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

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.     

Hopf Bifurcation for a Maturity Structured Population Dynamic Model

This article is devoted to investigate some dynamical properties of a structured population dynamic model with random walk on (0,+∞). This model has a nonlinear and nonlocal boundary condition. We reformulate the problem as an abstract non-densely defined Cauchy problem, and use integrated semigroup theory to study such a partial differential equation. Moreover, a Hopf bifurcation theorem is given for this model.     

Brusselator模型的扩散引起不稳定性和Hopf分支

研究了Brusselator常微分系统和相应的偏微分系统的Hopf分支,并用规范形理论和中心流形定理讨论了当空间的维数为1时Hopf分支解的稳定性．证明了:当参数满足某些条件时,Brusselator常微分系统的平衡解和周期解是渐近稳定的,而相应的偏微分系统的空间齐次平衡解和空间齐次周期解是不稳定的;如果适当选取参数,那么Brusselator常微分系统不出现Hopf分支,但偏微分系统出现Hopf分支,这表明,扩散可以导致Hopf分支．     

Hopf Bifurcation for a Susceptible-Infective Model with Infection-Age Structure

An SIS model is investigated in which the infective individuals are assumed to have an infection-age structure. The model is formulated as an abstract non-densely defined Cauchy problem. We study some dynamical properties of the model by using the theory of integrated semigroups, the Hopf bifurcation theory and the normal form theory for semilinear equations with non-dense domain. Qualitative analysis indicates that there exist some parameter values such that this SIS model has a non-trivial periodic solution which bifurcates from the positive equilibrium. Furthermore, the explicit formulae are given to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Numerical simulations are also carried out to support our theoretical results.     

多时滞捕食-食饵系统正平衡点的稳定性及全局Hopf分支

本文首先用Cooke等人建立的关于超越函数的零点分布定理,研究了一类多时滞捕食-食饵系统正平衡点的稳定性及局部Hopf分支,在此基础上再结合吴建宏等人用等变拓扑度理论建立起的一般泛函微分方程的全局Hopf分支定理,进一步研究了该系统的全局Hopf分支.     

Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with distributed delays

In this paper, a four-neuron BAM neural network with distributed delays is considered, where kernels are chosen as weak kernels. Its dynamics is studied in terms of local stability analysis and Hopf bifurcation analysis. By choosing the average delay as a bifurcation parameter and analyzing the associated characteristic equation, Hopf bifurcation occurs when the bifurcation parameter passes through some exceptive values. The stability of bifurcating periodic solutions and a formula for determining the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, numerical simulation results are given to validate the theorem obtained.     

中立型微分方程零解的稳定性与全局Hopf分支

本文用Ｒｏｕｃｈｅ定理建立起关于一般的超越函数的零点分布定理，以此定理为基础，结合应用吴建宏等用等变拓扑度理论建立起的一般泛函微分方程的Ｈｏｐｆ分支定理，研究了描述无损传输网络线路的中立型微分方程的零解的稳定性和全局Ｈｏｐｆ分支．     

A bifurcation theorem for potential operators

A general bifurcation theorem for potential operators is proved. It describes the possible behavior of the set of solutions of an operator equation as a function of the eigenvalue parameter in a neighborhood of the bifurcation point. The theorem applies in particular to buckling problems in elasticity theory as well as to other fields in which the bifurcation problems have a variational formulation.     

Hopf bifurcation in a size-structured population dynamic model with random growth

This paper is devoted to the study of a size-structured model with Ricker type birth function as well as random fluctuation in the growth process. The complete model takes the form of a reaction-diffusion equation with a nonlinear and nonlocal boundary condition. We study some dynamical properties of the model by using the theory of integrated semigroups. It is shown that Hopf bifurcation occurs at a positive steady state of the model. This problem is new and is related to the center manifold theory developed recently in [P. Magal, S. Ruan, Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age-structured models, Mem. Amer. Math. Soc., in press] for semilinear equation with non-densely defined operators.     

Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay

In this paper, we investigated Hopf bifurcation by analyzing the distributed ranges of eigenvalues of characteristic linearized equation. Using communication delay as the bifurcation parameter, linear stability criteria dependent on communication delay have also been derived, and, furthermore, the direction of Hopf bifurcation as well as stability of periodic solution for the exponential RED algorithm with communication delay is studied. We find that the Hopf bifurcation occurs when the communication delay passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, a numerical simulation is presented to verify the theoretical results.     

Hopf bifurcation in REM algorithm with communication delay

The purpose of this paper is to study bifurcation of an Internet congestion control algorithm, namely REM (Random Exponential Marking) algorithm, with communication delay. By choosing the delay constant as a bifurcation parameter, we prove that REM algorithm exhibits Hopf bifurcation. The formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by applying the center manifold theorem and the normal form theory. Finally, a numerical simulation is present to verify the theoretical results.     

