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On the asymptotic behaviour of nonlinear systems of ordinary differential equations

Published online by Cambridge University Press:  18 May 2009

Zhivko S. Athanassov
Zhivko S. Athanassov
Affiliation:
Institute of Mathematics, Poush Academy of Sciences, 00-950 Warsaw, Sniadeckich 8Poland
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In this paper we study the asymptotic behaviour of the following systems of ordinary differential equations:

where the identically zero function is a solution of (N) i.e. f(t, 0)=0 for all time t. Suppose one knows that all the solutions of (N) which start near zero remain near zero for all future time or even that they approach zero as time increases. For the perturbed systems (P) and (P1) the above property concerning the solutions near zero may or may not remain true. A more precise formulation of this problem is as follows: if zero is stable or asymptotically stable for (N), and if the functions g(t, x) and h(t, x) are small in some sense, give conditions on f(t, x) so that zero is (eventually) stable or asymptotically stable for (P) and (P1).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 26 , Issue 2 , July 1985 , pp. 161 - 170
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

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