Nonlinear Dynamics

, Volume 93, Issue 2, pp 749–766 | Cite as

Bifurcation and evolution of a forced and damped Duffing system in two-parameter plane

  • Jian-Fei Shi
  • Yan-Long Zhang
  • Xiang-Feng Gou
Original Paper


A general method to calculate multi-parameter bifurcation diagram in the parameter space is designed based on top Lyapunov exponent and Floquet multiplier to study the effect of different combinations of system parameters on the system’s dynamics. Bifurcation and chaos of the forced and damped Duffing system in two-parameter plane are investigated by using the method designed in this work. The correlation and matching laws of the Duffing system between dynamic performance and system parameters are analyzed. The effect of different types of bifurcation curves on the bifurcating of coexisting attractors is investigated according to basins of attraction, bifurcation diagrams, top Lyapunov exponent spectrums, phase portraits, Poincaré maps, and Floquet multipliers. The evolution of various bifurcation curves and codimension-two bifurcation in the parametric plane is studied as well. Coexisting attractors are found in the parameter plane. The results indicate that the different bifurcating curves are selective for the bifurcation of coexisting attractors. Both the pitchfork bifurcation curve and the period-doubling bifurcation curve just change the stability of some of the coexisting attractors, but have no effect on the stability of the other part of the attractors. The saddle-node bifurcation curve has an effect on the stability of all the coexisting attractors. A series of period-doubling bifurcation curves and codimension-two bifurcation points lead to chaos existence region in two-parameter plane. The special evolution of bifurcation points and bifurcation curves in two-parameter plane with the change of the system parameter is observed. The codimension-two bifurcation points and bifurcation curves play an important role in understanding nonlinear dynamics of the system in the parametric plane. The work in this study emphasizes the importance of the different combinations of system parameters on the system dynamics.


Duffing system Codimension-two bifurcation point Multi-parameter bifurcation Basins of attraction Floquet multiplier 



This investigation is financially supported by the National Natural Science Foundation of China (Grant No. 51365025) and by the Natural Science Key Foundation of Tianjin, China (Grant No. 16JCZDJC38500).

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest.


