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The Hopf bifurcation theorem in three dimensions

Published online by Cambridge University Press:  24 October 2008

Peter Swinnerton-Dyer
St Catharine's College, Cambridge


The Hopf bifurcation theorem describes the creation of a limit cycle from an isolated singular point of a system of first-order differential equations depending on a parameter. This paper describes a method for determining explicitly a range of values of the parameter throughout which the Hopf configuration continues to exist; only the three-dimensional case is described in this paper, but the method can be generalized.

Research Article
Copyright © Cambridge Philosophical Society 1977

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(1)Andronov, A. A., Leontovich, E. A., Gordon, I. I. and Maier, A. G.Qualitative theory of second-order dynamic systems (English translation, John Wiley and Sons, 1973).Google Scholar
(2)Marsden, J. E. and McCracken, M.The Hopf bifurcation and its applications (Springer-Verlag, 1976).CrossRefGoogle Scholar