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EQUIVALENCE OF DEGENERATE HOPF BIFURCATIONS

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Abstract

We prove the equivalence of degenerate Hopf bifurcations which have all their closed orbits at the bifurcation point. Although these Hopf bifurcations have infinity codimension, they can nevertheless occur generically in dynamical systems under constraint such as in the Hamiltonian systems or in the replicator equations; and so in these contexts a treatment of their equivalence is required. The analysis is rather delicate. The Poincare return maps of the flows give rise to a one-parameter family of one-dimensional maps and we start by determining the conjugacy classes of such families: There are surprisingly only two classes depending upon the finiteness or infiniteness of an integral modulus. The conjugacy class of the return maps is then used to show the equivalence of the Hopf bifurcations

Topics: QA, QC
Publisher: IOP PUBLISHING LTD
OAI identifier: oai:wrap.warwick.ac.uk:22496
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