  1. 1.
    Bi, Q.S., Yu, P.: Double Hopf bifurcations and Chaos of a nonlinear vibration system. Nonlin. Dyn. 19, 313–332 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Schupp, G.: Bifurcation analysis of railway vehicles. Multibody Syst. Dyn. 15, 25–50 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Xu, W., Li, R.H., Li, S.: Resonance and bifurcation in a nonlinear Duffing system with cubic coupled terms. Nonlin. Dyn. 46, 211–221 (2006)CrossRefzbMATHGoogle Scholar
  4. 4.
    Zhong, S., Chen, Y.S.: Bifurcation of piecewise-linear nonlinear vibration system of vehicle suspension. Appl. Math. Mech. 30, 677–684 (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Kaslik, E., Balint, St: Bifurcation analysis for a two-dimensional delayed discrete-time Hopf field neural network. Chaos Solitons Fractals 34, 1245–1253 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Peng, M.S., Jiang, Z.H., Jiang, X.X.: Multi-stability and complex dynamics in a simple discrete economic model. Chaos Solitons Fractals 41, 671–687 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Algaba, A., Gamero, E., Rodrguez, A.J.: A bifurcation analysis of a simple electronic circuit. Commun. Nonlin. Sci. Numer. Simul. 10, 169–178 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Yi, N., Zhang, Q.L., Liu, P., et al.: Codimension-two bifurcations analysis and tracking control on a discrete epidemic model. J. Syst. Sci. Complex. 24, 1033–1056 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ruan, S.G., Wang, W.D.: Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Differ. Equ. 188, 135–163 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Luo, G.W., Lv, X.H., Shi, Y.Q.: Vibro-impact dynamics of a two-degree-of freedom periodically-forced system with a clearance: diversity and parameter matching of periodic-impact motions. Int. J. Nonlin. Mech. 65, 173–195 (2014)CrossRefGoogle Scholar
  11. 11.
    Luo, G.W., Zhu, X.F., Shi, Y.Q.: Dynamics of a two-degree-of freedom periodically-forced system with a rigid stop: diversity and evolution of periodic-impact motions. J. Sound Vib. 334, 338–362 (2015)CrossRefGoogle Scholar
  12. 12.
    Wang, H., Yu, Y.G., Zhao, R., et al.: Two-parameter bifurcation in a two-dimension simplified Hodgkin–Huxley model. Commun. Nonlin. Sci. Numer. Simul. 18, 184–193 (2013)CrossRefzbMATHGoogle Scholar
  13. 13.
    Nguyen, V.L.: On two-parameter global bifurcation of periodic solutions to a class of differential variational inequalities. Nonlin. Anal. Theory Method Appl. 122, 83–99 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Govaerts, W., Khoshsiar, G.R., Kuznetsov, Y.A.: Numerical methods for two-parameter local bifurcation analysis of maps. SIAM J. Sci. Comput. 29, 2644–2667 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Duan, L.X., Lu, Q.S.: Condimension-two bifurcation analysis in Hindmarsh–Rose model with two parameters. Chin. Phys. Lett. 6, 1325 (2005)Google Scholar
  16. 16.
    Mason, J.F., Piiroinen, P.T.: Interactions between global and grazing bifurcation in an impacting system. Chaos 21, 013113 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mason, J.F., Piiroinen, P.T.: Saddle-point solutions and grazing bifurcations in an impacting system. Chaos 21, 013106 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mason, J.F., Humphries, N., Piiroinen, P.T.: Numerical analysis of codimension-one, -two, -three bifurcations in a periodically-forced impact oscillator with two discontinuity surfaces. Math. Comput. Simul. 95, 98–110 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Luo, G.W., Shi, Y.Q., Jiang, C.X., et al.: Diversity evolution and parameter matching of periodic-impact motions of a periodically forced system with a clearance. Nonlin. Dyn. 78, 2577–2604 (2014)CrossRefGoogle Scholar
  20. 20.
    Qin, Z.H., Chen, Y.S.: Sigularity analysis of Duffing-Van der pol system with two bifurcation parameters under multi-frequency excitations. Appl. Math. Mech. 31, 1019–1026 (2010)CrossRefzbMATHGoogle Scholar
  21. 21.
    Qin, Z.H., Chen, Y.S.: Sigular analysis of bifurcation systems with two parameters. Acta. Mechanica Sinica 26, 501–507 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhang, C., Bi, Q.S., Han, X.J.: On two-parameter bifurcation analysis of switched system composed of Duffing and Van der pol oscillators. Commun. Nonlin. Sci. Numer. Simul. 19, 750–757 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Brzeski, P., Perlikowski, P., Yanchuk, S., et al.: The dynamics of the pendulum suspended on the forced Duffing oscillator. J. Sound Vib. 331, 5347–5357 (2012)CrossRefGoogle Scholar
  24. 24.
    Dowell, E.H.: Chaotic oscillations in mechanical systems. Comput. Mech. 3, 199–216 (1988)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rusinek, R., Weremczuk, A., Kecik, K., et al.: Dynamics of a time delayed Duffing oscillator. Int. J. Nonlin. Mech. 65, 98–106 (2014)CrossRefGoogle Scholar
  26. 26.
    Yang, K.L., Wang, C.J.: Two-parameter bifurcations in a discontinuous map with a variable gap. Nonlin. Dyn. 87, 303–311 (2017)CrossRefGoogle Scholar
  27. 27.
    Jafari, S., Pham, V.T., Golpayegani, S.M.R.H., et al.: The relationship between Chaotic maps and some chaotic systems with hidden attractors. Int. J. Bifurc. Chaos 26, 650211 (2016)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Zhao, H.T., Lin, Y.P., Dai, Y.X.: Hopf bifurcation and hidden attractors of a delay-coupled Duffing oscillator. Int. J. Bifurc. Chaos 25, 1550162 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Brezetskyi, S., Dudkowski, D., Kapitaniak, T.: Rare and hidden attractors in Van der Pol-Duffing oscillators. Eur. Phys. J. Spec. Top. 224, 1459–1467 (2015)CrossRefGoogle Scholar
  30. 30.
    Barbara, B.O., Kapitaniak, T.: Co-existing attractors of impact oscillator. Chaos Solitons Fractals 9, 1439–1443 (1998)CrossRefzbMATHGoogle Scholar
  31. 31.
    Brzeski, P., Kapitaniak, T., Perlikowski, P.: Analysis of transitions between different ringing schemes of the church bell. Int. J. Impact Eng. 86, 57–66 (2015)CrossRefGoogle Scholar
  32. 32.
    de Souza, S.L.T., Caldas, I.L.: Basins of attraction and transient chaos in a gear-rattling model. J. Vib. Control. 7, 849–862 (2001)CrossRefzbMATHGoogle Scholar
  33. 33.
    Silvio, L.T., de Souza, S.L.T., Caldas, I.L.: Controlling chaotic orbits in mechanical systems with impacts. Chaos Solitons Fractals 19, 171–178 (2004)CrossRefzbMATHGoogle Scholar
  34. 34.
    Li, Q.H., Lu, Q.S.: Coexisting periodic orbits in vibro-impacting dynamical systems. Appl. Math. Mech. 24, 261–273 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Gou, X.F., Zhu, L.Y., Chen, D.L.: Bifurcation and chaos analysis of spur gear pair in two-parameter plane. Nonlin. Dyn. 79, 2225–2235 (2015)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Jian-Fei Shi
    • 1
    • 2
  • Yan-Long Zhang
    • 3
  • Xiang-Feng Gou
    • 1
    • 2
    • 3
  1. 1.School of Mechanical EngineeringTianjin Polytechnic UniversityTianjinChina
  2. 2.Tianjin Key Laboratory of Advanced Mechatronics Equipment TechnologyTianjinChina
  3. 3.School of Mechanical EngineeringLanzhou Jiaotong UniversityLanzhouChina

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