### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Differential Equations*,

*257*(4), 921-1011. https://doi.org/10.1016/j.jde.2014.04.018

**Normal forms for semilinear equations with non-dense domain with applications to age structured models.** / Liu, Zhihua; Magal, Pierre; Ruan, Shigui.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 257, no. 4, pp. 921-1011. https://doi.org/10.1016/j.jde.2014.04.018

}

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

VL - 257

SP - 921

EP - 1011

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 4

ER -