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Kukles Systems of Degree Three with Global Centers

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Abstract

A global center of a vector field in the plane is an equilibrium point such that the whole plane with the exception of the equilibrium point is filled with periodic orbits. We classify all Kukles systems of degree three that have global centers.

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References

  • Christopher, C.J., Lloyd, N.G.: On the paper of Jin and Wang concerning the conditions for a centre in certain cubic systems. Bull. Lond. Math. Soc. 22, 5–12 (1990)

    Article  MATH  Google Scholar 

  • Cima, A., Llibre, J.: Algebraic and topological classification of the homogeneous cubic vector fields in the plane. J. Math. Anal. Appl. 147, 420–448 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Conti, R.: Centers of planar polynomial systems: A review. Matematiche 53, 207–240 (1998)

    MathSciNet  MATH  Google Scholar 

  • Dias, F.S., Mello, L.F.: Globally asymptotically stable equilibrium points in Kukles systems. Qual. Theory Dyn. Syst. 19(94), 7 (2020)

    MathSciNet  MATH  Google Scholar 

  • Dias, F.S., Ribeiro, R.M., Valls, C.: Global phase portraits for a planar \(\mathbb{Z} _{2}\)-equivariant Kukles systems of degree \(3\). Int. J. Bifur. Chaos Appl. Sci. Eng. 30(2050164), 22 (2020)

    MATH  Google Scholar 

  • Dias, F.S., Ribeiro, R.M., Valls, C.: Global phase portraits for the Kukles systems of degree \(3\) with \(\mathbb{Z} _{2}\)-reversible symmetries. Int. J. Bifur. Chaos Appl. Sci. Eng. 31(2150083), 17 (2021)

    MATH  Google Scholar 

  • Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems. Springer, (2006)

  • Galeotti, M., Villarini, M.: Some properties of planar polynomial systems of even degree. Ann. Mat. Pura Appl. 161, 299–313 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • García-Saldaña, J.D., Llibre, J., Valls, C.: Linear type global centers of linear systems with cubic homogeneous nonlinearities. Rend. Circ. Mat. Palermo 69, 771–785 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Kukles, I.S.: Sur quelques cas de distinction entre un foyer et un centre. Dokl. Akad. Nauk. SSSR 42, 208–211 (1944)

    MathSciNet  MATH  Google Scholar 

  • Llibre, J., Valls, C.: Polynomial differential systems with even degree have no global centers. J. Math. Anal. Appl. 503(125281), 5 (2021)

    MathSciNet  MATH  Google Scholar 

  • Llibre, J., & Valls, C.: Reversible global centers with quintic homogeneous nonlinearities (Preprint) (2023)

  • Pearson, J.M., Lloyd, N.G.: Kukles revisited: Advances in computing techniques. Comput. Math. Appl. 60, 2797–2805 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the reviewer for his/her suggestions that helped them to improve the presentation of the paper. Fabio Scalco Dias and Luis Fernando Mello are partially supported by Fundação de Amparo à Pesquisa do Estado de Minas Gerais [Grant number APQ-01158-17] and Fundação de Amparo à Pesquisa do Estado de São Paulo [Grant number 2019/07316-0]. Luis Fernando Mello is partially supported by Fundação de Amparo à Pesquisa do Estado de Minas Gerais [Grant number APQ-01105-18] and by Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grant number 311921/2020-5]. Claudia Valls is partially supported by FCT/Portugal through CAMGSD, IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020.

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Authors and Affiliations

  1. Instituto de Matemática e Computação, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, Itajubá, Minas Gerais, 37500-903, Brazil

    Fabio Scalco Dias & Luis Fernando Mello

  2. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001, Lisbon, Portugal

    Claudia Valls

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  1. Fabio Scalco Dias
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  2. Luis Fernando Mello
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  3. Claudia Valls
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Correspondence to Fabio Scalco Dias.

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Dias, F.S., Mello, L.F. & Valls, C. Kukles Systems of Degree Three with Global Centers. Bull Braz Math Soc, New Series 54, 46 (2023). https://doi.org/10.1007/s00574-023-00363-7

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  • DOI: https://doi.org/10.1007/s00574-023-00363-7

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