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Hopf bifurcation for a class of differential equations with two delays

Mohamed El Azzouzi, A. Kaddar
2016 Nonlinear Analysis and Differential Equations  
In the present paper, we consider a particular kind of differential equation with two discrete time delays.  ...  When one of the delays is considered as a bifurcation parameter, the stability of the trivial steady state and the occurrence of Hopf bifurcation are investigated.  ...  Delay differential equations have also appeared in the study of ecological models [1, 13] .  ... 
doi:10.12988/nade.2016.51141 fatcat:6cpayl33sjfxvjpdl5y5bgpxgq

Delayed Feedback Control of a Delay Equation at Hopf Bifurcation

Bernold Fiedler, Sergio Muniz Oliva
2015 Journal of Dynamics and Differential Equations  
We embark on a case study for the scalar delay equatioṅ with odd nonlinearity f , real nonzero parameters λ, b, and three positive time delays 1, ϑ, p/2.  ...  Following an idea of [Pyr92], we seek parameter regions P of controls b = 0 and delays ϑ ≥ 0 such that the branch k becomes stable, locally at Hopf bifurcation.  ...  The only reason of our preference for the pure delay equation was the ease of a coherent treatment of all Hopf bifurcations, in section 2.  ... 
doi:10.1007/s10884-015-9456-8 fatcat:upq2q26fdfdeldk6pmtzkwtkaq

Numerical Hopf bifurcation for a class of delay differential equations

Volker Wulf, Neville J. Ford
2000 Journal of Computational and Applied Mathematics  
In this paper we consider discretization of parameter-dependent delay di erential equations of the form We show that, if the delay di erential equation undergoes a Hopf bifurcation, then the discrete scheme  ...  undergoes a Hopf bifurcation of the same type.  ...  Volker Wulf acknowledges with thanks the support of a Chester College Research Student Bursary.  ... 
doi:10.1016/s0377-0427(99)00181-8 fatcat:wbmm6vy5z5bsndype2yvvkucuq

Equivariant Hopf Bifurcation in a Time-Delayed Ring of Antigenic Variants

Israel Ncube
2016 International Journal of Differential Equations  
We establish conditions for the existence of an equivariant Hopf bifurcation in a ring of antigenic variants, characterised by time delay.  ...  We investigate the dynamics of the model, paying particular attention to bifurcation and stability of the uniform nonzero endemic equilibrium.  ...  Equivariant Hopf Bifurcation The extension of the theory of equivariant Hopf bifurcation to functional differential equations was established in the series of papers [18, 21, 22] .  ... 
doi:10.1155/2016/5915768 fatcat:bx6xa7n4lbhdbkqc2pehmxnz2q

Control by time delayed feedback near a Hopf bifurcation point

Sjoerd Verduyn Lunel, Babette de Wolff
2017 Electronic Journal of Qualitative Theory of Differential Equations  
the Floquet multipliers and 2) by use of the Hopf bifurcation theorem.  ...  We also propose an extension of the Pyragas control scheme for which the controlled system becomes a functional differential equation of neutral type.  ...  We first state Theorem X.2.7 and Theorem X.3.9 from [2] on the Hopf bifurcation for differential delay equations. Theorem 3.3 (Occurrence of a Hopf bifurcation).  ... 
doi:10.14232/ejqtde.2017.1.91 fatcat:uyec5hoazrdidhnvjw73advcji

Hopf Bifurcation for a Class of Partial Differential Equation with Delay

Katia A. G. Azevedo, Luiz A. C. Ladeira
2004 Funkcialaj Ekvacioj  
We show the existence of a sequence of values ft kn g n¼0; 1; 2;... of the parameter t such that a Hopf bifurcation occurs when the delay passes through each value ft kn g.  ...  We study a kind of delayed reaction-di¤usion equation with Dirichlet boundary condition.  ...  It is the purpose of this paper to prove the existence of a sequence of Hopf bifurcations which arise from positive equilibrium (spatially nonconstant) as the delay t varies. For us g : R 2 !  ... 
doi:10.1619/fesi.47.395 fatcat:krdpg5p4wfbgjppertvgn7muba

Hopf bifurcations in a reaction–diffusion population model with delay effect

Ying Su, Junjie Wei, Junping Shi
2009 Journal of Differential Equations  
From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained  ...  via the implicit function theorem, where the time delay is used as the bifurcation parameter.  ...  We also conjecture that a cascade of Hopf bifurcations occurs at (λ, u λ ) for a sequence of increasing delays τ λ n for each λ > λ * , and the values of τ λ n and time periods of solutions decrease in  ... 
doi:10.1016/j.jde.2009.04.017 fatcat:sjcdghn5kjgljjhmmiupfvfamy

Stability and Hopf Bifurcation for a Population Delay Model with Diffusion Effects

Stavros Busenberg, Wenzhang Huang
1996 Journal of Differential Equations  
Institute of Technology, by NSF Grant DEB-925370 to Carlos Castillo-Chavez, and by the U.S.  ...  Army Research Office through the Mathematical Sciences Institute of Cornell University (Contract DAAL03-91-C-0027). Both authors thank the referee for hisÂher useful suggestion.  ...  Hopf Bifurcation In this section we study the Hopf bifurcation which arises from the positive equilibrium U k as the delay r crosses r kn , n=0, 1, ...  ... 
doi:10.1006/jdeq.1996.0003 fatcat:fwkyrydog5efthguvrto3hwibq

