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Nonlinear Dynamics

, Volume 93, Issue 4, pp 2201–2212 | Cite as

Limit cycles for a class of discontinuous piecewise generalized Kukles differential systems

  • Ana C. Mereu
  • Regilene Oliveira
  • Camila A. B. Rodrigues
Original Paper
  • 91 Downloads

Abstract

The present paper is devoted to study an estimative to the number of limit cycles which bifurcate from the periodic orbits of the linear center \(\dot{x}=y, \dot{y}=-x\) by the averaging method of first order when it is perturbed inside a class of discontinuous generalized Kukles differential systems defined in 2l-zones, \(l=1,2,3,\ldots \), in the plane.

Keywords

Piecewise polynomial differential systems Limit cycles Averaging theory Kukles systems 

Mathematics Subject Classification

37G15 34A36 

Notes

Acknowledgements

The first author is partially supported by FAPESP Grant 2012/18780-0 and the CNPq Grant 449655/2014-8. The second author is partially supported by FAPESP Grant “Projeto Temático” 2014/00304-2. The third author was supported by CNPq Fellowship Number 140292/2017-9. The authors are grateful to the referees for their valuable suggestions to improve this manuscript.

References

  1. 1.
    Akhmet, M.U., Arugaslan, D.: Bifurcation of a non-smooth planar limit cycle from a vertex. Nonlinear Anal. 71, 2723–2733 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillators. Dover, New York (1966)zbMATHGoogle Scholar
  3. 3.
    Buica, A., Llibre, J.: Averaging methods for finding periodic orbits via Brouwer degree. Bull. Sci. Math. 128, 7–22 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Buica, A., Françoise, J.-P., Llibre, J.: Periodic solutions of nonlinear periodic differential systems with a small. Commun. Pure Appl. Anal. 6, 103–111 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Browder, F.: Fixed point theory and nonlinear problems. Bull. Am. Math. Soc. 9, 1–39 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chavarriga, J., Sáez, E., Szántó, I., Grau, M.: Coexistence of limit cycles and invariant algebraic curves for a Kukles system. Nonlinear Anal. 59, 673–693 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gradshteyn, I.S., Ryshik, I.M.: Indefinite Integrals of Elementary Functions, section 2.5-2.6 Trigonometric Functions., Chap. 2. In: Jeffrey, A. (ed.) Table of Integrals, Series and Products, vol. 5. Academic Press, New York (1994)Google Scholar
  8. 8.
    Henry, P.: Differential equations with discontinuous right-hand side for planning procedures. J. Econ. Theory 4, 545–551 (1972)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kukles, I.S.: Sur quelques cas de distinction entre un foyer et un centre. Dokl. Akad. Nauk. SSSR 43, 208–211 (1944)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kunze, M., Kupper, T.: Qualitative bifurcation analysis of a non-smooth friction-oscillator model. Z. Angew. Math. Phys. 48, 87–101 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Llibre, J., Mereu, A.C.: Limit cycles for generalized Kukles polynomial differential systems. Nonlinear Anal. 74, 1261–1271 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Llibre, J., Mereu, A.C.: Limit cycles for a class of discontinuous generalized Lienard polynomial differential equations. Electron. J. Differ. Equ. 195, 8 (2013)zbMATHGoogle Scholar
  13. 13.
    Llibre, J., Novaes, D., Teixeira, M.A.: Higher order averaging theory for finding periodic solutions via Brouwer degree. Nonlinearity 27, 563–583 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Llibre, J., Novaes, D., Teixeira, M.A.: Maximum number of limit cycles for certain piecewise linear dynamical systems. Nonlinear Dyn. 82(3), 11591175 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Llibre, J., Mereu, A.C., Novaes, D.: Averaging theory for discontinuous piecewise differential systems. J. Differ. Equ. 258, 4007–4032 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Llibre, J., Teixeira, M.A.: Limit cycles for \(m\)-piecewise discontinuous polynomial Liénard differential equations. Z. Angew. Math. Phys. 66, 51–66 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Applications. Applied Mathematical Sciences, vol. 19. Springer, New York (1976)CrossRefzbMATHGoogle Scholar
  18. 18.
    Poincaré, H.: Memoire sur les coubes definies par une equation differentielle I. J. Math. Pures Appl. 7, 375–422 (1881)zbMATHGoogle Scholar
  19. 19.
    Poincaré, H.: Memoire sur les coubes definies par une equation differentielle II. J. Math. Pures Appl. 8, 251–296 (1882)zbMATHGoogle Scholar
  20. 20.
    Poincaré, H.: Memoire sur les coubes definies par une equation differentielle III. J. Math. Pures Appl. 1, 167–244 (1885)Google Scholar
  21. 21.
    Poincaré, H.: Memoire sur les coubes definies par une equation differentielle IV. J. Math. Pures Appl. 2, 155–217 (1886)Google Scholar
  22. 22.
    Sadovskii, A.P.: Cubic systems of nonlinear oscillations with seven limit cycles. Differ. Uravn. SSSR 39, 472–481 (2003)MathSciNetGoogle Scholar
  23. 23.
    Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems. Applied Mathematical Sciences, vol. 59. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
  24. 24.
    Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems. Universitext. Springer, New York (1991)Google Scholar
  25. 25.
    Zang, H., Zhang, T., Tian, Y.-C., Tadé, M.: Limit cycles for the Kukles system. J. Dyn. Control Syst. 14, 283–298 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Física, Quimica e MatemáticaUFSCarSorocabaBrazil
  2. 2.Departamento de MatemáticaICMC-Universidade de São PauloSão CarlosBrazil

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