, 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
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.
Given a metric space
and a homeomorphism
, we say that
is structurally stable
if there is a neighborhood
(the space of all homeomorphisms mapping
to itself endowed with the compact-open topology
) such that every element of
is topologically conjugate
If is a compact smooth manifold, a diffeomorphism is said to be structurally stable if there is a neighborhood of in (the space of all diffeomorphisms from to itself endowed with the strong topology) in which every element is topologically conjugate to .
If is a vector field in the smooth manifold , we say that is -structurally stable if there is a neighborhood of in (the space of all vector fields on endowed with the strong topology) in which every element is topologically equivalent to , i.e. such that every other field in that neighborhood generates a flow on that is topologically equivalent to the flow generated by .