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Global Hopf Bifurcation for Differential-Algebraic Equations with State-Dependent Delay

Qingwen Hu
2018 Journal of Dynamics and Differential Equations  

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2017 pre-print arXiv:1705.03794v1 fatcat:5ykh7mtaxveajhone3iz2sydp4

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2017 accepted
arXiv:1705.03794v2 fatcat:zwa6czao55apdipgkqa6ez3n2m
We develop a global Hopf bifurcation theory for differential equations with a state-dependent delay governed by an algebraic equation, using the S^1-equivariant degree.  ...  We apply the global Hopf bifurcation theory to a model of genetic regulatory dynamics with threshold type state-dependent delay vanishing at the stationary state, for a description of the global continuation  ...  delay governed by an algebraic equation, we developed a global Hopf bifurcation theory for differential-algebraic equations with state-dependent delay, using the S 1 -equivariant degree.  ... 
doi:10.1007/s10884-017-9640-0 fatcat:ekizrylttbhrbedkvnaqe5hkny
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Qingwen Hu. "Global Hopf Bifurcation for Differential-Algebraic Equations with State-Dependent Delay." Journal of Dynamics and Differential Equations 31.1 (2018)

Global Hopf bifurcation for differential equations with state-dependent delay

Qingwen Hu, Jianhong Wu
2010 Journal of Differential Equations  

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We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay.  ...  and to describe the global continuation of periodic solutions for such a system with state-dependent delay.  ...  As such we show that information of the auxiliary equations can be used in a standard way to develop a local and global Hopf bifurcation theory for FDEs with state-dependent delay.  ... 
doi:10.1016/j.jde.2010.03.020 fatcat:l6f2wtkl35hknivtah64t6bkdi
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Qingwen Hu, Jianhong Wu. "Global Hopf bifurcation for differential equations with state-dependent delay." Journal of Differential Equations 248.12 (2010) 2801-2840

Global Hopf bifurcations of neutral functional differential equations with state-dependent delay

Xiuli Sun, Rong Yuan, Yunfei Lv
2018 Discrete and continuous dynamical systems. Series B  

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A global Hopf bifurcation theory for a system of neutral functional differential equations (NFDEs) with state-dependent delay is investigated by applying the S 1 -equivariant degree theory.  ...  The results are important in studying bifurcations of NFDEs with state-dependent delay.  ...  Global bifurcation of NFDEs with state-dependent delay.  ... 
doi:10.3934/dcdsb.2018038 fatcat:bywbwax5hfgy3hcmmkew75jc7e
Citation

Xiuli Sun, Rong Yuan, Yunfei Lv. "Global Hopf bifurcations of neutral functional differential equations with state-dependent delay." Discrete and continuous dynamical systems. Series B 23.2 (2018) 667-700

Global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback

Li Zhang, HuaiLei Wang, HaiYan Hu
2010 Science China Technological Sciences  

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Global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback.  ...  This paper presents both analytical and numerical studies on the global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback.  ...  Introduction Numerous retard phenomena in natural and social sciences can be modeled by delay differential equations, whose current state variation depends on the past states.  ... 
doi:10.1007/s11431-010-0073-9 fatcat:z6zrjcdzhjf2xb3tzp2tjbfggm
Citation

Li Zhang, HuaiLei Wang, HaiYan Hu. "Global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback." Science China Technological Sciences 53.3 (2010) 595-607

Advanced Nonlinear Dynamics of Population Biology and Epidemiology

Weiming Wang, Yun Kang, Malay Banerjee, Kaifa Wang
2014 Abstract and Applied Analysis  

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Zhai et al. investigate the behavior of price differential equation model based on economic theory with two delays and show the linear stability and local Hopf bifurcation. W. Wang and B.  ...  Ordinary differential equation models, partial differential equation models, delay differential equation models, stochastic differential equation models, difference equation models, and nonautonomous models  ...  Acknowledgments We would also like to thank the editorial board members of this journal, for their support and help throughout the preparation of this special issue.  ... 
doi:10.1155/2014/214514 fatcat:mcufvvxppjfujb7hiyy3fz4gim
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Weiming Wang, Yun Kang, Malay Banerjee, Kaifa Wang. "Advanced Nonlinear Dynamics of Population Biology and Epidemiology." Abstract and Applied Analysis (2014) 1-3

