# Stability

A term not having a clearly defined content.

1) Stability in connection with motion is a characteristic of the behaviour of the system in an infinite interval of time. This character of the motion can be elaborated in the following way.

a) As the property that the moving system in one sense or another has small deviations from a certain motion for small perturbations of the initial positions of the system (in phase space), where the smallness of the deviation is uniform for $t\geq0$ (cf. Lyapunov stability; Orbit stability; Uniform stability). In other arguments stability of motion is the property that the moving system only slightly deviates from a certain position for small perturbations in the initial position of the system (in phase space), as well as in the law of motion itself (cf. Stability in the presence of persistently acting perturbations). Sometimes the smallness of the deviation and the perturbation is measured only in certain parameters instead of all (cf. Stability for a part of the variables).

b) Stability of motion of a system as the property that the system preserves certain features of the phase portrait for small perturbations in the law of motion (cf. Stability theory; Rough system).

c) As the property that the system remains, in the process of motion, within a bounded region of the phase space (cf. Lagrange stability).

d) As the property that the system returns, in the process of motion, arbitrarily often and arbitrarily closely to its initial position (in the phase space; cf. Poisson stability).

2) Stability in connection with geometric and other objects depending on parameters — continuous dependence of these objects on the parameters (cf. [1], [2]).

However, all these senses of the term "stability" do not exhaust its meaning.

#### References

[1] | A.V. Pogorelov, "Geometric methods in the non-linear theory of elastic shells" , Moscow (1967) (In Russian) |

[2] | Yu.G. Reshet'nyak, "Stability theorems in geometry and analysis" , Novosibirsk (1982) (In Russian) |

#### Comments

Some other meanings of "stability" have to do with stabilization in the sense of algebraic topology and algebraic $K$-theory (cf. e.g. Stability theorems in algebraic $K$-theory; Stable homotopy group), stability of logical theories (cf. e.g. Stability theory (in logic); Stable and unstable theories) or stability in the sense of invariance (stability of a subset of a set under a group of transformations). Quite generally, stability refers to persistence of a certain property when something else changes.

**How to Cite This Entry:**

Stability.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Stability&oldid=31717