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The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p0724201.png" /> in the system of ordinary differential equations
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<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/p/p072/p072420/p0724202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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The mapping  $  f $
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in the system of ordinary differential equations
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$$ \tag{1 }
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\dot{x}  = A ( t) x + f ( x, t).
 +
$$
  
 
A perturbation is usually assumed to be small in some sense, for example
 
A perturbation is usually assumed to be small in some sense, for example
  
<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/p/p072/p072420/p0724203.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{2 }
  
The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p0724204.png" /> of the perturbed system (1) and the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p0724205.png" /> of the linear system
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\frac{| f ( x, t) | }{| x | }
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  \rightarrow  0 \ \
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\textrm{ if }  | x | \rightarrow 0.
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$$
  
<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/p/p072/p072420/p0724206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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The solution  $  \phi ( t) $
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of the perturbed system (1) and the solution  $  \Psi ( t) $
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of the linear system
  
with the same initial value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p0724207.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p0724208.png" />, are connected by the relation
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$$ \tag{3 }
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\dot{y}  = A ( t) y
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$$
  
<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/p/p072/p072420/p0724209.png" /></td> </tr></table>
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with the same initial value  $  y _ {0} $
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at  $  t = t _ {0} $,
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are connected by the relation
  
known as the formula of variation of constants, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p07242010.png" /> is the [[Fundamental matrix|fundamental matrix]] of the linear system (3).
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$$
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\phi ( t)  = \Psi ( t) \left ( y _ {0} +
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\int\limits _ {t _ {0} } ^ { t }
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\Psi  ^ {-1} ( \tau ) f ( \phi ( \tau ), \tau )  d \tau \right ) ,
 +
$$
  
It was shown by A.M. Lyapunov [[#References|[1]]] that the trivial solution of the system (1) is asymptotically stable (cf. [[Asymptotically-stable solution|Asymptotically-stable solution]]) if relation (2) is valid uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p07242011.png" />, the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p07242012.png" /> is constant and all real parts of the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p07242013.png" /> are negative; if only one such real part is positive, the trivial solution is not stable.
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known as the formula of variation of constants, where  $  \Psi ( t) $
 +
is the [[Fundamental matrix|fundamental matrix]] of the linear system (3).
  
The study of the periodic solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p07242014.png" /> of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p07242015.png" />, describing an oscillating process, reduces in the general case by the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p07242016.png" /> to the study of a perturbed linear system, the right-hand side of which is periodic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072420/p07242017.png" /> [[#References|[3]]].
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It was shown by A.M. Lyapunov [[#References|[1]]] that the trivial solution of the system (1) is asymptotically stable (cf. [[Asymptotically-stable solution|Asymptotically-stable solution]]) if relation (2) is valid uniformly in  $  t $,
 +
the matrix  $  A( t) $
 +
is constant and all real parts of the eigen values of  $  A( t) $
 +
are negative; if only one such real part is positive, the trivial solution is not stable.
 +
 
 +
The study of the periodic solution $  \phi $
 +
of the system $  \dot{x} = P( x, t) $,  
 +
describing an oscillating process, reduces in the general case by the transformation $  x = \phi ( t) + y $
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to the study of a perturbed linear system, the right-hand side of which is periodic in $  t $[[#References|[3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,  "Stability of motion" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.F. Bylov,  R.E. Vinograd,  D.M. Grobman,  V.V. Nemytskii,  "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.S. Pontryagin,  "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,  "Stability of motion" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.F. Bylov,  R.E. Vinograd,  D.M. Grobman,  V.V. Nemytskii,  "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.S. Pontryagin,  "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 07:39, 26 February 2022


The mapping $ f $ in the system of ordinary differential equations

$$ \tag{1 } \dot{x} = A ( t) x + f ( x, t). $$

A perturbation is usually assumed to be small in some sense, for example

$$ \tag{2 } \frac{| f ( x, t) | }{| x | } \rightarrow 0 \ \ \textrm{ if } | x | \rightarrow 0. $$

The solution $ \phi ( t) $ of the perturbed system (1) and the solution $ \Psi ( t) $ of the linear system

$$ \tag{3 } \dot{y} = A ( t) y $$

with the same initial value $ y _ {0} $ at $ t = t _ {0} $, are connected by the relation

$$ \phi ( t) = \Psi ( t) \left ( y _ {0} + \int\limits _ {t _ {0} } ^ { t } \Psi ^ {-1} ( \tau ) f ( \phi ( \tau ), \tau ) d \tau \right ) , $$

known as the formula of variation of constants, where $ \Psi ( t) $ is the fundamental matrix of the linear system (3).

It was shown by A.M. Lyapunov [1] that the trivial solution of the system (1) is asymptotically stable (cf. Asymptotically-stable solution) if relation (2) is valid uniformly in $ t $, the matrix $ A( t) $ is constant and all real parts of the eigen values of $ A( t) $ are negative; if only one such real part is positive, the trivial solution is not stable.

The study of the periodic solution $ \phi $ of the system $ \dot{x} = P( x, t) $, describing an oscillating process, reduces in the general case by the transformation $ x = \phi ( t) + y $ to the study of a perturbed linear system, the right-hand side of which is periodic in $ t $[3].

References

[1] A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)
[2] B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytskii, "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow (1966) (In Russian)
[3] L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)

Comments

Results of this type are usually called Poincaré–Lyapunov theorems. There are several extensions, described, e.g., in [a1]. A recent tutorial text containing these matters is [a2].

References

[a1] M. Roseau, "Vibrations non linéaires et théorie de la stabilité" , Springer (1966)
[a2] F. Verhulst, "Nonlinear differential equations and dynamical systems" , Springer (1989)
How to Cite This Entry:
Perturbation of a linear system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perturbation_of_a_linear_system&oldid=15561
This article was adapted from an original article by L.E. Reizin' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article