Abstract
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 language | English (US) |
|---|---|
| Pages (from-to) | 921-1011 |
| Number of pages | 91 |
| Journal | Journal of Differential Equations |
| Volume | 257 |
| Issue number | 4 |
| DOIs | |
| State | Published - Aug 15 2014 |
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Keywords
- Center manifold
- Hopf bifurcation
- Non-densely defined Cauchy problem
- Normal form
- Population dynamics
- Structured population
ASJC Scopus subject areas
- Analysis
Cite this
Normal forms for semilinear equations with non-dense domain with applications to age structured models. / Liu, Zhihua; Magal, Pierre; Ruan, Shigui.
In: Journal of Differential Equations, Vol. 257, No. 4, 15.08.2014, p. 921-1011.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Normal forms for semilinear equations with non-dense domain with applications to age structured models
AU - Liu, Zhihua
AU - Magal, Pierre
AU - Ruan, Shigui
PY - 2014/8/15
Y1 - 2014/8/15
N2 - 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.
AB - 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.
KW - Center manifold
KW - Hopf bifurcation
KW - Non-densely defined Cauchy problem
KW - Normal form
KW - Population dynamics
KW - Structured population
UR - http://www.scopus.com/inward/record.url?scp=84901624125&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84901624125&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2014.04.018
DO - 10.1016/j.jde.2014.04.018
M3 - Article
AN - SCOPUS:84901624125
VL - 257
SP - 921
EP - 1011
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 4
ER -