Chetaev instability theorem

The Chetaev instability theorem for dynamical systems states that if there exists, for the system x ˙ = X ( 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 G = { x ∣ V ( x ) > 0 } ;
2. there exists a neighborhood U of the origin such that V ˙ ( x ) > 0 for all x ∈ G ∩ 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 V ˙ both are of the same sign does not have to be produced.

It is named after Nicolai Gurevich Chetaev.