Chetaev instability theorem

The Chetaev instability theorem for dynamical systems states that if there exists, for the system ${\dot {\textbf {x}}}=X({\textbf {x}})$ 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 $G=\{\mathbf {x} \mid V(\mathbf {x} )>0\}$;
2. there exists a neighborhood $U$ of the origin such that ${\dot {V}}({\textbf {x}})>0$ for all $\mathbf {x} \in G\cap U$

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

It is named after Nicolai Gurevich Chetaev.

See also

• Lyapunov function — a function whose existence guarantees stability

References

• Rumyantsev, V. V. (2001) [1994], "Chetaev theorems", Encyclopedia of Mathematics, EMS Press

Further reading

• Chetaev function Emmanuil E. Shnol Scholarpedia 2(9):4672. doi:10.4249/scholarpedia.4672