Chetaev instability theorem

The Chetayev instability theorem for dynamical systems states that if there exists for the system ${\dot {\textbf {x}}}=X({\textbf {x}})$ a function V(x) such that

1. in any arbitrarily small neighborhood of the origin there is a region D₁ in which V(x) > 0 and on whose boundaries V(x) = 0;
2. at all points of the region in which V(x) > 0 the total time derivative ${\dot {V}}({\textbf {x}})$ assumes positive values along every trajectory of ${\dot {\textbf {x}}}=X({\textbf {x}})$
3. the origin is a boundary point of D₁;

then the trivial solution is unstable.

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