Dissipative system

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D-system, limit-bounded system

A system of ordinary differential equations

with continuous right-hand side, whose solutions satisfy the properties of uniqueness and infinite extendability to the right, and for which there exists a number such that for any solution it is possible to find a moment in time such that

In other words, each solution is immersed, sooner or later, in a fixed sphere . An important particular case of a dissipative system are the so-called systems with convergence, for which all solutions are defined for 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]).


[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)



[a1] J.K. Hale, "Ordinary differential equations" , Wiley (1980)
How to Cite This Entry:
Dissipative system. K.S. Sibirskii (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098