Definitions

# Structural stability

In mathematics, structural stability is an aspect of stability theory concerning whether a given function is sensitive to a small perturbation. The general idea is that a function or flow is structurally stable if any other function or flow close enough to it has similar dynamics (from the topological viewpoint, analogous to Lyapunov stability), which essentially means that the dynamics will not change under small perturbations.

## Definition

Given a metric space $\left(X,d\right)$ and a homeomorphism $fcolon Xto X$, we say that $f$ is structurally stable if there is a neighborhood $\left\{V\right\}$ of $f$ in $operatorname\left\{Homeo\right\}\left(X\right)$ (the space of all homeomorphisms mapping $X$ to itself endowed with the compact-open topology) such that every element of $\left\{V\right\}$ is topologically conjugate to $f$.

If $M$ is a compact smooth manifold, a $mathcal\left\{C\right\}^k$ diffeomorphism $f$ is said to be $mathcal\left\{C\right\}^k$ structurally stable if there is a neighborhood of $f$ in $operatorname\left\{Diff\right\}^k\left(M\right)$ (the space of all $mathcal\left\{C\right\}^k$ diffeomorphisms from $M$ to itself endowed with the strong $mathcal\left\{C\right\}^k$ topology) in which every element is topologically conjugate to $f$.

If $X$ is a vector field in the smooth manifold $M$, we say that $X$ is $mathcal\left\{C\right\}^k$-structurally stable if there is a neighborhood of $X$ in $\left\{X\right\}^k\left(M\right)$ (the space of all $mathcal\left\{C\right\}^k$ vector fields on $M$ endowed with the strong $mathcal\left\{C\right\}^k$ topology) in which every element is topologically equivalent to $X$, i.e. such that every other field $Y$ in that neighborhood generates a flow on $M$ that is topologically equivalent to the flow generated by $X$.