Chetayev 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..

See also

• Chetayev Nikolay Gurievich