Hopf bifurcation for non-densely defined Cauchy problems

Zhihua Liu; Pierre Magal; Shigui Ruan

doi:10.1007/s00033-010-0088-x

Hopf bifurcation for non-densely defined Cauchy problems

Zhihua Liu, Pierre Magal, Shigui Ruan

Mathematics

Research output: Contribution to journal › Article › peer-review

45 Scopus citations

Abstract

In this paper, we establish a Hopf bifurcation theorem for abstract Cauchy problems in which the linear operator is not densely defined and is not a Hille-Yosida operator. The theorem is proved using the center manifold theory for non-densely defined Cauchy problems associated with the integrated semigroup theory. As applications, the main theorem is used to obtain a known Hopf bifurcation result for functional differential equations and a general Hopf bifurcation theorem for age-structured models.

Original language	English (US)
Pages (from-to)	191-222
Number of pages	32
Journal	Zeitschrift fur Angewandte Mathematik und Physik
Volume	62
Issue number	2
DOIs	https://doi.org/10.1007/s00033-010-0088-x
State	Published - Apr 2011

Keywords

Age structured model
Cauchy problem
Functional differential equation
Hopf bifurcation
Non-dense domain
Periodic solution

ASJC Scopus subject areas

Mathematics(all)
Physics and Astronomy(all)
Applied Mathematics

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AU - Magal, Pierre

AU - Ruan, Shigui

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AB - In this paper, we establish a Hopf bifurcation theorem for abstract Cauchy problems in which the linear operator is not densely defined and is not a Hille-Yosida operator. The theorem is proved using the center manifold theory for non-densely defined Cauchy problems associated with the integrated semigroup theory. As applications, the main theorem is used to obtain a known Hopf bifurcation result for functional differential equations and a general Hopf bifurcation theorem for age-structured models.

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KW - Functional differential equation

KW - Hopf bifurcation

KW - Non-dense domain

KW - Periodic solution

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