Complex dynamics in Chen’s system

This paper characterizes some complex dynamics of Chen’s system. Some conditions of existence for pitchfork bifurcation and Hopf bifurcation are derived by using bifurcation theory and the center manifold theorem. Numerical simulation results not only show consistence with the theoretical analysis but also display some new and interesting dynamical behaviors including homoclinic bifurcation and the coexistence of two stable limit cycles and one chaotic attractor as well as some periodic solutions emerging from Hopf bifurcation but ending in homoclinic bifurcation, which are different from those reported in the literature before. All these show that Chen’s system has very rich nonlinear dynamics.     

Multiple bifurcations in a delayed predator–prey diffusion system with a functional response

The present paper is concerned with a delayed predator–prey diffusion system with a Beddington–DeAngelis functional response and homogeneous Neumann boundary conditions. If the positive constant steady state of the corresponding system without delay is stable, by choosing the delay as the bifurcation parameter, we can show that the increase of the delay can not only cause spatially homogeneous Hopf bifurcation at the positive constant steady state but also give rise to spatially heterogeneous ones. In particular, under appropriate conditions, we find that the system has a Bogdanov–Takens singularity at the positive constant steady state, whereas this singularity does not occur for the corresponding system without diffusion. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we give normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation and the explicit formula for determining the properties of spatial Hopf bifurcations.     

Hopf Bifurcation Analysis of a Class of Abstract Delay Differential Equation

The dynamics of a class of abstract delay differential equations are investigated. We prove that a sequence of Hopf bifurcations occur at the origin equilibrium as the delay increases. By using the theory of normal form and centre manifold, the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived. Then, the existence of the global Hopf bifurcation of the system is discussed by applying the global Hopf bifurcation theorem of general functional differential equation.     

Rhombic Periodicity Lattice in Bifurcational Symmetry Breaking Problems

General theory of symmetry breaking problems in branching theory is presented in the works [1–3]. Such problems are invariant relative to Euclidean space motions group and their solution which is invariant relatively this group is the rest state or uniform linear motion. At the stability loss cell structure solutions arise, which are invariant relative to the group of the definite period shifts along the definite directions, passing mutually under discrete subgroup transformations that is defined by the symmetry of elementary cell of the periodicity. Thus the Euclidean space motions group is changed by the symmetry of a certain crystallographic group. In this article for the case of 4–dimensional degeneracy of the linearized Fredholm operator abstract bifurcational symmetry breaking problems both for stationary and Andronov‐Hopf bifurcation with planar rhombic periodicity lattice are considered. Applications to hydrodynamical problems are given.     

Stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment

In this paper, the stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment is studied. It is shown that there exist 3 equilibria. The sufficient conditions for local asymptotic stability of the infection‐free equilibrium and no‐immune equilibrium are given. We also discussed the local stability of positive equilibrium and the existence of Hopf bifurcation. Moreover, the direction and stability of Hopf bifurcation is obtained by using standard form theory and the center manifold theorem. Finally, numerical simulations are performed to verify the theoretical conclusions.     

A symplectic perspective on constrained eigenvalue problems

The Maslov index is a powerful tool for computing spectra of selfadjoint, elliptic boundary value problems. This is done by counting intersections of a fixed Lagrangian subspace, which designates the boundary conditions, with the set of Cauchy data for the differential operator. We apply this methodology to constrained eigenvalue problems, in which the operator is restricted to a (not necessarily invariant) subspace. The Maslov index is defined and used to compute the Morse index of the constrained operator. We then prove a constrained Morse index theorem, which says that the Morse index of the constrained problem equals the number of constrained conjugate points, counted with multiplicity, and give an application to the nonlinear Schrödinger equation.     

Hopf Bifurcation of Delayed Predator-prey System with Reserve Area for Prey and in the Presence of Toxicity

A kind of three species delayed predator-prey system with reserve area for prey and in the presence of toxicity is proposed in this paper.Local stability of the coexistence equilibrium of the system and the existence of a Hopf bifurcation is established by choosing the time delay as the bifurcation parameter.Explicit formulas to determine the direction and stability of the Hopf bifurcation are obtained by means of the normal form theory and the center manifold theorem.Finally,we give a numerical example to illustrate the obtained results.     

Hopf bifurcation analysis of a general non-linear differential equation with delay

This work represents Hopf bifurcation analysis of a general non-linear differential equation involving time delay. A special form of this equation is the Hutchinson–Wright equation which is a mile stone in the mathematical modeling of population dynamics and mathematical biology. Taking the delay parameter as a bifurcation parameter, Hopf bifurcation analysis is studied by following the theory in the book by Hazzard et al. By analyzing the associated characteristic polynomial, we determine necessary conditions for the linear stability and Hopf bifurcation. In addition to this analysis, the direction of bifurcation, the stability and the period of a periodic solution to this equation are evaluated at a bifurcation value by using the Poincaré normal form and the center manifold theorem. Finally, the theoretical results are supported by numerical simulations.     

Hopf bifurcation in Cohen–Grossberg neural network with distributed delays

In this paper, we discuss the stability and bifurcation of the distributed delays Cohen–Grossberg neural networks with two neurons. By choosing the average delay as a bifurcation parameter, we prove that Hopf bifurcation occurs. The stability of bifurcating periodic solutions and the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, numerical simulation results are given to support the theoretical predictions.