Qualitative Theory of Dynamical Systems

, Volume 18, Issue 3, pp 1077–1112 | Cite as

Bifurcation of Periodic Orbits of a Three-Dimensional Piecewise Smooth System

  • Shenglan Xie
  • Maoan HanEmail author
  • Xuepeng Zhao


We consider a three-dimensional piecewise smooth system having an invariant surface and a closed orbit of multiplicity k on the invariant surface. By using bifurcation techniques and analyzing the number of solutions of bifurcation equations, we study certain bifurcation phenomena near the multiple k closed orbit and obtain some conditions for the existence of periodic orbits bifurcated from it. Moreover, our partial analytical results are demonstrated by some examples.


Bifurcation Three-dimensional system Piecewise smooth Closed orbit 



We would like to thank the referees for their valuable suggestions which helped to improve the presentation of the paper.

Compliance with Ethical Standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this article.


  1. 1.
    Carmona, V., Fernandez-Garcia, S., Freire, E.: Periodic orbits for perturbations of piecewise linear systems. J. Differ. Equ. 250, 2244–2266 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Carmona, V., Fernandez-Garcia, S., Freire, E.: Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete Continuous Dyn. Syst. Ser. A 35, 59–72 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chow, S.N., Hale, J.K.: Methods of Bifurcation Theory. Springer, New York (1982)CrossRefGoogle Scholar
  4. 4.
    di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise Smooth Dynamical Systems, Theory and Applications. Springer, London (2008)zbMATHGoogle Scholar
  5. 5.
    Du, Z., Li, Y., Zhang, W.: Bifurcation of periodic orbits in a class of planar Filippov systems. Nonlinear Anal. 69, 3610–3628 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fillppov, A.F.: Differential Equation with Discontinuous Right-Hand Sides. Kluwer Academic, Dordrecht (1988)CrossRefGoogle Scholar
  7. 7.
    Han, M.: Bifurcation of periodic orbits for three dimensional system. Acta Math. Appl. Sin. 18(4), 528–537 (1995). (in Chinese)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Han, M.: Theory of Periodic Solutions and Bifurcation of Dynamical Systems. Science Publishing House, Beijing (2002). (in Chinese)Google Scholar
  9. 9.
    Han, M.: Bifurcation Theory of Limit Cycles. Science Press, Beijing (2013)Google Scholar
  10. 10.
    Hale, J.K.: Ordinary Differential Equations. Robert E. Krieger Publishing Co, New York (1980)zbMATHGoogle Scholar
  11. 11.
    Kunze, M.: Non-Smooth Dynamical Systems, Theory and Applications. Springer, Berlin (2000)CrossRefGoogle Scholar
  12. 12.
    Liberzon, D.: Switching in Systems and Control. Birkhäuser, Boston (2003)CrossRefGoogle Scholar
  13. 13.
    Li, C., Ma, Z., Zhou, Y.: Periodic orbits in 3-dimensional systems and pplication to a perturbed Volterra system. J. Differ. Equ. 260, 2750–2762 (2016)CrossRefGoogle Scholar
  14. 14.
    Li, H., Ma, L., Zhu, W.: Chaotic behavior and subharmonic bifurcations for the Duffing-van Der Pol oscillator. J. Nonlinear Model. Anal. 1(2), 237–250 (2019)Google Scholar
  15. 15.
    Liang, F., Han, M.: Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems. Chaos Solitons Fractals 45, 454–464 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Liu, X., Han, M.: Bifurcation of periodic orbits of a three-dimensional system. Chin. Ann. Math. 26B(2), 253–274 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Liu, X., Han, M.: Bifurcation of limit cycles by perturbing piecewise Haniltionian systens. Int. J. Bifurc. Chaos Appl. Sci. Eng. 5, 1–12 (2010)Google Scholar
  18. 18.
    Llibre, J., Ponce, E., Ros, J.: Algebraic determination of limit cycles in a family of three-dimensional piecewise linear differential systems. Nonlinear Anal. 74, 6712–6727 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Llibre, J., Teruel, A.E.: Introduction to the Qualitative Theory of Differential Systems. Birkhäuser, Spain (2014)CrossRefGoogle Scholar
  20. 20.
    Perko, L.M.: Multiple limit cycle bifurcation surfaces and global families of multiple limit cycles. J. Differ. Equ. 122, 89–113 (1995)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tian, H., Han, M.: Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems. J. Differ. Equ. 263, 7448–7474 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Xiong, Y., Han, M.: Limit cycle bifurcations in a class of perturbed piecewise smooth systems. Appl. Math. Comput. 242, 47–64 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Yang, J., Han, M., Huang, W.: On Hopf bifurcation of piecewise planar Hamiltonian systems. J. Differ. Equ. 26B(250), 1026–1051 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Yu, P., Han, M., Bai, Y.: Dynamics and bifurcation study on an extended Lorenz system. J. Nonlinear Model. Anal. 1(1), 107–128 (2019)Google Scholar
  25. 25.
    Zhang, J., Feng, B.: Geometric Theory and Bifurcation Problems in Ordinary Differetial Equations. Science Publishing House, Beijing (1980). (in Chinese)Google Scholar
  26. 26.
    Zhang, Z., Ding, T., Huang, W., Dong, Z.: Qualitative Theory of Differetial Equations, Translations of Mathematical Monographs. American Mathematical Society, Providence, Rl (1992)Google Scholar
  27. 27.
    Zhao, A., Li, M., Han, M.: The Basic Theory of Differential Equation. Science Publishing House, Beijing (2011). (in Chinese)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaPeople’s Republic of China
  2. 2.Department of MathematicsShanghai Normal UniversityShanghaiPeople’s Republic of China
  3. 3.Department of Mathematics and Physics, Nanhu CollegeJiaxing UniversityJiaxingPeople’s Republic of China

Personalised recommendations