- Structural stability
mathematics, structural stability is an aspect of stability theoryconcerning 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 topologicalviewpoint, analogous to Lyapunov stability), which essentially means that the dynamics will not change under small perturbations.
metric spaceand a homeomorphism, we say that is structurally stable if there is a neighborhood of in (the space of all homeomorphisms mapping to itself endowed with the compact-open topology) such that every element of is topologically conjugateto .
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 fieldin 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 .
sensitive dependence on initial conditions
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