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The Chetaev instability theorem for dynamical systems states that if there exists, for the system x ˙ ˙ = X ( x ) {\displaystyle {\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 = { x ∣ ∣ V ( x ) > 0 } {\displaystyle G=\{\mathbf {x} \mid V(\mathbf {x} )>0\}} ;

2. there exists a neighborhood U {\displaystyle U} of the origin such that V ˙ ˙ ( x ) > 0 {\displaystyle {\dot {V}}({\textbf {x}})>0} for all x ∈ ∈ G ∩ ∩ U {\displaystyle \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 {\displaystyle V} and V ˙ ˙ {\displaystyle {\dot {V}}} both are of the same sign does not have to be produced.

It is named after Nicolai Gurevich Chetaev.

• Applications

• See also

• References

• Further reading

Chetaev instability theorem has been used to analyze the unfolding dynamics of proteins under the effect of optical tweezers.cite-ref-1[1]

• Lyapunov function — a function whose existence guarantees stability

cite-note-11. ↑ citerefmohammadispong2022Mohammadi, A.; Spong, Mark W. (2022). "Chetaev Instability Framework for Kinetostatic Compliance-Based Protein Unfolding". IEEE Control Systems Letters. 6: 2755–2760. arXiv:2205.07375. doi:10.1109/LCSYS.2022.3176433. ISSN 2475-1456.

• citerefrumyantsev2001Rumyantsev, V. V. (2001) [1994]. "Chetaev theorems". Encyclopedia of Mathematics. EMS Press.

• citerefshnol2007Shnol, Emmanuil (2007). "Chetaev function". Scholarpedia. 2 (9): 4672. Bibcode:2007SchpJ...2.4672S. doi:10.4249/scholarpedia.4672.