Dadda multiplier: Difference between revisions
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The '''Chetayev instability theorem''' for [[dynamical system]]s states that if there exists for the system <math>\dot{\textbf{x}} = X(\textbf{x})</math> a function V('''x''') such that | |||
# in any arbitrarily small neighborhood of the origin there is a region D<sub>1</sub> 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 derivative|total time derivative]] <math>\dot{V}(\textbf{x})</math> assumes positive values along every trajectory of <math>\dot{\textbf{x}} = X(\textbf{x})</math> | |||
# the origin is a [[Boundary (topology)|boundary point]] of D<sub>1</sub>; | |||
then the trivial solution is unstable. | |||
This theorem is somewhat less restrictive than the [[Lyapunov instability theorem]]s, since a complete sphere (circle) around the origin for which V and <math>\dot{V}</math> both are of the same sign does not have to be produced.. | |||
== See also == | |||
* [[Chetayev Nikolay Gurievich]] | |||
[[Category:Theorems in dynamical systems]] | |||
Revision as of 21:02, 20 January 2014
The Chetayev instability theorem for dynamical systems states that if there exists for the system 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 assumes positive values along every trajectory of
- 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 both are of the same sign does not have to be produced..