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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 characteristic exponent]]; [[Lyapunov stability|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 [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Andronov,   "Collected works" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,   "Ordinary differential equations" , Addison-Wesley  (1962)  pp. 264  (Translated from Russian)</TD></TR></table>

====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 [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]]. In [[#References|[a2]]] the theorem under discussion occurs as a statement about periodic attractors, cf. pp. 277-278. The original Andronov–Witt paper is [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Hahn,   "Stability of motion" , Springer  (1967)  pp. 422</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.W. Hirsch,   S. Smale,   "Differential equations, dynamic systems and linear algebra" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.A. Coddington,   N. Levinson,   "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. 323</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.A. Andronov,   A. Witt,   "Zur Stabilität nach Liapounov"  ''Physikal. Z. Sowjetunion'' , '''4'''  (1933)  pp. 606–608</TD></TR></table>