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''D-system, limit-bounded system''
 
''D-system, limit-bounded system''
  
 
A system of ordinary differential equations
 
A system of ordinary differential equations
  
<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/d/d033/d033440/d0334401.png" /></td> </tr></table>
+
$$
 +
\dot{x}  = f ( t , x ) ,\  x \in \mathbf R  ^ {n} ,
 +
$$
  
with continuous right-hand side, whose solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033440/d0334402.png" /> satisfy the properties of uniqueness and infinite extendability to the right, and for which there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033440/d0334403.png" /> such that for any solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033440/d0334404.png" /> it is possible to find a moment in time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033440/d0334405.png" /> such that
+
with continuous right-hand side, whose solutions $  x ( t ;  t _ {0} , x _ {0} ) $
 +
satisfy the properties of uniqueness and infinite extendability to the right, and for which there exists a number $  \rho > 0 $
 +
such that for any solution $  x ( t ;  t _ {0} , x _ {0} ) $
 +
it is possible to find a moment in time $  T ( t _ {0} , x _ {0} ) \geq  t _ {0} $
 +
such 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/d/d033/d033440/d0334406.png" /></td> </tr></table>
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$$
 +
\| x ( t ; t _ {0} , x _ {0} ) \|
 +
< \rho \  \textrm{ for  all  }  t \geq  T ( t _ {0} , x _ {0} ) .
 +
$$
  
In other words, each solution is immersed, sooner or later, in a fixed sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033440/d0334407.png" />. An important particular case of a dissipative system are the so-called systems with convergence, for which all solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033440/d0334408.png" /> are defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033440/d0334409.png" /> and, in addition, there exists a unique bounded solution on the entire axis which is asymptotically stable in the large. Such systems have been thoroughly studied (see, for example, [[#References|[1]]]).
+
In other words, each solution is immersed, sooner or later, in a fixed sphere $  \| x \| < \rho $.  
 +
An important particular case of a dissipative system are the so-called systems with convergence, for which all solutions $  x ( t ;  t _ {0} , x _ {0} ) $
 +
are defined for $  t _ {0} \leq  t < \infty $
 +
and, in addition, there exists a unique bounded solution on the entire axis which is asymptotically stable in the large. Such systems have been thoroughly studied (see, for example, [[#References|[1]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Pliss,  "Nonlocal problems of the theory of oscillations" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.P. Demidovich,  "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Pliss,  "Nonlocal problems of the theory of oscillations" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.P. Demidovich,  "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.K. Hale,  "Ordinary differential equations" , Wiley  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.K. Hale,  "Ordinary differential equations" , Wiley  (1980)</TD></TR></table>

Latest revision as of 19:36, 5 June 2020


D-system, limit-bounded system

A system of ordinary differential equations

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

with continuous right-hand side, whose solutions $ x ( t ; t _ {0} , x _ {0} ) $ satisfy the properties of uniqueness and infinite extendability to the right, and for which there exists a number $ \rho > 0 $ such that for any solution $ x ( t ; t _ {0} , x _ {0} ) $ it is possible to find a moment in time $ T ( t _ {0} , x _ {0} ) \geq t _ {0} $ such that

$$ \| x ( t ; t _ {0} , x _ {0} ) \| < \rho \ \textrm{ for all } t \geq T ( t _ {0} , x _ {0} ) . $$

In other words, each solution is immersed, sooner or later, in a fixed sphere $ \| x \| < \rho $. An important particular case of a dissipative system are the so-called systems with convergence, for which all solutions $ x ( t ; t _ {0} , x _ {0} ) $ are defined for $ t _ {0} \leq t < \infty $ and, in addition, there exists a unique bounded solution on the entire axis which is asymptotically stable in the large. Such systems have been thoroughly studied (see, for example, [1]).

References

[1] V.A. Pliss, "Nonlocal problems of the theory of oscillations" , Acad. Press (1966) (Translated from Russian)
[2] B.P. Demidovich, "Lectures on the mathematical theory of stability" , Moscow (1967) (In Russian)

Comments

References

[a1] J.K. Hale, "Ordinary differential equations" , Wiley (1980)
How to Cite This Entry:
Dissipative system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dissipative_system&oldid=46751
This article was adapted from an original article by K.S. Sibirskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article