Nonlinear Dynamics

, Volume 92, Issue 2, pp 721–739 | Cite as

Bifurcation and number of subharmonic solutions of a 4D non-autonomous slow–fast system and its application

  • Jing Li
  • Tingting Quan
  • Wei Zhang
Original Paper


In this paper, we study the existence and bifurcation of subharmonic solutions of a four-dimensional slow–fast system with time-dependent perturbations for the unperturbed system in two cases: one is a Hamilton system and the other has a singular periodic orbit, respectively. We perform the curvilinear coordinate transformation and construct a Poincaré map for both cases. Then some of sufficient conditions and necessary conditions of the existence and bifurcation of subharmonic solutions are obtained by analyzing the Poincaré map. We apply them to study the bifurcation of multiple subharmonic solutions of a honeycomb sandwich plate dynamics system and to discuss the number of subharmonic solutions in different bifurcation regions induced by two parameters. The maximum number of subharmonic solutions of the honeycomb sandwich plate system is 7 and the relative parameter control condition is obtained.


Slow–fast system Non-autonomous Curvilinear coordinate Bifurcation Subharmonic solutions 



The authors gratefully acknowledge the support of the National Natural Science Foundation of China through Grant nos. 11072007, 11372014, 11290152, 11772007 and 11072008, the Natural Science Foundation of Beijing through Grant nos. 1122001 and 1172002, the International Science and Technology Cooperation Program of China through Grant no. 2014DFR61080, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality(PHRIHLB), Beijing Key Laboratory on Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P. R. China.


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Authors and Affiliations

  1. 1.College of Applied SciencesBeijing University of TechnologyBeijingPeople’s Republic of China
  2. 2.School of ScienceTianjin Chengjian UniversityTianjinPeople’s Republic of China
  3. 3.College of Mechanical EngineeringBeijing University of TechnologyBeijingPeople’s Republic of China

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