Chetaev function

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)