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)