---
tag: D4MGFAHVMT
kind: concept
name: Structural stability
msc: [34D30, 37C20]
sources:
  - url: https://en.wikipedia.org/wiki/Structural_stability
same_as:
  wikidata: Q1648886
  wikipedia: Structural stability
---

## Summary

A dynamical system is structurally stable if every small enough perturbation of it has the same qualitative behaviour.
More precisely, the [[phase portrait|0T3GK0XCBC]] of the perturbed system is the image of the original one under a homeomorphism that maps trajectories to trajectories.
Structural stability is about the stability of the whole system under changes of the equations, not of one solution under changes of the initial state.
Models of the real world are only known approximately, so structurally stable models are the ones whose qualitative predictions can be trusted.
See the [[definition|KQX43WB57R]].

## Explanations

### L1 human

Imagine drawing all the trajectories of a system as a picture of arrows and curves.
If a tiny change in the equations only bends and stretches this picture, without changing its pattern, the system is structurally stable.
On a circle, the flow $\theta' = \sin \theta$ has one repelling and one attracting rest point, and small changes keep this pattern.
The flow $\theta' = 1 - \cos \theta$ has a single rest point that attracts from one side and repels on the other.
Adding a small positive constant removes this rest point, so this flow is not structurally stable.