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

Original Paper
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Abstract

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.

Keywords

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

Notes

Acknowledgements

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.

References

  1. 1.
    Artstein, Z., Slemrod, M.: On singularly perturbed retarded functional differential equations. J. Differ. Equ. 171(1), 88–109 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Zheng, Y.G., Wang, Z.H.: Relaxation oscillation and attractive basins of a two-neuron Hopfield network with slow and fast variables. Nonlinear Dyn. 70(2), 1231–1240 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Yu, Y., Gao, Y.B., Han, X.J.: Modified function projective bursting synchronization for fast-slow systems with uncertainties and external disturbances. Nonlinear Dyn. 79(4), 2359–2369 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Maree, G.J.M.: Slow periodic crossing of a Pitchfork bifurcation in an oscillating system. Nonlinear Dyn. 12(1), 1–37 (1996)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Krupa, M., Szmolyan, A.P.: Extending geometric singular perturbation theory to nonhyperbolic points-fold and Canard points in two dimensions. Siam J. Appl. Math. 33(2), 286–314 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Verhulst, F.: Singular perturbation methods for slow-fast dynamics. Nonlinear Dyn. 50(4), 747–753 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Zheng, S., Han, X., Bi, Q.: Bifurcations and fast-slow behaviors in a hyperchaotic dynamical system. Commun. Nonlinear Sci. 16(4), 1998–2005 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Maesschalck, P.D., Schecter, S.: The entry-exit function and geometric singular perturbation theory. J. Differ. Equ. 260(8), 6697–6715 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Wiggins, S., Holmes, P.: Homoclinic orbits in slowly varying oscillators. Siam J. Appl. Math. 18(18), 592–611 (1987)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Stiefenhofer, M.: Singular perturbation with Hopf points in the fast dynamics. Z. Angew. Math. Phys. 49(4), 602–629 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Han, M.A., Jiang, K., Green Jr., D.: Bifurcations of periodic orbits, subharmonic solutions and invariant Tori of high-dimensional systems. Nonlinear Anal. TMA 36(3), 319–329 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ye, Z.Y., Han, M.A.: Singular limit cycle bifurcations to closed orbits and invariant tori. Chaos Solitons Fract. 27(3), 758–767 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ye, Z.Y., Han, M.A.: Bifurcations of invariant Tori and subharmonic solutions of singularly perturbed system. Chin. Ann. Math. Ser. B 28(2), 135–148 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Chiba, H.: Periodic orbits and chaos in fast-slow systems with Bogdanov-Takens type fold points. J. Differ. Equ. 250(1), 112–160 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Sourdis, C.: On periodic orbits in a slow-fast system with normally elliptic slow manifold. Math. Mehtods Appl. Sci. 37(2), 270–276 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Han, M.A., Li, S.M.: Perturbations of parallel flows on the sphere in \(R^{3}\). J. Math. Anal. Appl. 351(1), 224–231 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Cima, A., Llibre, J., Teixeira, M.A.: Limit cycles of some polynomial differential systems in dimension 2, 3 and 4, via averaging theory. Appl. Anal. 87(2), 149–164 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Llibre, J., Teixeira, M.A., Zeli, I.O.: Birth of limit cycles for a class of continuous and discontinuous differential systems in \((d+2)\)-dimension. Dyn. Syst. 31(3), 237–250 (2016)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Li, C.Z., Lu, K.N.: Slow divergence integral and its application to classical Liénard equations of degree 5. J. Differ. Equ. 257(12), 4437–4469 (2014)CrossRefMATHGoogle Scholar
  20. 20.
    Bobieński, M., Gavrilov, L.: Finite cyclicity of slow-fast Darboux systems with a two-saddle loop. Proc. Am. Math. Soc. 144, 4205–4219 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Huzak, R., Maesschalck, P.D., Dumortier, F.: Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations. J. Differ. Equ. 255(11), 4012–4051 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Huzak, R.: Regular and slow-fast codimension 4 saddle-node bifurcations. J. Differ. Equ. 262(2), 1119–1154 (2017)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Llibre, J., Rodrigues, A.: A non-autonomous kind of Duffing equation. Appl. Math. Comput. 251, 669–674 (2015)MathSciNetMATHGoogle Scholar
  24. 24.
    Liu C.L.: The vibration and chaos of honeycomb sandwich plate with in-plane and transverse excitations. In: Master Thesis, Beijing University of Technology (2010)Google Scholar

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

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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