Andronov-Pontryagin criterion

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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.

[edit] Statement

A vector field \dot{x} = v(x) with x\in \mathbb{R}^2 is orbitally topologically stable if and only if:

  1. All singular points of v are hyperbolic, and
  2. All periodic orbits of v are hyperbolic
  3. There exist no saddle connections.

[edit] 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 x0 is said to be hyperbolic if the eigenvalues of the linearization of v at x0 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.

[edit] References

  • Andronov, A. A. & Pontryagin, L. S. (1937), “Systèmes Grossières”, C.R. (Dokl.) Acad. Sci. URSS (N.S.) 14: 247–251 . Cited in Kuznetsov (2004).
  • Kuznetsov, Yuri A. (2004), Elements of Applied Bifurcation Theory, Springer, ISBN 978-0-387-21906-6 . See Theorem 2.5.