# Stability theory

A collection of views, presentations, ideas, concepts, arguments, methods, theories (containing definitions, lemmas, theorems, and proofs) arising from and having as its aim the study of the stability of motion (understood in the same wide form). Thus, stability theory is a theory in the widest sense of this word. Among the different concepts of the stability of motion the best known are the following:

1) The concept of stability introduced by A.M. Lyapunov, and its modifications: Lyapunov stability (in particular, asymptotic stability and exponential stability); conditional stability (in particular, asymptotic conditional stability and exponential conditional stability); stability for a part of the variables; uniform stability; stability in the presence of persistently acting perturbations; orbit stability; the presence of attractors (cf. Limit cycle; Lorenz attractor); stochastic stability; absolute stability (cf. Stability, absolute). Cf. also Stability criterion; Stability region.

3) Poisson stability and the concepts related to it (wandering point; complete instability).

4) Structural stability (cf. Rough system) — a concept introduced by A.A. Andronov and L.S. Pontryagin.

5) Preservation of most of the invariant tori of an integrable Hamiltonian system for small perturbations of the Hamiltonian, discovered by A.N. Kolmogorov (cf. Small denominators).

In Lyapunov stability theory (cf. , Vol. 2 and also ) one selects questions connected with Lyapunov's first method. Here it is usual to refer to the theory of linear systems of differential equations (cf. Variational equations; Linear system of differential equations with periodic coefficients; Linear system of differential equations with almost-periodic coefficients; Regular linear system of differential equations; Irregularity indices; Almost-reducible linear system of differential equations; Reducible linear system of differential equations; Multipliers; Hamiltonian system, linear) and having a large intersection with the theory of linear systems, the theory of Lyapunov characteristic exponents (cf. Lyapunov characteristic exponent; cf. also Singular exponents; Central exponents; Integral separation condition; Stability of characteristic exponents). For Lyapunov's second method see Lyapunov function, and also .

In the theory of structural stability one singles out the theory of Anosov systems (cf. \$Y\$-system, ) as well as criteria of structural stability (cf. , ).

For a study of Lyapunov stability in mechanics one touches upon the following questions: the stability of equilibrium shapes of rotating fluids (cf. , Vols. 3–4), of other gravitational systems (cf. ), the stability of the motion of fluids (cf. , ), the stability of motion of deformable rigid bodies (cf. Stability of an elastic system, as well as ), the stability of motion of bodies with cavities containing a fluid , the stability of automatic control systems , and the stability of the solutions of equations with delay .

How to Cite This Entry:
Stability theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stability_theory&oldid=31718
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article