Normal forms for semilinear equations with non-dense domain with applications to age structured models

Zhihua Liu, Pierre Magal, Shigui Ruan

Research output: Contribution to journalArticle

14 Scopus citations


Normal form theory is very important and useful in simplifying the forms of equations restricted on the center manifolds in studying nonlinear dynamical problems. In this paper, using the center manifold theorem associated with the integrated semigroup theory, we develop a normal form theory for semilinear Cauchy problems in which the linear operator is not densely defined and is not a Hille-Yosida operator and present procedures to compute the Taylor expansion and normal form of the reduced system restricted on the center manifold. We then apply the main results and computation procedures to determine the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions in a structured evolutionary epidemiological model of influenza A drift and an age structured population model.

Original languageEnglish (US)
Pages (from-to)921-1011
Number of pages91
JournalJournal of Differential Equations
Issue number4
StatePublished - Aug 15 2014



  • Center manifold
  • Hopf bifurcation
  • Non-densely defined Cauchy problem
  • Normal form
  • Population dynamics
  • Structured population

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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