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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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012480/a0124801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \tag{1 }
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 +
\frac{d x _ {i} }{dt}
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  = X _ {i} ( x _ {1} \dots x _ {n} ) ,
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\  i = 1 \dots n .
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$$
  
 
Let
 
Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012480/a0124802.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{2 }
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x _ {i}  = \phi _ {i} ( t )
 +
$$
  
 
be a periodic solution of the system (1), and let
 
be a periodic solution of the system (1), and let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012480/a0124803.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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$$ \tag{3 }
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\dot \xi  _ {i}  = \sum _ {j = 1 } ^ { n } 
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\frac{\partial  X _ {i} ( \phi _ {1} \dots \phi _ {n} ) }{\partial  x _ {j} }
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\xi _ {j} ,
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\  i = 1 \dots n ,
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$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012480/a0124804.png" /> 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]]).
+
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]]].
 
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]]].
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====References====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 18:47, 5 April 2020


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)

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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Andronov-Witt_theorem&oldid=45186
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