structural stability

Given a metric space (X,d) and an homeomorphismPlanetmathPlanetmath f:XX, we say that f is structurally stableMathworldPlanetmath if there is a neighborhoodMathworldPlanetmathPlanetmath 𝒱 of f in Homeo(X) (the space of all homeomorphisms mapping X to itself endowed with the compact-open topologyMathworldPlanetmath) such that every element of 𝒱 is topologically conjugateMathworldPlanetmath to f.

If M is a compactPlanetmathPlanetmath smooth manifoldMathworldPlanetmath, a 𝒞k diffeomorphism f is said to be 𝒞k structurally stable if there is a neighborhood of f in Diffk(M) (the space of all 𝒞k diffeomorphisms from M to itself endowed with the strong 𝒞k topologyMathworldPlanetmath) 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 𝒞k-structurally stable if there is a neighborhood of X in 𝒳k(M) (the space of all 𝒞k vector fields on M endowed with the strong 𝒞k 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.

Remark. The concept of structural stability may be generalized to other spaces of functions with other topologies; 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), which essentially means that the dynamics will not change under small perturbations.

Title structural stability
Canonical name StructuralStability
Date of creation 2013-03-22 13:48:45
Last modified on 2013-03-22 13:48:45
Owner Koro (127)
Last modified by Koro (127)
Numerical id 8
Author Koro (127)
Entry type Definition
Classification msc 37C20
Classification msc 34D30
Synonym structurally stable