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Chetaev function

From Encyclopedia of Mathematics
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A function , defined in a neighbourhood of a fixed point of a system of ordinary differential equations

(*)

and satisfying the two conditions: 1) there exists a domain with the point on its boundary in which , and on the boundary of the domain close to ; and 2) in the derivative along the flow of the system (*) (cf. Differentiation along the flow of a dynamical system) satisfies .

Chetaev's theorem [1] holds: If there is a Chetaev function for the system (*), then the fixed point is Lyapunov unstable.

A Chetaev function is a generalization of a Lyapunov function and gives a convenient way of proving instability (cf. [2]). For example, for the system

where , a Chetaev function is for any . Generalizations of Chetaev functions have been suggested, in particular for non-autonomous systems (cf. [3]).

References

[1] N.G. Chetaev, "A theorem on instability" Dokl. Akad. Nauk SSSR , 1 : 9 (1934) pp. 529–531 (In Russian)
[2] N.G. Chetaev, "Stability of motion" , Moscow (1965) (In Russian)
[3] N.N. Krasovskii, "Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay" , Stanford Univ. Press (1963) (Translated from Russian)
[4] N. Rouche, P. Habets, M. Laloy, "Stability theory by Liapunov's direct method" , Springer (1977)
How to Cite This Entry:
Chetaev function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chetaev_function&oldid=32994
This article was adapted from an original article by A.D. Bryuno (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article