Page 247 of Mathematical Reviews Vol. , Issue 2001A [page]

2001 Mathematical Reviews  

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In this paper, the authors study a family of delay differential equations (DDE) which model a simple harmonic oscillator with delayed forcing, with forcing depending on the position or the velocity.  ...  positive steady state remains globally stable.” 2001a:34127 34K20 Dolgii, Yu.  ... 
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Mathematical Reviews (2001) 247

Page 8760 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  

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The authors consider the Hopf bifurcation of a Hopfield neural network of two neurons with a single delay.  ...  Summary: “For a class of neural systems with time-varying per- turbations in the time-delayed state, this paper studies the periodic solution and global robust exponential stability.  ... 
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Mathematical Reviews (2004) 8760

Mapping dynamical systems with distributed time delays to sets of ordinary differential equations [article]

Daniel Henrik Nevermann, Claudius Gros
2023 arXiv   pre-print

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2023 pre-print arXiv:2303.14815v1 fatcat:hzhh7f22znevlcymfgizuxj3xi
An exact mapping onto a set of N+1 ordinary differential equations exists when the respective delay distribution is given in terms of a gamma distribution with discrete exponents.  ...  We find that local properties, f.i. the locus of a Hopf bifurcation, are robust against the introduction of broadened memory kernels.  ...  The Hopf bifurcation point is then T Hopf = 4.708. The kernel series framework (9) for the linear delay system (12) leads to a linear system of N +1 coupled ordinary differential equations.  ... 
arXiv:2303.14815v2 fatcat:jzwrohu2zfhgdjglfynxuxigtq
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Daniel Henrik Nevermann, Claudius Gros. "Mapping dynamical systems with distributed time delays to sets of ordinary differential equations." arXiv (2023)

Stability and Bifurcations in an Epidemic Model with Varying Immunity Period

Konstantin B. Blyuss, Yuliya N. Kyrychko
2009 Bulletin of Mathematical Biology  

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2012 pre-print
arXiv:1201.4587v1 fatcat:mpp75pmhpra4zogzighzit5pdi
An epidemic model with distributed time delay is derived to describe the dynamics of infectious diseases with varying immunity.  ...  Bifurcation analysis is performed using DDE-BIFTOOL and traceDDE to investigate different dynamical regimes in the model using numerical continuation for different values of system parameters and different  ...  Figure 1 shows the boundary of Hopf bifurcation of the endemic steady state E in the space of parameters.  ... 
doi:10.1007/s11538-009-9458-y pmid:19898905 fatcat:mcg3g2jlbrf7jc7i6xieyw4b6a
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Konstantin B. Blyuss, Yuliya N. Kyrychko. "Stability and Bifurcations in an Epidemic Model with Varying Immunity Period." Bulletin of Mathematical Biology 72.2 (2009) 490-505

Hopf bifurcation and period functions for Wright type delay differential equations [article]

István Balázs, Gergely Röst
2018 arXiv   pre-print

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We present the simplest criterion that determines the direction of the Hopf bifurcations of the delay differential equation x'(t)=-μ f(x(t-1)), as the parameter μ passes through the critical values μ_k  ...  Using this information and the Cooke-transformation, we obtain local estimates and monotonicity properties of the periods of the bifurcating limit cycles along the Hopf-branches.  ...  Introduction The appearance of limit cycles around equilibria via Hopf bifurcations is a common phenomenon for delay differential equations, when a parameter of the equation is passing through a critical  ... 
arXiv:1801.04325v1 fatcat:wbyksug4xvatzi4ufduswspx5a
Citation

István Balázs, Gergely Röst. "Hopf bifurcation and period functions for Wright type delay differential equations." arXiv (2018)

Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with Two Time Delays

Huitao Zhao, Yiping Lin, Yunxian Dai
2013 Abstract and Applied Analysis  

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Using a global Hopf bifurcation result of Wu (1998) for functional differential equations, the global existence of the periodic solutions is obtained.  ...  By using the theory of functional equation and Hopf bifurcation, the conditions on which positive equilibrium exists and the quality of Hopf bifurcation are given.  ...  This work is supported by the National Natural Science Foundation of China (no. 11061016), Science and Technology Department of Henan Province (no. 122300410417) and Education Department of Henan Province  ... 
doi:10.1155/2013/321930 fatcat:4wxxrelcdnejrmu5fc2iapx7zm
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Huitao Zhao, Yiping Lin, Yunxian Dai. "Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with Two Time Delays." Abstract and Applied Analysis (2013) 1-15

Hopf Bifurcation Analysis of a Three-Stage-Structured Prey-Predator System with Multi-Delays

Shun Yi Li, Wen Wu Liu
2013 Advanced Materials Research  

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The characteristic equations and local stability of the equilibrium are analyzed, and the conditions for the positive equilibrium occurring Hopf bifurcation are obtained by applying the theorem of Hopf  ...  bifurcation.  ...  of the positive equilibrium occurring Hopf bifurcation.  ... 
doi:10.4028/www.scientific.net/amr.756-759.2857 fatcat:tuj3qo3ukne2rkmsmvjzob7t3i
Citation

Shun Yi Li, Wen Wu Liu. "Hopf Bifurcation Analysis of a Three-Stage-Structured Prey-Predator System with Multi-Delays." Advanced Materials Research (2013) 2857-2862

Stabilization of complex spatio-temporal dynamics near a subcritical Hopf bifurcation by time-delayed feedback

M. Kehrt, P. Hövel, V. Flunkert, M. A. Dahlem, P. Rodin, E. Schöll
2009 European Physical Journal B : Condensed Matter Physics  

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Global time-delayed feedback control is applied to stabilize unstable spatio-temporal patterns which are created in a subcritical Hopf bifurcation.  ...  PACS . 05.45.Gg Control of chaos, applications of chaos -02.30.Ks Delay and functional equations -47.20.Ky Nonlinearity, bifurcation, and symmetry breaking -87.19.lr Control theory and feedback  ...  This work was supported by DFG in the framework of Sfb 555 (Complex Nonlinear Processes). We acknowledge stimulating discussions with Bernold Fiedler, Niels Majer, and Grischa Stegemann.  ... 
doi:10.1140/epjb/e2009-00132-5 fatcat:vfsyszh2wfhttjfrspdtpol4hi
Citation

M. Kehrt, P. Hövel, V. Flunkert, M. A. Dahlem, P. Rodin, E. Schöll. "Stabilization of complex spatio-temporal dynamics near a subcritical Hopf bifurcation by time-delayed feedback." European Physical Journal B : Condensed Matter Physics 68.4 (2009) 557-565

Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks

Yongli Song, Valeri A. Makarov, Manuel G. Velarde
2009 Biological cybernetics  

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Then we investigate the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric local Hopf bifurcation theory of delay differential equations and derive the equation describing  ...  the flow on the center manifold that enables us determining the direction of Hopf bifurcations and stability of the bifurcating periodic orbits.  ...  Then using the symmetric bifurcation theory of delay differential equations together with representation theory of cyclic groups we investigated the spatio-temporal patterns of these bifurcating periodic  ... 
doi:10.1007/s00422-009-0326-5 pmid:19629517 fatcat:asw2hzrrtfddhdnnmhg6bk6yta
Citation

Yongli Song, Valeri A. Makarov, Manuel G. Velarde. "Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks." Biological cybernetics 101.2 (2009) 147-167

Page 6375 of Mathematical Reviews Vol. , Issue 94k [page]

1994 Mathematical Reviews  

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The authors use the notion of integrated semigroups to prove a global Hopf bifurcation result for retarded functional-differential equations.  ...  (English and French summaries) [Hopf global bifurcation for delay differential equations using integrated semigroups] C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 8, 767-772.  ... 
Citation

Mathematical Reviews (1994) 6375

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