What is Chetaev instability theorem?

The Chetaev instability theorem for dynamical systems states that if there exists, for the system x ˙ = X {\displaystyle {\dot {\textbf {x}}}=X} with an equilibrium point at the origin, a continuously differentiable function V such that

then the origin is an unstable equilibrium point of the system.

This theorem is somewhat less restrictive than the Lyapunov instability theorems, since a complete sphere around the origin for which V {\displaystyle V} and V ˙ {\displaystyle {\dot {V}}} both are of the same sign does not have to be produced.

It is named after Nicolai Gurevich Chetaev.