Structural stability

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.


Given a metric space (X,d) and a homeomorphism fcolon X o X, we say that f is structurally stable if there is a neighborhood {V} of f in operatorname{Homeo}(X) (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.

If M is a compact smooth manifold, a mathcal{C}^k diffeomorphism f is said to be mathcal{C}^k structurally stable if there is a neighborhood of f in operatorname{Diff}^k(M) (the space of all mathcal{C}^k diffeomorphisms from M to itself endowed with the strong mathcal{C}^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{C}^k-structurally stable if there is a neighborhood of X in {X}^k(M) (the space of all mathcal{C}^k vector fields on M endowed with the strong mathcal{C}^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.

ee also

*sensitive dependence on initial conditions
*self-stabilization, superstabilization

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • structural stability — struktūrinis stabilumas statusas T sritis automatika atitikmenys: angl. structural stability; structure stability vok. Gefügebeständigkeit, f; strukturelle Stabilität, f rus. структурная устойчивость, f; устойчивость структуры, f pranc. stabilité …   Automatikos terminų žodynas

  • Stability — can refer to: *Aircraft flight Stability (aircraft) *In atmospheric fluid dynamics, atmospheric stability, a measure of the turbulence in the ambient atmosphere *BIBO stability (Bounded Input, Bounded Output stability), in signal processing and… …   Wikipedia

  • Stability theory — In mathematics, stability theory deals with the stability of solutions (or sets of solutions) for differential equations and dynamical systems. Definition Let (R, X, Φ) be a real dynamical system with R the real numbers, X a locally compact… …   Wikipedia

  • structural — adjective Date: circa 1827 1. of or relating to the physical makeup of a plant or animal body < structural defects of the heart > 2. a. of, relating to, or affecting structure < structural stability > b. used in building structures < structural… …   New Collegiate Dictionary

  • structural — Relating to the structure of a part; having a structure. SYN: anatomical (2). * * * struc·tur·al strək chə rəl, strək shrəl adj 1) of or relating to the physical makeup of a plant or animal body <structural defects of the heart> compare… …   Medical dictionary

  • Structural Integration — is a type of Alternative medicine which aims to align the human body in the gravitational field. [ [ What is Structural Integration? ] ] The claimed benefit is that the increased use of balance at finer levels of… …   Wikipedia

  • Structural geology — is the study of the three dimensional distribution of rock bodies and their planar or folded surfaces, and their internal fabrics . Structural geology includes features of and overlaps with facets of geomorphology, metamorphism and geotechnical… …   Wikipedia

  • Structural adjustment — is a term used to describe the policy changes implemented by the International Monetary Fund (IMF) and the World Bank (the Bretton Woods Institutions) in developing countries. These policy changes are conditions (Conditionalities) for getting new …   Wikipedia

  • Structural analysis — comprises the set of physical laws and mathematics required to study and predict the behavior of structures. The subjects of structural analysis are engineering artifacts whose integrity is judged largely based upon their ability to withstand… …   Wikipedia

  • stability — is best defined as the opposite of instability, which is the occurrence of large structural deformations which are not the result of material failure. Note: passenger vehicles exhibit varying characteristics depending upon test conditions and… …   Mechanics glossary

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.