---
tag: 7AMQ8GS78Y
kind: concept
name: Morse–Smale system
msc: [37D15]
specializes: [116TPXR5FS]
sources:
  - url: https://en.wikipedia.org/wiki/Morse%E2%80%93Smale_system
same_as:
  wikidata: Q6914276
  wikipedia: Morse–Smale system
---

## Summary

A Morse–Smale system is a smooth dynamical system whose non-wandering set consists of finitely many hyperbolic periodic orbits, with stable and unstable manifolds meeting transversally.
These are the dynamical systems with the simplest possible recurrent behaviour.
They are [[structurally stable|D4MGFAHVMT]], and on a compact manifold they have zero topological entropy.
The gradient flow of a [[Morse function|KPQ96640ZJ]] for a generic metric is Morse–Smale, which links them to Morse theory and Morse homology.

## Explanations

### L1 human

Think of water flowing on a landscape with finitely many peaks, passes and pits.
Every drop starts at one of these rest points and ends at another.
A Morse–Smale system is a motion of this kind.
All its rest points and periodic orbits are of the clean, non-degenerate type, and the paths between them cross in general position.
For example, on a round sphere, the flow from the north pole down to the south pole is Morse–Smale.
See the [[definition|10B1TS96VA]].