Chetaev function

A function $v(x)$, defined in a neighbourhood of a fixed point $x=0$ of a system of ordinary differential equations

$$\dot x=F(x),\quad x\in\mathbf R^n,\quad F(0)=0,\label{*}\tag{*}$$

and satisfying the two conditions: 1) there exists a domain $G$ with the point $x=0$ on its boundary in which $v>0$, and $v=0$ on the boundary of the domain close to $x=0$; and 2) in $G$ the derivative along the flow of the system \eqref{*} (cf. Differentiation along the flow of a dynamical system) satisfies $\dot v>0$.

Chetaev's theorem [1] holds: If there is a Chetaev function $v$ for the system \eqref{*}, then the fixed point $x=0$ 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

$$\dot x=ax+o(|x|+|y|),$$

$$\dot y=-by+o(|x|+|y|),$$

where $a,b>0$, a Chetaev function is $v=x^2-c^2y^2$ for any $c\neq0$. 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)