# Andronov-Witt theorem

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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)

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. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Andronov-Witt_theorem&oldid=15354
This article was adapted from an original article by E.A. Leontovich-Andronova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article