stable and unstable manifolds of the singular points and periodic orbits intersect transversely. In other words, if is a singular point (or periodic orbit) and (respectively, ) its stable (respectively, unstable) manifold, then implies that the corresponding tangent spaces of the stable and unstable manifold satisfy .
Examples
Any
Morse functionf on a
compactRiemannian manifoldM defines a gradient vector field. If one imposes the condition that the
unstable and
stablemanifolds of the
critical points intersect transversely, then the gradient vector field and the corresponding smooth
flow form a Morse–Smale system. The finite set of
critical points of f forms the non-wandering set, which consists entirely of fixed points.
For Morse–Smale systems on the 2D-sphere all equilibrium points and periodical orbits are
hyperbolic; there are no
separatrice loops.
Properties
By
Peixoto's theorem, the vector field on a 2D manifold is structurally stable if and only if this field is Morse-Smale.
Consider a Morse-Smale system defined on compact differentiable manifold M with dimension n, and let the index of an equilibrium point (or a periodic orbit) be defined as the dimension of its associated unstable manifold. In Morse-Smale systems, the indices of the equilibrium points (and periodic orbits) are related with the topology of M by the Morse-Smale inequalities. Precisely, define mi as the sum of the number of equilibrium points with index i and the number of periodic orbits with indices i and i + 1, and bi as the i-th
Betti number of M. Then the following inequalities are valid:[1]