Limit Cycles for a Discontinuous Quintic Polynomial Differential System

  • Bo HuangEmail author


In this article, we study the maximum number of limit cycles for a discontinuous quintic differential system. Using the first-order averaging method, we explain how limit cycles can bifurcate from the period annulus around the center of the considered system when it is perturbed inside a class of discontinuous quintic polynomial differential systems. Our results show that the lower bound and the upper bound of the number of limit cycles, 8 and 10 respectively, that can bifurcate from the period annulus around the center.


Averaging method Center Discontinuous quintic system Limit cycle Period annulus 

Mathematics Subject Classification

34C05 34C07 



I wish to thank Xiuli Cen for her helpful suggestions and to Dongming Wang for his profound concern and encouragement. I also wish to thank the reviewers for their valuable comments that have helped to improve the presentation of the paper.


  1. 1.
    Buicǎ, A., Llibre, J.: Averaging methods for finding periodic orbits via Brouwer degree. Bull. Sci. Math. 128, 7–22 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Binyamini, G., Novikov, D., Yakovenko, S.: On the number of zeros of Abelian integrals. Invent. Math. 181, 227–289 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Coll, B., Gasull, A., Prohens, R.: Bifurcation of limit cycles from two families of centers. Discrete Contin. Dyn. Syst. 12, 275–287 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chen, F.D., Li, C., Llibre, J., Zhang, Z.H.: A unified proof on the weak Hilbert 16th problem for \(n=2\). J. Differ. Equ. 221, 309–342 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cen, X., Li, S., Zhao, Y.: On the number of limit cycles for a class of discontinuous quadratic differential systems. J. Math. Anal. Appl. 449, 314–342 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    da Cruz, L.P.C., Novaes, D.D., Torregrosa, J.: New lower bound for the Hilbert number in piecewise quadratic differential systems. J. Differ. Equ. 7, 1 (2018). Google Scholar
  7. 7.
    Euzébio, R.D., Llibre, J.: On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line. J. Math. Anal. Appl. 424, 475–486 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gavrilov, L.: The infinitesimal 16th Hilbert problem in the quadratic case. Invent. Math. 143, 449–497 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gavrilov, L., Iliev, I.D.: Quadratic perturbations of quadratic codimension-four centers. J. Math. Anal. Appl. 357, 69–76 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gautier, S., Gavrilov, L., Iliev, I.D.: Perturbations of quadratic center of genus one. Discrete Contin. Dyn. Syst. 25, 511–535 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hilbert, D.: Mathematical problems. In: Second International Congress of Mathematics (Paris 1900), Nachr. Ges. Wiss. Göttingen Math. Phys. KL. 253–297 (1900) (English transl: Bulletin American Mathematical Society), vol. 8, pp. 437–479 (1902)Google Scholar
  12. 12.
    Huang, B.: Bifurcation of limit cycles from the center of a quintic system via the averaging method. Int. J. Bifur. Chaos 27, 1750072-1–1750072-16 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ilyashenko, Y.: Centennial history of Hilbert’s 16th problem. Bull. (New Series) Am. Math. Soc. 39, 301–354 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Itikawa, J., Llibre, J., Novaes, D.D.: A new result on averaging theory for a class of discontinuous planar differential systems with applications. Rev. Mat. Iberoam. 33, 1247–1265 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Karlin, J., Studden, W.J.: T-Systems: With Applications in Analysis and Statistics. Pure and Applied Mathematics Interscience Publishers, New York (1966)zbMATHGoogle Scholar
  16. 16.
    Li, C., Llibre, J., Zhang, Z.: Weak focus, limit cycles and bifurcations for bounded quadrtic systems. J. Differ. Equ. 115, 193–223 (1995)CrossRefzbMATHGoogle Scholar
  17. 17.
    Li, J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifur. Chaos 13, 47–106 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Llibre, J., Mereu, A.C.: Limit cycles for discontinuous quadratic differential systems with two zones. J. Math. Anal. Appl. 413, 763–775 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Llibre, J., Lopes, B.D., Moraes, J.R.D.: Limit cycles for a class of continuous and discontinuous cubic polynomial differential systems. Qual. Theory Dyn. Syst. 13, 129–148 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Llibre, J., Novaes, D.D., Teixeira, M.A.: On the birth of limit cycles for non-smooth dynamical systems. Bull. Sci. Math. 139, 229–244 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Llibre, J., Mereu, A.C., Novaes, D.D.: Averaging theory for discontinuous piecewise differential systems. J. Differ. Equ. 258, 4007–4032 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Liang, H., Llibre, J., Torregrosa, J.: Limit cycles coming from some uniform isochronous centers. Adv. Nonlinear Stud. 16, 197–220 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Novaes, D.D., Torregrosa, J.: On extended Chebyshev systems with positive accuracy. J. Math. Anal. Appl. 448, 171–186 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Peng, L., Feng, Z.: Bifurcation of limit cycles from a quintic center via the second order averaging method. Int. J. Bifur. Chaos 25, 1550047-1–18 (2015)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Wang, D.: Mechanical manipulation for a class of differential systems. J. Symb. Comput. 12, 233–254 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.LMIB-School of Mathematics and Systems ScienceBeihang UniversityBeijingPeople’s Republic of China
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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