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Dissipative system

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D-system, limit-bounded system

A system of ordinary differential equations

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