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Zero singularities in a ring network with two delays

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Abstract

In this paper, we consider a neural network model consisting of three neurons with delayed self- and nearest-neighbor connections. We provide multiple bifurcations of the zero solution of the system near zero eigenvalue singularity. Taking the coupling coefficients as the bifurcation parameters, four kinds of zero singularities are demonstrated through center manifold reduction and normal form calculation.

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References

  1. Bogdanov R.I.: Versal deformations of a singular point on the plane in the case of zero eigenvalues. Funct. Anal. Appl. 9, 144–145 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bungay S.D., Campbell S.A.: Patterns of oscillation in a ring of identical cells with delayed coupling. Int. J. Bifurcat. Chaos 17, 3109–3125 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Campbell S.A., Ncube I., Wu J.: Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system. Physica D 214, 101–119 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Campbell S.A., Yuan Y.: Zero singularities of codimension two and three in delay differential equations. Nonlinearity 21, 2671–2691 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Campbell S.A., Yuan Y., Bungay S.D.: Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling. Nonlinearity 18, 2827–2846 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chow S., Hale J.: Methods of Bifurcation Theory. Springer, New York (2002)

    Google Scholar 

  7. Diekmann O., Gils S.A., Verduyn Lunel S.M., Walther H.-O.: Delay Equations, Functional-, Complex-, and Nonlinear Analysis. Springer, New York (1995)

    MATH  Google Scholar 

  8. Elphick C., Tirapegui E., Brachet M.E., Coullet P., Iooss G.: A simple global characterization for normal forms of singular vector fields. Physica D 29, 95–127 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ermentrout G.B.: The behavior of rings of coupled oscillators. J. Math. Biol. 23, 55–74 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Linkens D.A.: Stability of entrainment conditions for a particular form of mutually coupled van der Pol oscillators. IEEE Trans. Circuits Syst. 23, 113–121 (1976)

    Article  Google Scholar 

  11. Takens F.: Singularities of vector fields. Publ. Math. Inst. Hautes Études Sci 43, 47–100 (1974)

    Article  MathSciNet  Google Scholar 

  12. Turing A.M.: The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. [Biol] 237, 37–72 (1952)

    Article  Google Scholar 

  13. Faria T., Magalháes L.T.: Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. J. Differ. Equ. 122, 181–200 (1995)

    Article  MATH  Google Scholar 

  14. Faria T., Magalháes L.T.: Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity. J. Differ. Equ. 122, 201–224 (1995)

    Article  MATH  Google Scholar 

  15. Golubitsky M., Stewart I., Schaeffer D.G.: Singularities and Groups in Bifurcation Theory. Springer, New York (1988)

    Book  MATH  Google Scholar 

  16. Golubitsky M., Swift J.W., Knobloch E.: Symmetries and pattern selection in Rayleigh-Bénard convection. Physica D 10, 249–276 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Guckenheimer J., Holmes P.J.: Nonlinear Oscillations: Dynamical System and Bifurcations of Vector Fields. Springer, New York (1983)

    Google Scholar 

  18. Guo S., Lamb J.S.W.: Equivariant Hopf bifurcation for neutral functional differential equations. Proc. Am. Math. Soc. 136, 2031–2041 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hale J.K., Verduyn Lunel S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)

    MATH  Google Scholar 

  20. Iooss G., Adelmeyer M.: Topics in Bifurcation Theory and Applications, volume 3 of Advanced series in Nonlinear Dynamics. World Scientific, Singapore (1992)

    Google Scholar 

  21. Kuznetsov Y.A.: Elements of Applied Bifurcation Theory. 2nd edn. Springer, New York (1998)

    MATH  Google Scholar 

  22. Matthies K.: A subshift of finite-type in the Takens-Bogdanov bifurcation with D 3 symmetry. Documenta Mathematica 4, 463–485 (1999)

    MathSciNet  MATH  Google Scholar 

  23. Yuan Y., Campbell S.A.: Stability and synchronization of a ring of identical cells with delayed coupling. J. Dyn. Differ. Equ. 16, 709–744 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yuan Y., Wei J.: Multiple bifurcation analysis in a neural network model with delays. Int. J. Bifurcat. Chaos 16, 2903–2913 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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  1. College of Mathematics and Econometrics, Hunan University Changsha, 410082, Hunan, People’s Republic of China

    Shangjiang Guo

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  1. Shangjiang Guo
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Correspondence to Shangjiang Guo.

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Guo, S. Zero singularities in a ring network with two delays. Z. Angew. Math. Phys. 64, 201–222 (2013). https://doi.org/10.1007/s00033-012-0247-3

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  • DOI: https://doi.org/10.1007/s00033-012-0247-3

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