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Zubov's condition revisited

Published online by Cambridge University Press:  20 January 2009

Ronald A. Knight
Affiliation:
Northeast Missouri State University
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Zubov states an elegant necessary and sufficient limit set condition for positive orbital stability of compact invariant sets in his book “Metody A. M. Lyapunova i ih Primenenie” [11]. Stated in terms of our terminology of L for the negative limit set, Zubov's proposition is as follows: A necessary and sufficient condition for positive stability of a compact invariant set M isL(X\M)∩M=Ø. Unfortunately, Zubov's condition L(X\M)∩M=Ø has subsequently been shown to be necessary but not sufficient (see [9]). Bass and Ura devote considerable effort in [2] and [9[ to correcting Zubov's proposition and Desbrow obtains additional results principally concerning unstable sets in [6] and [7]. Ura gives his classical corrected prolongational version of Zubov's assertion on locally compact phase spaces in [9] and extends it to any closed invariant set with compact boundary on such spaces in [10].

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

REFERENCES

1.Ahmad, S., Strong attraction and classification of certain continuous flows, Math. Systems Theory 5 (1971), 157163.CrossRefGoogle Scholar
2.Bass, R., Zubov's stability criterion, Bol. Soc. Mat. Mexicana 4 (1959), 2639.Google Scholar
3.Bhatia, N. and Hajek, O., Theory of dynamical systems, Part II (Technical Note BN-606, Univ. of Maryland, 1969).Google Scholar
4.Bhatia, N. and Hajek, O., Local semi-dynamical systems (Lecture Notes in Math., vol. 90, Springer-Verlag, Berlin and New York, 1969).CrossRefGoogle Scholar
5.Bhatia, N. and Szego, G., Stability Theory of Dynamical Systems (Springer-Verlag, Berlin and New York, 1970).CrossRefGoogle Scholar
6.Desbrow, D., On unstable invariant sets, Funcialaj Ekvacioj 13 (1970), 109126.Google Scholar
7.Desbrow, D., On asymptotically stable sets, Proc. Edinburgh Math. Soc. 17 (1970), 181186.CrossRefGoogle Scholar
8.Hajek, O., Compactness and asymptotic stability, Math. Systems Theory 4 (1970), 154156.CrossRefGoogle Scholar
9.Ura, T., On the flow outside a closed invariant set; stability, relative stability, and saddle sets, Contra, to Diff. Eq. 3 (1964), 249294.Google Scholar
10.Ura, T., Sur le courant exterieur a une region invariante; Prolongements d'unecharacteristique et l'order de stabilite, Funcialaj Ekvacioj 2 (1959), 143200; nouv. edition, 105-143.Google Scholar
11.Zubov, V., Metody A. M. Lyapunova i ih Primenenie, (Izdatel'stvo LeningradskogoUniversiteta, Moscow, 1957). English translation: Methods of Lyapunov, A. M. and their Application, AEC-tr-4439, 1961.Google Scholar