Stability of Motion
The theory of the stability of motion has gained increasing signifi cance in the last decades as is apparent from the large number of publi cations on the subject. A considerable part of this work is concerned with practical problems, especially problem
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H ertlll.rgegebell
VOll
]. L. Doob . E. Heinz· F. Hirzcbruch . E. Hopf . H. Hopf
W. Maak . S. Mac Lane· W. Magnus. D. Mumford Iv1. M. Postnikov . F. K. Schmidt· D. S. Scott· K. Stein
Ge.rchdjiJji(/, rellde H erall.rgeber B. Eckmann und B. L. van der Waerdcn
Wolfgang Hahn
Stability of Motion Translated by
Arne P. Baartz
With 63 Figures
Springer-Verlag New York Inc. 1967
Professor Dr. phi!. \:Volfgang Hahn Technische Huchschtlle Graz Graz (.\115tria)
Professor _,'une P. Baartz, Ph. D. Univcrsity of Vktoria Department of Mathcmatirs, \'jetoria (British CnhtllllJia:'
Geschdftsführende Herausgcut.'r:
Professor Dr. B. Eckmann Eidgenössische Technische Hochschule Zürich
Professor Dr. B. 1.. van der \Vaerdcn Mathematisches Institut der Cnivefsitüt Zürich
ISBN 978-3-642-50087-9 ISBN 978-3-642-50085-5 (eBook) DOI 10.1007/978-3-642-50085-5
All rights rcserved, especially that of translation into forcign lallguages. It is abo forbidden to reproducc this bOGk, cithcr wholc 01' in part, uy phntolllcchanical means (phOt05t to' Fig. 2.1. Trajectories in the (I, x)-plane possibly requiring a different value for O. For the spherical neighborhood of zero at to is mapped by the graphs of the solutions onto a neighborhood of zero at tl which of course contains a certain ball Ix I< 01 entirely in its interior. If we choose IXII < 01 , (2.2) implies that Ip (t, xl> t l ) I < e for t ;::::: t l . It is therefore unnecessary to require in Def. 2.1 that the desired property hold for all to. We must however keep in mind that the number 0 depends on to (cf. sec. 3u, uniform stability) . Def. 2.2. The equilibrium of the differential equation (2.1) is called attractive if there exists a number 'Y} > 0 having the property : ~
(2.4)
lim p (t, x o' to) = 0
t-+~
whenever
IXo I
