Chetaev instability theorem

The Chetaev instability theorem for <a href="/facts/Dynamical_system/782gREl5">dynamical systems</a> states that if there exists, for the system 
 
 
 
 
 
 
 
 x
 
 ˙
 
 
 
 =
 X
 (
 
 
 x
 
 
 )
 
 
 {\displaystyle {\dot {\textbf {x}}}=X({\textbf {x}})}
 
 with an <a href="/facts/Equilibrium_point/DgbGMPkg">equilibrium point</a> at the origin, a continuously differentiable function V(x) such that

<ol><li>the origin is a <a href="/facts/Boundary_(topology)/WgdH18kp">boundary point</a> of the set 
 
 
 
 G
 =
 {
 
 x
 
 ∣
 V
 (
 
 x
 
 )
 >
 0
 }
 
 
 {\displaystyle G=\{\mathbf {x} \mid V(\mathbf {x} )>0\}}
 
;</li>
<li>there exists a <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighborhood</a> 
 
 
 
 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}
 
</li></ol>
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 <a href="/facts/Nicolai_Gurevich_Chetaev/l2xCAPST">Nicolai Gurevich Chetaev</a>.

Chetaev instability theorem open-in-new

Chetaev instability theorem