# 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

$$\tag{1 } \frac{d x _ {i} }{dt} = X _ {i} ( x _ {1} \dots x _ {n} ) , \ i = 1 \dots n .$$

Let

$$\tag{2 } x _ {i} = \phi _ {i} ( t )$$

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

$$\tag{3 } \dot \xi _ {i} = \sum _ {j = 1 } ^ { n } \frac{\partial X _ {i} ( \phi _ {1} \dots \phi _ {n} ) }{\partial x _ {j} } \xi _ {j} , \ i = 1 \dots n ,$$

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 $n - 1$ 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://encyclopediaofmath.org/index.php?title=Andronov%E2%80%93Witt_theorem&oldid=22024