The Chetaev instability theorem for

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...

s states that if there exists, for the system

\dot = X(\textbf)

with an

equilibrium point In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation. Formal definition The point \tilde\in \mathbb^n is an equilibrium point for the differential equation :\frac = \ma ...

at the origin, a continuously differentiable function V(x) such that # the origin is a

boundary point In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term boun ...

of the set

G = \

; # there exists a

neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

U

of the origin such that

\dot(\textbf)>0

for all

\mathbf \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 theorem Lyapunov (, in old-Russian often written Лепунов) is a Russian surname that is sometimes also romanized as Ljapunov, Liapunov or Ljapunow. Notable people with the surname include: * Alexey Lyapunov (1911–1973), Russian mathematician * Alek ...

s, since a complete sphere (circle) around the origin for which

V

and

\dot

both are of the same sign does not have to be produced. It is named after

Nicolai Gurevich Chetaev Nikolay Gur'yevich Chetaev (23 November 1902 – 17 October 1959) was a Russian Soviet mechanician and mathematician. He was born in Karaduli, Laishevskiy uyezd, Kazan province, Russian Empire (now Tatarstan of Russian Federation) and died in Mos ...

Applications

Chetaev instability theorem has been used to analyze the unfolding dynamics of proteins under the effect of optical tweezers.

Applications

See also

References

Further reading