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''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, [[#References|[1]]]).

====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>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.K. Hale,   "Ordinary differential equations" , Wiley  (1980)</TD></TR></table>