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

Hande Akkocaoğlu, Hüseyin Merdan, Canan Çelik
2013 Journal of Computational and Applied Mathematics  
This work represents Hopf bifurcation analysis of a general non-linear differential equation involving time delay.  ...  Taking the delay parameter as a bifurcation parameter, Hopf bifurcation analysis is studied by following the theory in the book by Hazzard et al.  ...  In this paper, we worked on a general nonlinear differential equation involving a discrete time delay.  ... 
doi:10.1016/j.cam.2012.06.029 fatcat:nmc2tt72xrchnffa37pwuipt6i

Hopf Bifurcation in a Diffusive Logistic Equation with Mixed Delayed and Instantaneous Density Dependence

Ying Su, Junjie Wei, Junping Shi
2012 Journal of Dynamics and Differential Equations  
The stability of the unique positive steady state solution and the occurrence of Hopf bifurcation from this positive steady state solution are obtained by a detailed analysis of the characteristic equation  ...  A diffusive logistic equation with mixed delayed and instantaneous density dependence and Dirichlet boundary condition is considered.  ...  Acknowledgements The authors thank an anonymous referee and the editor for careful reading of the manuscript and some helpful comments.  ... 
doi:10.1007/s10884-012-9268-z fatcat:xd7arp4cg5dglnxe6xsy5zdrdq

Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect

Shanshan Chen, Junping Shi
2012 Journal of Differential Equations  
The stability of the positive spatially nonhomogeneous positive equilibrium and associated Hopf bifurcation are investigated for the case of near equilibrium bifurcation point and the case of spatially  ...  A reaction-diffusion model with logistic type growth, nonlocal delay effect and Dirichlet boundary condition is considered, and combined effect of the time delay and nonlocal spatial dispersal provides  ...  Busenberg and Huang [4] studied the Hopf bifurcation of the diffusive logistic equation with delay effect and Dirichlet boundary condition proposed in Green and Stech [21] : ∂u(x, t) ∂t = d u(x, t)  ... 
doi:10.1016/j.jde.2012.08.031 fatcat:lr3ezahdyne4rk72op6t4y7nii

Change in criticality of Hopf bifurcation in a time-delayed cancer model

Israel Ncube, Kiara Martin
2019 Electronic Journal of Qualitative Theory of Differential Equations  
Khajanchi et al. [12] looked at the effects of discrete time delay in a chaotic mathematical model of cancer, and studied the ensuing Hopf bifurcation problem with the time delay used as the bifurcation  ...  In this work, we perform a linear stability analysis of the model's three equilibria, and formulate a local Hopf bifurcation theorem for one of the two endemic equilibria.  ...  Acknowledgements We would like to express our gratitude to the anonymous referee whose feedback significantly helped us improve the presentation and exposition of our work.  ... 
doi:10.14232/ejqtde.2019.1.84 fatcat:5axlymck6ffolhej7oyuxxvepu

Direction and Stability of Hopf Bifurcation in a Delayed Solow Model with Labor Demand

Sanaa ElFadily, Abdelilah Kaddar, Khalid Najib
2019 International Journal of Differential Equations  
or subcritical Hopf bifurcation and the bifurcating periodic solution is stable or unstable in a neighborhood of some bifurcation points, depending on the choice of parameters.  ...  By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points and we show that the system under consideration can undergo a supercritical  ...  Conflicts of Interest The authors declare that they have no conflicts of interest regarding the publication of this paper.  ... 
doi:10.1155/2019/7609828 fatcat:mtq7y6i7p5bnjeepgbsaiy27ra

How do numerical methods perform for delay differential equations undergoing a Hopf bifurcation?

Neville J. Ford, Volker Wulf
2000 Journal of Computational and Applied Mathematics  
In this paper we consider the numerical solution of delay di erential equations (DDEs) undergoing a Hopf bifurcation. Some authors use special methods to calculate bifurcating periodic solutions.  ...  We present three distinctive and complementary approaches to the analysis which together provide us with the result that #-methods applied to a DDE will retain Hopf bifurcations and preserve their type  ...  Volker Wulf was supported in this work by a Chester College Research Student Bursary.  ... 
doi:10.1016/s0377-0427(00)00473-8 fatcat:7z6xefczljdmzleezne37c3bqe

Stability and Hopf bifurcation of a diffusive Gompertz population model with nonlocal delay effect

Xiuli Sun, Luan Wang, Baochuan Tian
2018 Electronic Journal of Qualitative Theory of Differential Equations  
The stability of the positive spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcations with the change of the time delay are discussed by analyzing the distribution of eigenvalues  ...  In this paper, we investigate the dynamics of a diffusive Gompertz population model with nonlocal delay effect and Dirichlet boundary condition.  ...  existence of Hopf bifurcation with the changes of the time delay.  ... 
doi:10.14232/ejqtde.2018.1.22 fatcat:o4uwglp7ajc7fhjvvjzt6o6smy
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