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 ${\displaystyle V}$ and ${\displaystyle {\dot {V}}}$ both are of the same sign does not have to be produced.

It is named after Nicolai Gurevich Chetaev.

Chetaev instability theorem has been used to analyze the unfolding dynamics of proteins under the effect of optical tweezers.¹