# Chetaev instability theorem

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

1. the origin is a boundary point of the set ${\displaystyle G=\{\mathbf {x} \mid V(\mathbf {x} )>0\}}$;
2. there exists a neighborhood ${\displaystyle U}$ of the origin such that ${\displaystyle {\dot {V}}({\textbf {x}})>0}$ for all ${\displaystyle \mathbf {x} \in G\cap U}$

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 (circle) around the origin for which V and ${\displaystyle {\dot {V}}}$ both are of the same sign does not have to be produced.

It is named after Nicolai Gurevich Chetaev.