The Andronov–Pontryagin criterion is a necessary and sufficient condition for the stability of

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

s in the plane. It was derived by

Aleksandr Andronov Aleksandr Aleksandrovich Andronov (russian: Алекса́ндр Алекса́ндрович Андро́нов; , Moscow – October 31, 1952, Gorky) was a Soviet physicist and member of the Soviet Academy of Sciences (1946). He worked exten ...

and Lev Pontryagin in 1937.

Statement

A dynamical system :

\dot = v(x),

where

v

is a

C^

- vector field on the plane,

x \in \mathbb^

, is orbitally topologically stable if and only if the following two conditions hold: # All

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 = \m ...

s and

periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given ...

s are ''hyperbolic''. # There are no ''saddle connections''. The same statement holds if the vector field

v

is defined on the

unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose ...

and is transversal to the boundary.

Clarifications

Orbital topological stability of a dynamical system means that for any sufficiently small perturbation (in the ''C''¹-metric), there exists a

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

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''₀ where ''v''(''x''₀)=0, is said to be hyperbolic if none of the

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...

s of the linearization of ''v'' at ''x''₀ is purely imaginary. A periodic orbit of a flow is said to be hyperbolic if none of the

s of the 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 separatrices are connected (cf

homoclinic orbit In mathematics, a homoclinic orbit is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold o ...

and

heteroclinic orbit In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end ...

References

* Cited in . * . See Theorem 2.5. {{DEFAULTSORT:Andronov-Pontryagin criterion Dynamical systems