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Normal form of Bogdanov–Takens: styles and a hypernormalization method

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Abstract

A brief review is stated on a hypernormalization method of autonomous differential equations in the vicinity of an equilibrium. The formalization is obtained from the method of cohomology spectral sequences. We skip the technical concepts such that it can be accessible for a broad audience. Three normal form styles are demonstrated by computing three different and known first-level normal forms of Bogdanov–Takens singularity.

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References

  1. Algaba, A., Gamero, E., Garcia, C., Merino, M.: A degenerate Hopf–saddle-node bifurcation analysis in a family of electronic circuits. Nonlinear Dyn. 48, 55–76 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Algaba, A., Freire, E., Gamero, E., Garcia, C.: Quasi-homogeneous normal forms. J. Comp. and Appl. Math. 150, 193–216 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnol’d, V.I.: A spectral sequence for the reduction of functions to normal form. Funkcional. Anal. Priložen. 9, 81–82 (1975)

    Google Scholar 

  4. Arnol’d, V.I.: Spectral sequences for the reduction of functions to normal forms. In: Dobolev, S.L., Suslov, A.I. (eds.) Problems in Mechanics and Mathematical Physics, pp. 7–20. Nauka, Moscow (1976) (Russian)

    Google Scholar 

  5. Baider, A.: Unique normal forms for vector fields and Hamiltonians. J. Differential Equations 78, 33–52 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baider, A., Churchill, R.C.: The Campbell–Hausdorff group and a polar decomposition of graded algebra authomorphisms. Pacif. J. Math. 131, 219–235 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. Baider, A., Sanders, A.J.: Unique normal forms: The nilpotent Hamiltonian case. J. Differential Equations 92, 282–304 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Baider, A., Sanders, A.J.: Further reductions of the Takens–Bogdanov normal form. J. Differential Equations 99, 205–244 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bendersky, M., Churchill, R.: A spectral sequence approach to normal forms. In: Recent developments in algebraic topology. Contemp. Math., vol. 407, pp. 27–81. Amer. Math. Soc., Providence (2006)

    Chapter  Google Scholar 

  10. Bi, Q., Yu, P.: Double Hopf bifurcations and chaos of a nonlinear vibration system. Nonlinear Dyn. 19, 313–332 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, G., Della Dora, J.: Further Reductions of Normal Forms for Dynamical Systems. J. Differential Equations 166, 79–106 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, G., Wang, D., Wang, X.: Unique normal forms for nilpotent planer vector fields. Int. J. Bifurcation Chaos 10, 2159–2174 (2002)

    Article  Google Scholar 

  13. Ding, Y., Jiang, W., Yu, P.: Hopf-zero bifurcation in a generalized Gopalsamy neural network model. Nonlinear Dyn. 70, 1037–1050 (2012)

    Article  MathSciNet  Google Scholar 

  14. Gazor, M., Mokhtari, F.: Normal forms of Hopf-Zero singularity. Preprint, arXiv:1210.4467 (October 16, 2012)

  15. Gazor, M., Mokhtari, F.: Volume-preserving normal forms of Hopf-zero singularity. Preprint, arXiv:1203.1807v1 (March 8, 2012)

  16. Gazor, M., Mokhtari, F., Sanders, J.A.: Normal forms for Hopf-zero singularities with nonconservative nonlinear part. J. Differential Equations 254, 1571–1581 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gazor, M., Yu, P.: Infinite order parametric normal form of Hopf singularity. Int. J. Bifurcation Chaos 11, 3393–3408 (2008)

    Article  MathSciNet  Google Scholar 

  18. Gazor, M., Yu, P.: Formal decomposition method and parametric normal forms. Int. J. Bifurcation Chaos 11, 3487–3515 (2010)

    Article  MathSciNet  Google Scholar 

  19. Gazor, M., Yu, P.: Spectral sequences and parametric normal forms. J. Differential Equations 252, 1003–1031 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kokubu, H., Oka, H., Wang, D.: Linear grading function and further reduction of normal forms. J. Differential Equations 132, 293–318 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. McCleary, J.: A User’s Guide to Spectral Sequences, 2nd edn. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  22. Murdock, J.: Normal Forms and Unfoldings for Local Dynamical Systems. Springer, New York (2003)

    MATH  Google Scholar 

  23. Murdock, J.: Hypernormal form theory: Foundations and algorithms. J. Differential Equations 2, 424–465 (2004)

    Article  MathSciNet  Google Scholar 

  24. Sanders, J.A.: Normal form theory and spectral sequences. J. Differential Equations 192, 536–552 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sanders, J.A.: Normal form in filtered Lie algebra representations. Acta Appl. Math. 87, 165–189 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang, D., Li, J., Huang, M., Jiang, Y.: Unique normal form of Bogdanov–Takens singularities. J. Differential Equations 163, 223–238 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, W., Zhang, Q.C.: Computation of the simplest normal form of a resonant double Hopf bifurcation system with the complex normal form method. Nonlinear Dyn. 57, 219–229 (2009)

    Article  MATH  Google Scholar 

  28. Weibel, C.A.: An Introduction to Homological Algebra. Cambridge University Press, Cambridge (1994)

    Book  MATH  Google Scholar 

  29. Yu, P.: Analysis on double Hopf bifurcation using computer algebra with the aid of multiple scales. Nonlinear Dyn. 27, 19–53 (2002)

    Article  MATH  Google Scholar 

  30. Yu, P., Bi, Q.: Symbolic computation of normal forms for semi-simple cases. Nonlinear Dyn. 27, 19–53 (2002)

    Article  MATH  Google Scholar 

  31. Yu, P., Chen, G.: The simplest parameterized normal forms of Hopf and generalized Hopf bifurcations. Nonlinear Dyn. 50, 297–313 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yu, P., Yuan, Y.: Computation of simplest normal forms of differential equations associated with a double zero eigenvalue. Int. J. Bifurcation Chaos 5, 1307–1330 (2001)

    MathSciNet  Google Scholar 

  33. Yu, P., Yuan, Y.: A matching pursuit technique for computing the simplest normal forms of vector fields. J. Symbolic Computation 35, 591–615 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to acknowledge Majid Gazor’s useful discussions.

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  1. Department of Mathematics, Majlesi Branch, Islamic Azad University, Isfahan, Iran

    Hamid Gazor

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  1. Hamid Gazor
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Correspondence to Hamid Gazor.

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Gazor, H. Normal form of Bogdanov–Takens: styles and a hypernormalization method. Nonlinear Dyn 72, 499–505 (2013). https://doi.org/10.1007/s11071-012-0739-x

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