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

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Abstract

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 of the periodic oscillations.

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References

  1. Balanov, Z., Hu, Q., Krawcewicz, W.: Global Hopf bifurcation of differential equations with threshold-type state-dependent delay. J. Differ. Equ. 257, 2622–2670 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cooke, K., Huang, W.: On the problem of linearization for state-dependent delay differential equations. Proc. Am. Math. Soc. 124(5), 1417–1426 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Goodwin, B.: Oscillatory behavior in enzymatic control processes. Adv. Enzym. Regul. 3, 425–438 (1965)

    Article  Google Scholar 

  4. Hartung, F., Krisztin, T., Walther, H.-O., Wu, J.: Chapter 5: Functional differential equations with state-dependent delays: theory and applications. In: CaÑada, P.D.A., Fonda, A. (eds.) Handbook of Differential Equations: Ordinary Differential Equations, vol. 3, pp. 435–545. North-Holland, Amsterdam (2006)

    Chapter  Google Scholar 

  5. Hu, Q.: A model of regulatory dynamics with threshold type state-dependent delay. Math. Biosci. Eng. (2017, accepted)

  6. Hu, Q., Wu, J.: Global Hopf bifurcation for differential equations with state-dependent delay. J. Differ. Equ. 248(12), 2801–2840 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Insperger, T., Stépán, G., Turi, J.: State-dependent delay in regenerative turning processes. Nonlinear Dyn. 47, 275–283 (2007)

    Article  MATH  Google Scholar 

  8. Krawcewicz, W., Wu, J.: Theory of Degrees with Applications to Bifurcations and Differential Equations, Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1997)

    Google Scholar 

  9. Mallet-Paret, J., Yorke, A.J.: Snakes: oriented families of periodic orbits, their sources, sinks, and continuation. J. Differ. Equ. 43, 419–450 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mallet-Paret, J., Nussbaum, R.: Boundary layer phenomena for differential-delay equations with state-dependent time lags, I. Arch. Ration. Mech. Anal. 120(2), 99–146 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mallet-Paret, J., Nussbaum, R.: Boundary layer phenomena for differential-delay equations with state dependent time lags: II. J. Reine Angew. Math. 477, 129–197 (1996)

    MathSciNet  MATH  Google Scholar 

  12. Mallet-Paret, J., Nussbaum, R.: Boundary layer phenomena for differential-delay equations with state-dependent time lags: III. J. Differ. Equ. 189(2), 640–692 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mallet-Paret, J.: The Fredholm alternative for functional differential equations of mixed type. J. Dyn. Differ. Equ. 11, 1–47 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rustichini, A.: Hopf bifurcation for functional differential equations of mixed type. J. Dyn. Differ. Equ. 1, 145–177 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Smith, H.: Structured population models, threshold-type delay and functional-differential equations. Delay Differ. Equ. AMS, IA 1992, 52–64 (1991)

    Google Scholar 

  16. Smith, H.L.: Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study. Math. Biosci. 113(1), 1–23 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Vidossich, V.: On the structure of periodic solutions of differential equations. J. Differ. Equ. 21, 263–278 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  18. Walther, H.: Stable periodic motion of a system using echo for position control. J. Dyn. Differ. Equ. 15(1), 143–223 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wu, J.: Global continua of periodic solutions to some differential equations of neutral type. Tôhoku Math. J. 45, 67–88 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank an anonymous referee for the detailed and constructive comments.

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  1. Department of Mathematical Sciences, The University of Texas at Dallas, 800 West Campbell Road, Richardson, TX, 75080, USA

    Qingwen Hu

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  1. Qingwen Hu
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Correspondence to Qingwen Hu.

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Hu, Q. Global Hopf Bifurcation for Differential-Algebraic Equations with State-Dependent Delay. J Dyn Diff Equat 31, 93–128 (2019). https://doi.org/10.1007/s10884-017-9640-0

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  • DOI: https://doi.org/10.1007/s10884-017-9640-0

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