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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, stability theory deals with the stability of the solutions of differential equations and dynamical systems. ... Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, flow refers to the group action of a one-parameter group on a set. ... Topology (Greek topos = place and logos = word) is a branch of mathematics concerned with the study of topological spaces. ... In mathematics, the notion of Lyapunov stability occurs in the study of dynamical systems. ...

Definition

Given a metric space (X,d) and a homeomorphism , we say that f is structurally stable if there is a neighborhood V of f in (the space of all homeomorphisms mapping X to itself endowed with the compact-open topology) such that every element of V is topologically conjugate to f. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. ... In mathematics, two functions are said to be topologically conjugate to one another if there exists a homeomorphism that will conjugate the one into the other. ...

If M is a compact smooth manifold, a diffeomorphism f is said to be structurally stable if there is a neighborhood of f in (the space of all diffeomorphisms from M to itself endowed with the strong topology) in which every element is topologically conjugate to f. On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...

If X is a vector field in the smooth manifold M, we say that X is -structurally stable if there is a neighborhood of X in X^k(M) (the space of all vector fields on M endowed with the strong 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. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, flow refers to the group action of a one-parameter group on a set. ...

See also

• sensitive dependence on initial conditions
• stability
• homeostasis
• self-stabilization, superstabilization

This article incorporates material from StructuralStability on PlanetMath, which is licensed under the GFDL. Point attractors in 2D phase space. ... Look up stability in Wiktionary, the free dictionary. ... Homeostasis is the property of either an open system or a closed system,[1] especially a living organism, to regulate its internal environment to maintain a stable, constant condition. ... Self-stabilization is a concept from computer science. ... Superstabilization is a specialization of the concept of self-stabilization. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

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