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''of ordinary differential equations''
 
''of ordinary differential equations''
  
 
A system of the form
 
A system of the form
  
<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/r/r080/r080750/r0807501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\dot{x}  = A( t) x,\  x \in \mathbf R  ^ {n}
 +
$$
  
(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r0807502.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r0807503.png" /> that is summable on every interval and has the property that
+
(where $  A( \cdot ) $
 +
is a mapping $  \mathbf R  ^ {+} \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {m} ) $
 +
that is summable on every interval and has the property that
  
<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/r/r080/r080750/r0807504.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow \infty } 
 +
\frac{1}{t}
 +
\int\limits _ { 0 } ^ { t }
 +
\mathop{\rm Tr}  A( \tau )  d \tau
 +
$$
  
exists and is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r0807505.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r0807506.png" /> are the characteristic Lyapunov exponents (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]) of the system (1)).
+
exists and is equal to $  \sum _ {i= 1 }  ^ { n }  \lambda _ {i} ( A) $,  
 +
where $  \lambda _ {1} ( A) \geq  \dots \geq  \lambda _ {n} ( A) $
 +
are the characteristic Lyapunov exponents (cf. [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]) of the system (1)).
  
 
For a triangular system
 
For a triangular 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/r/r080/r080750/r0807507.png" /></td> </tr></table>
+
$$
 +
\dot{u}  ^ {i}  = \sum _ {j= i } ^ { n }  p _ {ij} ( t) u
 +
^ {j} ,\  i= 1 \dots n,
 +
$$
  
 
to be regular it is necessary and sufficient that the limits
 
to be regular it is necessary and sufficient that the limits
  
<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/r/r080/r080750/r0807508.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow \infty } 
 +
\frac{1}{t}
 +
\int\limits _ { 0 } ^ { t }  p _ {ii} ( \tau )  d \tau ,\  i= 1 \dots n,
 +
$$
  
 
exist (Lyapunov's criterion). Every [[Reducible linear system|reducible linear system]] and every [[Almost-reducible linear system|almost-reducible linear system]] is regular.
 
exist (Lyapunov's criterion). Every [[Reducible linear system|reducible linear system]] and every [[Almost-reducible linear system|almost-reducible linear system]] is regular.
  
The role of the concept of a regular linear system is clarified by the following theorem of Lyapunov. Let the system (1) be regular and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r0807509.png" /> of its characteristic Lyapunov exponents be negative:
+
The role of the concept of a regular linear system is clarified by the following theorem of Lyapunov. Let the system (1) be regular and let $  k $
 +
of its characteristic Lyapunov exponents be negative:
  
<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/r/r080/r080750/r08075010.png" /></td> </tr></table>
+
$$
 +
> \lambda _ {n-} k+ 1 ( A)  \geq  \dots \geq  \lambda _ {n} ( A).
 +
$$
  
 
Then for every system
 
Then for every 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/r/r080/r080750/r08075011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\dot{x}  = A( t) x + g( t, x),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075012.png" /> satisfies the following conditions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075014.png" /> are continuous, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075017.png" />, there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075018.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075019.png" /> containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075020.png" />, such that every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075021.png" /> of (2) starting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075022.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075023.png" />) exponentially decreases as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075024.png" />; more precisely, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075025.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080750/r08075026.png" /> such that the inequality
+
where $  g( t, x) $
 +
satisfies the following conditions: $  g $
 +
and $  g _ {x}  ^  \prime  $
 +
are continuous, and $  g( t, 0)= 0 $,  
 +
$  \sup _ {t \geq  0 }  \| g _ {x}  ^  \prime  ( t, x) \| = O( | x |  ^  \epsilon  ) $,  
 +
where $  \epsilon = \textrm{ const } > 0 $,  
 +
there is a $  k $-
 +
dimensional manifold $  V  ^ {k} \subset  \mathbf R  ^ {n} $
 +
containing the point $  x= 0 $,  
 +
such that every solution $  x( t) $
 +
of (2) starting on $  V  ^ {k} $(
 +
i.e. $  x( 0) \in V  ^ {k} $)
 +
exponentially decreases as $  t \rightarrow \infty $;  
 +
more precisely, for every $  \delta > 0 $
 +
there is a $  C _  \delta  $
 +
such that the inequality
  
<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/r/r080/r080750/r08075027.png" /></td> </tr></table>
+
$$
 +
| x( t) |  \leq  C _  \delta  e ^ {[ \lambda _ {n-} k+ 1 ( A)+
 +
\delta ] t } | x( 0) |
 +
$$
  
 
is satisfied.
 
is satisfied.

Latest revision as of 08:10, 6 June 2020


of ordinary differential equations

A system of the form

$$ \tag{1 } \dot{x} = A( t) x,\ x \in \mathbf R ^ {n} $$

(where $ A( \cdot ) $ is a mapping $ \mathbf R ^ {+} \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {m} ) $ that is summable on every interval and has the property that

$$ \lim\limits _ {t \rightarrow \infty } \frac{1}{t} \int\limits _ { 0 } ^ { t } \mathop{\rm Tr} A( \tau ) d \tau $$

exists and is equal to $ \sum _ {i= 1 } ^ { n } \lambda _ {i} ( A) $, where $ \lambda _ {1} ( A) \geq \dots \geq \lambda _ {n} ( A) $ are the characteristic Lyapunov exponents (cf. Lyapunov characteristic exponent) of the system (1)).

For a triangular system

$$ \dot{u} ^ {i} = \sum _ {j= i } ^ { n } p _ {ij} ( t) u ^ {j} ,\ i= 1 \dots n, $$

to be regular it is necessary and sufficient that the limits

$$ \lim\limits _ {t \rightarrow \infty } \frac{1}{t} \int\limits _ { 0 } ^ { t } p _ {ii} ( \tau ) d \tau ,\ i= 1 \dots n, $$

exist (Lyapunov's criterion). Every reducible linear system and every almost-reducible linear system is regular.

The role of the concept of a regular linear system is clarified by the following theorem of Lyapunov. Let the system (1) be regular and let $ k $ of its characteristic Lyapunov exponents be negative:

$$ 0 > \lambda _ {n-} k+ 1 ( A) \geq \dots \geq \lambda _ {n} ( A). $$

Then for every system

$$ \tag{2 } \dot{x} = A( t) x + g( t, x), $$

where $ g( t, x) $ satisfies the following conditions: $ g $ and $ g _ {x} ^ \prime $ are continuous, and $ g( t, 0)= 0 $, $ \sup _ {t \geq 0 } \| g _ {x} ^ \prime ( t, x) \| = O( | x | ^ \epsilon ) $, where $ \epsilon = \textrm{ const } > 0 $, there is a $ k $- dimensional manifold $ V ^ {k} \subset \mathbf R ^ {n} $ containing the point $ x= 0 $, such that every solution $ x( t) $ of (2) starting on $ V ^ {k} $( i.e. $ x( 0) \in V ^ {k} $) exponentially decreases as $ t \rightarrow \infty $; more precisely, for every $ \delta > 0 $ there is a $ C _ \delta $ such that the inequality

$$ | x( t) | \leq C _ \delta e ^ {[ \lambda _ {n-} k+ 1 ( A)+ \delta ] t } | x( 0) | $$

is satisfied.

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] N.A. Izobov, "Linear systems of ordinary differential equations" J. Soviet Math. , 5 : 1 (1976) pp. 46–96 Itogi Nauk. i Tekhn. Mat. Anal. , 12 : 1 (1974) pp. 71–146
How to Cite This Entry:
Regular linear system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_linear_system&oldid=48482
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article