Hostname: page-component-84b7d79bbc-l82ql Total loading time: 0 Render date: 2024-07-28T09:24:10.170Z Has data issue: false hasContentIssue false
  • English
  • Français

On asymptotically stable sets

Published online by Cambridge University Press:  20 January 2009

D. Desbrow
D. Desbrow
Affiliation:
University of Edinburgh, Edinburgh EH1 1HZ
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 closed sets having a neighbourhood with compact closure which are positively asymptotically stable under a flow on a metric space X. For an understanding of this and the rest of the introduction it is sufficient for the reader to have in mind as an example of a flow a system of first order, autonomous ordinary differential equations describing mathematically a time-independent physical system; in short a dynamical system. In a flow a set M is positively stable if the trajectories through all points sufficiently close to M remain in the future in a given neighbourhood of M. The set M is positively asymptotically stable if it is positively stable and, in addition, trajectories through all points of some neighbourhood of M approach M in the future.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 17 , Issue 2 , December 1970 , pp. 181 - 186
Copyright
Copyright © Edinburgh Mathematical Society 1970

References

REFERENCES

(1) Bhatia, N. P., Lazer, A. C. and Szegö, G. P., On global weak attractors in dynamical systems, J. Math. Anal. Appl. 16 (1966), 544–552.CrossRefGoogle Scholar
(2) Bhatia, N. P. and Szegö, G. P., Dynamical systems: stability theory and applications, Lecture notes in mathematics no. 35 (Springer-Verlag, Berlin-Heidelberg- New York, 1967).CrossRefGoogle Scholar
(3) Desbrow, D., On connexion, invariance and stability in certain flows, Proc. Cambridge Philos. Soc. 60 (1964), 51–55.CrossRefGoogle Scholar
(4) Desbrow, D., On unstable invariant sets, Ftmkcial. Ekvac. 13 (1970), 109–126.Google Scholar
(5) Lefschetz, S., Liapunov and stability in dynamical systems, Bol. Soc. Mat. Mexicana (2) 3 (1958), 25–39.Google Scholar
(6) Saito, T., On a compact invariant set isolated from minimal sets, Funkcial. Ekvac. 12 (1969), 193–203.Google Scholar
(7) Ura, T., On the flow outside a closed, invariant set; stability, relative stability and saddle sets, Contributions to Differential Equations, 3 (1964), 249–294.Google Scholar