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Explicit formulas for computing the normal form of Bogdanov–Takens bifurcation in delay differential equations

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Abstract

This paper considers the computation of normal form associated with codimension-two Bogdanov–Takens (BT) bifurcation in delay differential equations. The main attention is focused on dynamical systems described by delay differential equations having a double-zero eigenvalue with geometric multiplicity one, which is usually called non-semisimple double-zero eigenvalue. Explicit formulas for computing the normal form of such systems with two unfolding parameters are obtained by applying center manifold reduction and the method of normal forms. In particular, the normal form associated with the flow on a center manifold up to third-order terms are derived. As an application, an Oregonator oscillator having such a BT singularity with delay is analyzed using our explicit formulas.

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Acknowledgements

The authors would like to express their gratitude for valuable comments on this manuscript from the referees and the editor.

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Authors and Affiliations

  1. Department of Mathematics, Northeast Forestry University, Harbin, 150040, People’s Republic of China

    Chunrui Zhang

  2. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Baodong Zheng

Authors
  1. Chunrui Zhang
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  2. Baodong Zheng
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Correspondence to Chunrui Zhang.

Additional information

This work is supported by the Natural Science Foundation of Heilongjiang Province (No. A2015016).

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Zhang, C., Zheng, B. Explicit formulas for computing the normal form of Bogdanov–Takens bifurcation in delay differential equations. Nonlinear Dyn 89, 1187–1194 (2017). https://doi.org/10.1007/s11071-017-3509-y

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  • DOI: https://doi.org/10.1007/s11071-017-3509-y

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