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Revisiting the analysis of a codimension-three Takens–Bogdanov bifurcation in planar reversible systems

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Abstract

A family of planar nilpotent reversible systems with an equilibrium point located at the origin has been studied in the recent paper Algaba et al. (Nonlinear Dyn 87:835–849, 2017). The authors investigate the candidate for an universal unfolding of a codimension-three degenerate case which exhibits a rich bifurcation scenario. However, a codimension-two point is missed in one of the two cases considered. In this paper, we complete the bifurcation set demonstrating the existence of this new organizing center and analyzing the dynamics generated in this case. Moreover, by means of the Melnikov theory, we study analytically four different global connections present in the system under consideration. Numerical continuation of the bifurcation curves illustrates that the first-order analytical approximation is valid in a large region of the parameter space.

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References

  1. Algaba, A., Freire, E., Gamero, E., García, C.: A bifurcation analysis of planar nilpotent reversible systems. Nonlinear Dyn. 87, 835–849 (2017)

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Authors and Affiliations

  1. Departamento de Ciencias Integradas, Centro de Investigación de Física Teórica y Matemática FIMAT, Universidad de Huelva, 21071, Huelva, Spain

    Antonio Algaba

  2. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    Kwok-Wai Chung & Bo-Wei Qin

  3. Departamento de Matemática Aplicada II, E.T.S. Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Seville, Spain

    Alejandro J. Rodríguez-Luis

Authors
  1. Antonio Algaba
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  2. Kwok-Wai Chung
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  3. Bo-Wei Qin
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  4. Alejandro J. Rodríguez-Luis
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Corresponding author

Correspondence to Alejandro J. Rodríguez-Luis.

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This work has been partially supported by the Ministerio de Economía y Competitividad, Plan Nacional I+D+I co-financed with FEDER funds, in the frame of the projects MTM2014-56272-C2 and MTM2017-87915-C2-1-P and by the Consejería de Economía, Innovación, Ciencia y Empleo de la Junta de Andalucía (FQM-276, TIC-0130 and P12-FQM-1658). It was also supported by the Strategic Research Grant of the City University of Hong Kong (Grant No. 7004671). B.W.Q. is also grateful to the Instituto de Matemáticas de la Universidad de Sevilla (IMUS) and to the Centro de Estudios Avanzados en Física, Matemática y Computación de la Universidad de Huelva (CEAFMC) for collaborating in the financing of his research stays in Seville and Huelva.

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Algaba, A., Chung, KW., Qin, BW. et al. Revisiting the analysis of a codimension-three Takens–Bogdanov bifurcation in planar reversible systems. Nonlinear Dyn 96, 2567–2580 (2019). https://doi.org/10.1007/s11071-019-04941-7

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  • DOI: https://doi.org/10.1007/s11071-019-04941-7

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