# Dissipative system

*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]).

#### 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=13165