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The '''Andronov–Pontryagin criterion''' is a necessary and sufficient condition for the stability of [[dynamical system]]s in the plane. It was derived by [[Aleksandr Andronov]] and [[Lev Pontryagin]] in 1937.
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==Statement==
 
A dynamical system
 
: <math>\dot{x} = v(x), </math>
 
where ''v'' is a ''C''<sup>1</sup>-[[vector field]] on the [[Euclidean plane|plane]], ''x''&isin;'''R'''<sup>2</sup> ,
is [[topological conjugacy|orbitally topologically stable]] if and only if the following two conditions hold:
 
# All [[equilibrium point]]s and [[periodic orbit]]s are ''hyperbolic''.
# There are no ''saddle connections''.
 
The same statement holds if the vector field ''v'' is defined on the [[unit disk]] and is transversal to the boundary.
 
==Clarifications==
 
'''Orbital topological stability''' of a dynamical system means that for any sufficiently small perturbation (in the ''C''<sup>1</sup>-metric), there exists a [[homeomorphism]] close to the identity map which transforms the orbits of the original dynamical system to the orbits of the perturbed system (cf [[structural stability]]).
 
The first condition of the theorem is known as '''global hyperbolicity'''. A zero of a vector field ''v'', i.e. a point ''x''<sub>0</sub> where ''v''(''x''<sub>0</sub>)=0, is said to be '''hyperbolic''' if none of the [[eigenvalue]]s of the linearization of ''v'' at ''x''<sub>0</sub> is purely imaginary. A periodic orbit of a flow is said to be hyperbolic if none of the [[eigenvalue]]s of the [[Poincaré map|Poincaré return map]] at a point on the orbit has absolute value one.
 
Finally, '''saddle connection''' refers to a situation where an orbit from one saddle point enters the same or another saddle point, i.e. the unstable and stable [[separatrix|separatrices]] are connected (cf [[homoclinic orbit]] and [[heteroclinic orbit]]).
 
== References ==
* {{cite journal| last=Andronov | first=Aleksandr A. | authorlink=Aleksandr Andronov | coauthors=[[Lev Pontryagin|Lev S. Pontryagin]] | year=1937 | title=Грубые системы | trans_title=Coarse systems | journal=[[Doklady Akademii Nauk SSSR]] | volume=14 | issue=5 | pages=247–250| ref=harv}} Cited in {{harvtxt|Kuznetsov|2004}}.
* {{Cite book | first1 = Yuri A. | last1 = Kuznetsov | year = 2004 | title = Elements of Applied Bifurcation Theory | publisher = Springer | isbn = 978-0-387-21906-6 | ref = harv | postscript = <!--None--> }}. See Theorem 2.5.
 
{{DEFAULTSORT:Andronov-Pontryagin criterion}}
[[Category:Dynamical systems]]

Latest revision as of 17:08, 3 November 2014

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