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Bifurcation of an attracting invariant circle: a Denjoy attractor

Published online by Cambridge University Press:  19 September 2008

Glen R. Hall
Glen R. Hall
Affiliation:
Mathematics Research Center, 610 Walnut Street, University of Wisconsin, Madison, Wisconsin, 53705
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Abstract

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We construct an example of a C∞ diffeomorphism of an annulus into itself which has an attracting invariant circle such that the map restricted to this circle has no periodic points and no dense orbits. By studying two parameter families of maps of the plane which undergo Hopf bifurcation, particularly the set of parameter values for which the rotation number is irrational, we see that the above example can be considered as a ‘worst case’ of the loss of smoothness of an attracting invariant circle without periodic orbits.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 1 , March 1983 , pp. 87 - 118
Copyright
Copyright © Cambridge University Press 1983

References

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