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Remark on the Hopf Bifurcation Theorem

✍ Scribed by A. M. Krasnosel'skii; D. I. Rachinskii

Publisher: John Wiley and Sons
Year: 2004
Tongue: English
Weight: 164 KB
Volume: 272
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.200310190

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✦ Synopsis

Abstract

A simple generalization of the Hopf Bifurcation Theorem for scalar higher order ordinary differential equations is suggested. We study the degenerate case where several roots of the characteristic polynomial cross the imaginary axis at the same point for some value λ~0~ of the parameter λ. The main result is that if N~1~ roots cross the imaginary axis from the left to the right and N~2~ roots cross it from the right to the left, then λ~0~ is a Hopf bifurcation point whenever N~1~ ≠ N~2~. In particular, in the classical Hopf Bifurcation Theorem the numbers N~j~ are 0 and 1. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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