Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-22T00:18:47.106Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCE

1.Athanassov, Zh. S., Perturbation theorems for nonlinear systems of ordinary differential equations. J. Math. Anal. Appl. 86 (1982), 194–207.Google Scholar
2.Brauer, F., Perturbations of nonlinear systems of differential equations, I. J. Math., Anal. Appl. 14 (1967), 198–206.CrossRefGoogle Scholar
3.Brauer, F., Perturbations of nonlinear systems of differential equations, II. J. Math. Anal. Appl. 17 (1967), 418–434.CrossRefGoogle Scholar
4.Brauer, F. and Wong, J. S. W., On the asymptotic relationships between solutions of two systems of ordinary differential equations, J. Differential Equations 6 (1969), 527–543.CrossRefGoogle Scholar
5.Coddington, E. A. and Levinson, N., Theory of ordinary differential equations (McGraw-Hill, 1965).Google Scholar
6.Duboshin, G. N., On the problem of stability of a motion under constantly acting perturbations. Trudy Gos. Astron. Inst. P. K. Sternberg 14 (1940).Google Scholar
7.Gorshin, S. I., On the stability of motion under constantly acting perurbations. Izv. Akad. Nauk Kazakh. SSR. Ser. Mat. Mekh. (2), 56 (1948), 46–73.Google Scholar
8.Malkin, I. G., Stability in the case of constantly acting disturbances. Prikl. Mat. Mekh. 8 (1944), 241–245.Google Scholar
9.Malkin, I. G., Theory of stability of motion second edition (Nauka, 1966).Google Scholar
10.Markus, L., Asymptotically autonomous differential systems. Contributions to the theory of nonlinear oscillations, Annals of Mathematics Studies No. 36 (Princeton University Press, 1956), 17–29.Google Scholar
11.Massera, J. L., Contributions to stability theory. Ann. of Math. 64 (1956), 182–206; Erratum, 68 (1956), 202.CrossRefGoogle Scholar
12.Salvadori, L. and Schiaffino, A., On the problem of total stability. Nonlinear Anal. 1 (1977), 207–213.Google Scholar
13.Seibert, P., Estabilidad bajo perturbaciones sostenidas y su generalizacion en flujos continuos. Acta Mexicana Cene. Teen. 11 (1968), 154–165.Google Scholar
14.Strauss, A. and Yorke, J. A., Perturbation theorems for ordinary differential equations. J Differential Equations 3 (1967), 15–30.CrossRefGoogle Scholar
15.Yoshizawa, T., Stability theory by Liapunov's second method (Math. Soc. Japan, 1966).Google Scholar