---
tag: 10B1TS96VA
kind: definition
name: Morse–Smale system
msc: [37D15]
defines: 7AMQ8GS78Y
uses: [EY2W34JZA4, T75CZ5H47N, EWVATW6EYZ, 0Z1Y7JHGBR]
sources:
  - url: https://en.wikipedia.org/wiki/Morse%E2%80%93Smale_system
---

## Definition

Let $M$ be a compact smooth manifold and let $f \colon M \to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]].
A [[periodic point|T75CZ5H47N]] $p$ of period $n$ is hyperbolic if no eigenvalue of $Df^n_p$ has modulus $1$.
Such a point has a stable manifold $W^s(p)$ and an unstable manifold $W^u(p)$.
The diffeomorphism $f$ is *Morse–Smale* if the following hold.

1. The [[non-wandering set|EY2W34JZA4]] $\Omega(f)$ is finite, so it consists of finitely many periodic orbits.
2. Every periodic point of $f$ is hyperbolic.
3. For all periodic points $p$ and $q$, the manifolds $W^s(p)$ and $W^u(q)$ intersect [[transversally|EWVATW6EYZ]].

## 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.