Dissipative system

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)

References

[a1]	J.K. Hale, "Ordinary differential equations" , Wiley (1980)