Nonlinear Dynamics

, Volume 82, Issue 1–2, pp 643–662 | Cite as

Subharmonic Melnikov method of four-dimensional non-autonomous systems and application to a rectangular thin plate

  • M. Sun
  • W. Zhang
  • J. E. Chen
  • M. H. Yao
Original Paper


The existence and bifurcations of the subharmonic orbits for four-dimensional non-autonomous nonlinear systems are investigated in this paper. The improved subharmonic Melnikov method is presented by using the periodic transformations and Poincaré map. The theoretical results and the formulas are obtained, which can be used to analyze the subharmonic dynamic responses of four-dimensional non-autonomous nonlinear systems. The improved subharmonic Melnikov method is used to investigate the subharmonic orbits of a simply supported rectangular thin plate under combined parametric and external excitations for verifying the validity and applicability of the method. The theoretical results indicate that the subharmonic orbits can occur in the rectangular thin plate with 1:1 and 1:2 internal resonances. The results of numerical simulation also indicate the existence of the subharmonic orbits for the rectangular thin plate, which can verify the analytical predictions.


Subharmonic Melnikov method Bifurcation Periodic solution Rectangular plate 



This research was supported by the National Natural Science Foundation of China (No. 11402165 and 11402170). The authors would like to thank the referees for their valuable suggestions.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of ScienceTianjin Chengjian UniversityTianjinPeople’s Republic of China
  2. 2.College of Mechanical EngineeringBeijing University of TechnologyBeijingPeople’s Republic of China
  3. 3.Tianjin Key Laboratory of the Design and Intelligent Control of the Advanced Mechatronical SystemTianjin University of TechnologyTianjinPeople’s Republic of China

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