Chetaev instability theorem

The Chetaev 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

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;
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 \(\dot{\textbf{x}} = X(\textbf{x}) \)
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.

It is named after Nicolai Gurevich Chetaev.
References

Rumyantsev, V. V. (2001), "Chetaev theorems", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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