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Topological Hopf bifurcation in the plane

Published online by Cambridge University Press:  24 October 2008

Dieter Erle
Dieter Erle
Affiliation:
Universitüt Dortmund, Fed. Rep., Germany
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Extract

Classical bifurcation theorems for a 1 -parameter family of plane dynamical systems

assert the presence of closed orbits clustering at some distinguished parameter value (∈ = 0, say). Here, for any ∈, the origin is the only stationary point. The topological content of the mostly analytic hypotheses imposed is some change in the stability behaviour of the origin at ∈ = 0, roughly the passing of a kind of stability to a kind of instability. Topologically speaking, e.g. some of the conditions demanded are asymptotic stability of the origin for the negative system at ∈ > 0 and asymptotic stability of the origin for at ∈ < 0 (Hopf (8), Ruelle and Takens(11)) or ∈ = 0 (Chafee(2)).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 1 , January 1983 , pp. 113 - 119
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

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