IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v232y2025icp427-453.html
   My bibliography  Save this article

Global dynamics in the lateral oscillation model of pedestrian walking on a vibrating surface

Author

Listed:
  • Hu, Sengen
  • Zhou, Liangqiang

Abstract

This paper studies the lateral oscillations of pedestrian walking on a vibrating ground with a known motion, which can be modeled by a hybrid Rayleigh–van der Pol–Duffing oscillator with quintic nonlinearity and dual parametric excitations. The focus of the work is on the global dynamics of the oscillator, including chaos and subharmonic bifurcations. It reveals that the system can be subdivided into three categories in the undisturbed case: single well, double hump, and triple well. Specifically, the exact solutions for homoclinic, heteroclinic and subharmonic orbits in triple-well case are obtained analytically. The Melnikov method is employed to investigate the chaotic phenomena resulting from different orbits. Compared to a single self-excited oscillator, this hybrid oscillator exhibits higher sensitivity to external excitation and strong nonlinear terms. By adjusting the system parameters, the peak value of the chaos threshold can be controlled to avoid the occurrence of chaos. Based on the subharmonic Melnikov method, the subharmonic bifurcations of the system are examined and the extreme case is discussed. Some nonlinear phenomena are discovered. The system only exhibits chaotic behavior when there is a strong resonance, that is, when there is an integer-order subharmonic bifurcation. Furthermore, we find the pathways to chaos though subharmonic bifurcations encompass two distinct mechanisms: odd and even finite bifurcation sequences. The numerical simulation serves to verify the findings of the preceding analysis, while simultaneously elucidating a number of additional dynamic phenomena, including multi-stable state motion, bursting oscillations, and the coexistence of attractors.

Suggested Citation

  • Hu, Sengen & Zhou, Liangqiang, 2025. "Global dynamics in the lateral oscillation model of pedestrian walking on a vibrating surface," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 232(C), pages 427-453.
    • Handle: RePEc:eee:matcom:v:232:y:2025:i:c:p:427-453
    DOI: 10.1016/j.matcom.2024.12.026
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S037847542400510X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2024.12.026?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Zhang, Yufeng & Li, Jing & Zhu, Shaotao & Ma, Zerui, 2024. "Harmonic resonance and bifurcation of fractional Rayleigh oscillator with distributed time delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 221(C), pages 281-297.
    2. Safartoobi, Masoumeh & Dardel, Morteza & Daniali, Hamidreza Mohammadi, 2024. "Piezoelectric energy harvesting from walking motion of a passive biped robot model with flexible legs," Chaos, Solitons & Fractals, Elsevier, vol. 188(C).
    3. He, Ji-Huan & Jiao, Man-Li & Gepreel, Khaled A. & Khan, Yasir, 2023. "Homotopy perturbation method for strongly nonlinear oscillators," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 243-258.
    4. Chian, Abraham C.-L. & Rempel, Erico L. & Rogers, Colin, 2006. "Complex economic dynamics: Chaotic saddle, crisis and intermittency," Chaos, Solitons & Fractals, Elsevier, vol. 29(5), pages 1194-1218.
    5. Zhou, Liangqiang & Liu, Shanshan & Chen, Fangqi, 2017. "Subharmonic bifurcations and chaotic motions for a class of inverted pendulum system," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 270-277.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Dominique Guégan & Justin Leroux, 2008. "Local Lyapunov exponents: Zero plays no role in Forecasting chaotic systems," Cahiers de recherche 08-10, HEC Montréal, Institut d'économie appliquée.
    2. Richter, Hendrik, 2008. "On a family of maps with multiple chaotic attractors," Chaos, Solitons & Fractals, Elsevier, vol. 36(3), pages 559-571.
    3. Ding, Yuting & Jiang, Weihua & Wang, Hongbin, 2012. "Hopf-pitchfork bifurcation and periodic phenomena in nonlinear financial system with delay," Chaos, Solitons & Fractals, Elsevier, vol. 45(8), pages 1048-1057.
    4. Dominique Guegan & Justin Leroux, 2009. "Forecasting chaotic systems: The role of local Lyapunov exponents," PSE-Ecole d'économie de Paris (Postprint) halshs-00431726, HAL.
    5. Chen, Wei-Ching, 2008. "Nonlinear dynamics and chaos in a fractional-order financial system," Chaos, Solitons & Fractals, Elsevier, vol. 36(5), pages 1305-1314.
    6. Chagas, T.P. & Toledo, B.A. & Rempel, E.L. & Chian, A.C.-L. & Valdivia, J.A., 2012. "Optimal feedback control of the forced van der Pol system," Chaos, Solitons & Fractals, Elsevier, vol. 45(9), pages 1147-1156.
    7. Vasily E. Tarasov, 2019. "Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models," Mathematics, MDPI, vol. 7(6), pages 1-50, June.
    8. Ke, Qiaoqiao & Wang, Hailing & Chen, Zhang & Li, Junhua & Lin, Yezhi, 2024. "Quantitative analysis of limit cycles in two-stroke oscillators with exponential functions based on Perturbation Incremental Method," Chaos, Solitons & Fractals, Elsevier, vol. 188(C).
    9. Phukan, Animesh & Sarmah, Hemanta Kumar, 2025. "Bifurcation analysis of a non linear 6D financial system with three time delay feedback," Chaos, Solitons & Fractals, Elsevier, vol. 194(C).
    10. Li, Jiaorui & Ren, Zhengzheng & Wang, Zuoren, 2008. "Response of nonlinear random business cycle model with time delay state feedback," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(23), pages 5844-5851.
    11. Dominique Guegan & Justin Leroux, 2009. "Forecasting chaotic systems: The role of local Lyapunov exponents," Post-Print halshs-00431726, HAL.
    12. Shoji, Isao & Nozawa, Masahiro, 2022. "Geometric analysis of nonlinear dynamics in application to financial time series," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    13. Feryal Abdullah Aladsani & Ghulam Muhammad & Sayed K. Elagan, 2025. "Granular Fuzzy Fractional Financial Systems Governed by Granular Caputo Fractional Derivative," Mathematics, MDPI, vol. 13(8), pages 1-24, April.
    14. Valls, Claudia, 2012. "Rational integrability of a nonlinear finance system," Chaos, Solitons & Fractals, Elsevier, vol. 45(2), pages 141-146.
    15. Chen, Wei-Ching, 2008. "Dynamics and control of a financial system with time-delayed feedbacks," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 1198-1207.
    16. Roy, Tapas & Maiti, Dilip K., 2024. "General approach on the best fitted linear operator and basis function for homotopy methods and application to strongly nonlinear oscillators," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 44-64.
    17. Alim, Md. Abdul & Kawser, M. Abul, 2023. "Illustration of the homotopy perturbation method to the modified nonlinear single degree of freedom system," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    18. Son, Woo-Sik & Park, Young-Jai, 2011. "Delayed feedback on the dynamical model of a financial system," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 208-217.
    19. Dominique Guégan & Justin Leroux, 2007. "Forecasting chaotic systems: The role of local Lyapunov exponents," Cahiers de recherche 07-12, HEC Montréal, Institut d'économie appliquée.
    20. Zemtchou, Francis Rolphe & Mabekou Takam, Jeanne Sandrine & Louodop Fotso, Patrick Hervé & Talla, Pierre Kisito, 2025. "Piezoelectric energy harvesting from beam vibrations induced by an aerodynamic force generated by a fluctuating wind," Chaos, Solitons & Fractals, Elsevier, vol. 194(C).

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:232:y:2025:i:c:p:427-453. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.