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A nonlinear time transformation method to compute all the coefficients for the homoclinic bifurcation in the quadratic Takens–Bogdanov normal form

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Abstract

In this paper, we present an algorithm based on the nonlinear time transformation method to approximate homoclinic orbits in planar autonomous nonlinear oscillators. With this approach, a unique perturbation solution up to any desired order can be obtained for them using trigonometric functions. To demonstrate its efficiency, the method is applied to calculate the homoclinic connection, both in the phase space and in the parameter space, of the versal unfolding of the nondegenerate Takens–Bogdanov singularity. Our approach considerably improves the results obtained so far by other methods (Melnikov, Poincaré–Lindstedt, regular perturbations, multiple scales, etc.). The approximations achieved to different orders are confirmed by numerical continuation.

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Authors and Affiliations

  1. Departamento de Ciencias Integradas, Centro de Estudios Avanzados en Física, Matemática y Computación, 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, Sevilla, Spain

    Alejandro J. Rodríguez-Luis

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  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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Correspondence to Alejandro J. Rodríguez-Luis.

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The authors thank the reviewers for their careful reading of the manuscript and their constructive remarks. 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. A nonlinear time transformation method to compute all the coefficients for the homoclinic bifurcation in the quadratic Takens–Bogdanov normal form. Nonlinear Dyn 97, 979–990 (2019). https://doi.org/10.1007/s11071-019-05025-2

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