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Zero-Hopf Periodic Orbit of a Quadratic System of Differential Equations Obtained from a Third-Order Differential Equation

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Abstract

We study the zero-Hopf bifurcation of the third-order differential equations

$$\begin{aligned} x^{\prime \prime \prime }+ (a_{1}x+a_{0})x^{\prime \prime }+ (b_{1}x+b_{0})x^{\prime }+x^{2} =0, \end{aligned}$$

where \(a_{0}\), \(a_{1}\), \(b_{0}\) and \(b_{1}\) are real parameters. The prime denotes derivative with respect to an independent variable t. We also provide an estimate of the zero-Hopf periodic solution and their kind of stability. The Hopf bifurcations of these differential systems were studied in [5], here we complete these studies adding their zero-Hopf bifurcations.

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References

  1. Bamon, R.: Quadratic vector fields in the plane have a finite number of limit cycles. Inst. Hautes Etudes Sci. Publ. Mat. 64, 111–142 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Buica, A.: On the equivalence of the Melnikov functions method and the averaging method. Qual. Theory Dyn. Syst. doi:10.1007/s12346-016-0216-x (in press )

  3. Buzzi, C.A., Llibre, J., Medrado, J.C., Torregrosa, J.: Bifurcation of limit cycles from a center in \({\mathbb{R}}^{4}\) in resonance 1: N. Dyn. Sys. Int. J. 24, 123–137 (2009)

    Article  MATH  Google Scholar 

  4. Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifur. Chaos 9, 1465–1466 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dias, F.S., Mello, L.F.: Nonlinear analysis of a quadratic system obtained from a scalar third order differential equation. Electron. J. Diff. Equ. 161, 1–25 (2010)

    MATH  Google Scholar 

  6. Ecalle, J.: Introduction Aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac (French). Hermann, Paris (1992)

    MATH  Google Scholar 

  7. Ferragut, A., Llibre, J., Pantazi, C.: Polynomial vector fields in \({\mathbb{R}}^{3}\) with infinitely many limit cycles. Int. J. Bifur. Chaos 23, 1350029–1350110 (2013)

    Article  MATH  Google Scholar 

  8. Giné, J., Grau, M., Llibre, J.: Averaging theory at any order for computing periodic orbits. Phys. D 250, 58–65 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Giné, J., Llibre, J.: Limit cycles of cubic polynomial vector fields via the averaging theory. Nonlinear Anal. 66, 1707–1721 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Han, M., Romanovski, V.G., Zhang, X.: Equivalence of the Melnikov function method and the averaging method. Qual. Theory Dyn. Syst. 15, 471–479 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ilyashenko, Y.: Finiteness Theorems for Limit Cycles. American Mathematical Society, Providence (1993)

    Google Scholar 

  12. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  13. Lu, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifur. Chaos 12, 659–661 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, C., Liu, T., Liu, L., Liu, K.: A new chaotic. Chaos Solitons Fractals 22, 1031–1038 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Llibre, J., Makhlouf, A.: Periodic solutions for a class of non-autonomous Newton differential equations. Differ. Equ. Dyn. Syst. (2017). doi:10.1007/s12591-016-0333-7

    MATH  Google Scholar 

  16. Llibre, J., Mereu, A.C., Teixeira, M.A.: Limit cycles of resonant four-dimensional polynomial system. Dyn. Syst. Int. J. 25, 145–158 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Llibre, J., Novaes, D.D., Teixeira, M.A.: Higher order averaging theory for finding periodic solutions via Brouwer degree. Nonlinearity 27, 563–583 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Llibre, J., Moeckel, R., Simó, C.: Central Configurations, Periodic Orbits and Hamiltonian Systems, Advances Courses in Math. CRM Barcelona, Bellaterra (2015)

    Book  Google Scholar 

  19. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  MATH  Google Scholar 

  20. Rikitake, T.: Oscillations of a system of disk dynamos. Proc. R. Camb. Philos. Soc. 54, 89–105 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rössler, E.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976)

    Article  MATH  Google Scholar 

  22. Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems. Applied Mathematical Sciences 59, 2nd edn. Springer, New York (2007)

    MATH  Google Scholar 

  23. Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems. Springer, New York (1996)

    Book  MATH  Google Scholar 

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Acknowledgements

We thank to the reviewers their comments which help us to improve the presentation of this paper. The first author is partially supported by a FEDER-MINECO Grant MTM2016-77278-P, a MINECO Grant MTM2013-40998-P, and an AGAUR Grant number 2014SGR-568.

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Authors and Affiliations

  1. Departament de Matematiques, Universitat Autònoma de Barcelona, Bellaterra, 08193, Barcelona, Catalonia, Spain

    Jaume Llibre

  2. Department of Mathematics, University of Annaba, Elhadjar 23, Annaba, Algeria

    Ammar Makhlouf

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  1. Jaume Llibre
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  2. Ammar Makhlouf
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Correspondence to Jaume Llibre.

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Llibre, J., Makhlouf, A. Zero-Hopf Periodic Orbit of a Quadratic System of Differential Equations Obtained from a Third-Order Differential Equation. Differ Equ Dyn Syst 27, 75–82 (2019). https://doi.org/10.1007/s12591-017-0375-5

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