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Analytical approximation of cuspidal loops using a nonlinear time transformation method

Author

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  • Qin, Bo-Wei
  • Chung, Kwok-Wai
  • Algaba, Antonio
  • Rodríguez-Luis, Alejandro J.

Abstract

In this work we consider cuspidal loops, i.e., homoclinic orbits to cuspidal singular points. We develop an iterative procedure, founded on the nonlinear time transformation method, to estimate such codimension-three global bifurcations up to any wanted order, not only in the space of parameters but also in the phase plane. As far as we know, this is the first time in the literature that this theoretical result is achieved for these global connections. The existence and uniqueness of the perturbed solution obtained are proved. To illustrate the effectiveness of the method we study cuspidal loops in two normal forms of degenerate Takens–Bogdanov bifurcations. Excellent agreement is found between our analytical predictions and the corresponding numerical continuations.

Suggested Citation

  • Qin, Bo-Wei & Chung, Kwok-Wai & Algaba, Antonio & Rodríguez-Luis, Alejandro J., 2020. "Analytical approximation of cuspidal loops using a nonlinear time transformation method," Applied Mathematics and Computation, Elsevier, vol. 373(C).
    • Handle: RePEc:eee:apmaco:v:373:y:2020:i:c:s0096300320300114
    DOI: 10.1016/j.amc.2020.125042
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    References listed on IDEAS

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    1. Wu, Zeyan & Li, Jianjuan & Liu, Shuying & Zhou, Liuting & Luo, Yang, 2019. "A spatial predator–prey system with non-renewable resources," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 381-391.
    2. Yang, Junmin & Han, Maoan, 2011. "Limit cycle bifurcations of some Liénard systems with a cuspidal loop and a homoclinic loop," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 269-289.
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    Cited by:

    1. Xue, Miao & Gou, Junting & Xia, Yibo & Bi, Qinsheng, 2021. "Computation of the normal form as well as the unfolding of the vector field with zero-zero-Hopf bifurcation at the origin," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 377-397.

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