Abstract
The term Hopf bifurcation refers to a phenomenon in which a steady state of an evolution equation evolves into a periodic orbit as a bifurcation parameter is varied. The Hopf bifurcation theorem (Theorem 3.2) provides sufficient conditions for determining when this behavior occurs. In this chapter, we study Hopf bifurcation for systems of ODE using singularity theory methods. The principal advantage of these methods is that they adapt well to degenerate Hopf bifurcations; i.e., cases where one or more of the hypotheses of the traditional theory fail. The power of these methods is illustrated by Case Study 2, where we present the analysis by Labouriau [1983] of degenerate Hopf bifurcation in the clamped Hodgkin-Huxley equations.
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© 1985 Springer Science+Business Media New York
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Golubitsky, M., Schaeffer, D.G. (1985). The Hopf Bifurcation. In: Singularities and Groups in Bifurcation Theory. Applied Mathematical Sciences, vol 51. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5034-0_8
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DOI: https://doi.org/10.1007/978-1-4612-5034-0_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-9533-4
Online ISBN: 978-1-4612-5034-0
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