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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
  • 300 Downloads

Abstract

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.

Keywords

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

Notes

Acknowledgements

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.

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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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