# Andronov-Witt theorem

A modification of Lyapunov's theorem (on the stability of a periodic solution of a non-autonomous system of differential equations) for the autonomous system

(1) |

Let

(2) |

be a periodic solution of the system (1), and let

(3) |

be the corresponding system of variational equations which always has, in the case here considered, one zero characteristic exponent. The Andronov–Witt theorem is then valid: If characteristic exponents of the system (3) have negative real parts, a periodic solution (2) of the system (1) is stable according to Lyapunov (cf. Lyapunov characteristic exponent; Lyapunov stability).

The Andronov–Witt theorem was first formulated by A.A. Andronov and A.A. Witt in 1930 and was proved by them in 1933 [1].

#### References

[1] | A.A. Andronov, "Collected works" , Moscow (1976) (In Russian) |

[2] | L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) pp. 264 (Translated from Russian) |

#### Comments

The Andronov–Witt theorem is usually found in the Western literature under some heading like "hyperbolic periodic attractorhyperbolic periodic attractor" .

Good additional general references are [a1], [a2], [a3]. In [a2] the theorem under discussion occurs as a statement about periodic attractors, cf. pp. 277-278. The original Andronov–Witt paper is [a4].

#### References

[a1] | W. Hahn, "Stability of motion" , Springer (1967) pp. 422 |

[a2] | M.W. Hirsch, S. Smale, "Differential equations, dynamic systems and linear algebra" , Acad. Press (1974) |

[a3] | E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. 323 |

[a4] | A.A. Andronov, A. Witt, "Zur Stabilität nach Liapounov" Physikal. Z. Sowjetunion , 4 (1933) pp. 606–608 |

**How to Cite This Entry:**

Andronov–Witt theorem.

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