Andronov-Pontryagin criterion

The Andronov–Pontryagin criterion is a necessary and sufficient condition for the stability of dynamical systems in the plane. It was derived by Aleksandr Andronov and Lev Pontryagin in 1937.

tatement

A vector field $dot\{x\}\; =\; v(x)$ with $xin\; mathbb\{R\}^2$ is orbitally topologically stable if and only if:
# All singular points of $v$ are hyperbolic, and
# All periodic orbits of $v$ are hyperbolic
# There exist no saddle connections.

Clarifications

Orbital topological stability means that for any other dynamical system sufficiently close to the original one, there exists a homeomorphism which maps the orbits of one dynamical system to orbits of the other.

The first two criteria of the theorem are known as "global hyperbolicity". A singular point $x\_0$ is said to be hyperbolic if the eigenvalues of the linearization of $v$ at $x\_0$ have non-zero real parts.A periodic orbit is said to be hyperbolic if none of the eigenvalues of the Poincaré map of $v$ at a point on the orbit have modulus 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.

References

*. Cited in harvtxt|Kuznetsov|2004.
*. See Theorem 2.5.