IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i13p2161-d1692941.html
   My bibliography  Save this article

The Stochastic Hopf Bifurcation and Vibrational Response of a Double Pendulum System Under Delayed Feedback Control

Author

Listed:
  • Ruichen Qi

    (Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China)

  • Shaoyi Chen

    (Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China)

  • Caiyun Huang

    (Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China)

  • Qiubao Wang

    (Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China)

Abstract

In this paper, we investigate the nonlinear dynamic behavior of a cart–double pendulum system with single time delay feedback control. Based on the center manifold theorem and stochastic averaging method, a reduced-order dynamic model of the system is established, with a focus on analyzing the influence of time delay parameters and stochastic excitation on the system’s Hopf bifurcation characteristics. By introducing stochastic differential equation theory, the system is transformed into the form of an Itô equation, revealing bifurcation phenomena in the parameter space. Numerical simulation results demonstrate that decreasing the time delay and increasing the time delay feedback gain can effectively enhance system stability, whereas increasing the time delay and decreasing the time delay feedback gain significantly improves dynamic performance. Additionally, it is observed that Gaussian white noise intensity modulates the bifurcation threshold. This study provides a novel theoretical framework for the stochastic stability analysis of time delay-controlled multibody systems and offers a theoretical basis for subsequent research.

Suggested Citation

  • Ruichen Qi & Shaoyi Chen & Caiyun Huang & Qiubao Wang, 2025. "The Stochastic Hopf Bifurcation and Vibrational Response of a Double Pendulum System Under Delayed Feedback Control," Mathematics, MDPI, vol. 13(13), pages 1-25, July.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:13:p:2161-:d:1692941
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/13/2161/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/13/2161/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Burić, Nikola & Todorović, Kristina & Vasović, Nebojša, 2009. "Dynamics of noisy FitzHugh–Nagumo neurons with delayed coupling," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2405-2413.
    2. Mihaela Sterpu & Carmen Rocşoreanu & Raluca Efrem & Sue Ann Campbell, 2023. "Stability and Bifurcations in a Nutrient–Phytoplankton–Zooplankton Model with Delayed Nutrient Recycling with Gamma Distribution," Mathematics, MDPI, vol. 11(13), pages 1-24, June.
    3. Yu, Dong & Wang, Guowei & Ding, Qianming & Li, Tianyu & Jia, Ya, 2022. "Effects of bounded noise and time delay on signal transmission in excitable neural networks," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    4. Din, Anwarud, 2024. "Bifurcation analysis of a delayed stochastic HBV epidemic model: Cell-to-cell transmission," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    5. Ai, Hao & Yang, GuiJiang & Liu, Wei & Wang, Qiubao, 2023. "A fast search method for optimal parameters of stochastic resonance based on stochastic bifurcation and its application in fault diagnosis of rolling bearings," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    6. Liancheng Wang & Min Wang, 2024. "Bifurcation Analysis for an OSN Model with Two Delays," Mathematics, MDPI, vol. 12(9), pages 1-17, April.
    7. Yurong Dong & Hua Liu & Yumei Wei & Qibin Zhang & Gang Ma, 2024. "Stability and Hopf Bifurcation Analysis of a Predator–Prey Model with Weak Allee Effect Delay and Competition Delay," Mathematics, MDPI, vol. 12(18), pages 1-24, September.
    8. Chiarella, Carl & He, Xue-Zhong & Wang, Duo & Zheng, Min, 2008. "The stochastic bifurcation behaviour of speculative financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(15), pages 3837-3846.
    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. Ping, Zhu, 2023. "Analytical equivalent transformation method for nonlinear stochastic dynamics with multiple noises in high dimensions," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    2. Rao, Feng & Wang, Weiming & Li, Zhenqing, 2009. "Spatiotemporal complexity of a predator–prey system with the effect of noise and external forcing," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1634-1644.
    3. Geoffrey Poitras & John Heaney, 2015. "Classical Ergodicity and Modern Portfolio Theory," Post-Print hal-03680380, HAL.
    4. Yu, Dong & Wu, Yong & Yang, Lijian & Zhao, Yunjie & Jia, Ya, 2023. "Effect of topology on delay-induced multiple resonances in locally driven systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    5. Xu, Yong & Feng, Jing & Li, JuanJuan & Zhang, Huiqing, 2013. "Stochastic bifurcation for a tumor–immune system with symmetric Lévy noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(20), pages 4739-4748.
    6. Li, Mengdi & Huang, Jinfeng & Shi, Peiming & Zhang, Feibin & Gu, Fengshou & Chu, Fulei, 2024. "A secondary optimization strategy in stochastic resonance modelling for the detection of unknown bearing faults," Chaos, Solitons & Fractals, Elsevier, vol. 189(P1).
    7. Zhao, Xiangkui & Li, Ting, 2025. "Extinction and persistence of a stochastic HBV model," Chaos, Solitons & Fractals, Elsevier, vol. 196(C).
    8. Poitras, Geoffrey, 2018. "The pre-history of econophysics and the history of economics: Boltzmann versus the marginalists," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 507(C), pages 89-98.
    9. Yu, Xingwang & Wang, Shengdan & Yang, Yanhua & Ma, Yuanlin & Liu, Tiantian & Wei, Yifan, 2025. "P-bifurcation and bistability arising from cross-correlated sine-Wiener bounded noises: A stochastic single-species model incorporating double Allee effects," Chaos, Solitons & Fractals, Elsevier, vol. 193(C).
    10. Dieci, Roberto & Schmitt, Noemi & Westerhoff, Frank, 2018. "Interactions between stock, bond and housing markets," Journal of Economic Dynamics and Control, Elsevier, vol. 91(C), pages 43-70.
    11. Hu, Yipeng & Ding, Qianming & Wu, Yong & Jia, Ya, 2023. "Polarized electric field-induced drift of spiral waves in discontinuous cardiac media," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    12. Xiang, Jiawei & Guo, Jianchun & Li, Xiaoqi, 2024. "A two-stage Duffing equation-based oscillator and stochastic resonance for mechanical fault diagnosis," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    13. Feifei Yang & Xikui Hu & Guodong Ren & Jun Ma, 2023. "Synchronization and patterns in a memristive network in noisy electric field," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 96(6), pages 1-14, June.
    14. Adam Majewski & Stefano Ciliberti & Jean-Philippe Bouchaud, 2018. "Co-existence of Trend and Value in Financial Markets: Estimating an Extended Chiarella Model," Papers 1807.11751, arXiv.org.
    15. Li, Tianyu & Wu, Yong & Yang, Lijian & Zhan, Xuan & Jia, Ya, 2022. "Spike-timing-dependent plasticity enhances chaotic resonance in small-world network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 606(C).
    16. Schmitt, Noemi & Westerhoff, Frank, 2017. "On the bimodality of the distribution of the S&P 500's distortion: Empirical evidence and theoretical explanations," Journal of Economic Dynamics and Control, Elsevier, vol. 80(C), pages 34-53.
    17. Majewski, Adam A. & Ciliberti, Stefano & Bouchaud, Jean-Philippe, 2020. "Co-existence of trend and value in financial markets: Estimating an extended Chiarella model," Journal of Economic Dynamics and Control, Elsevier, vol. 112(C).
    18. Li, Tianyu & Wu, Yong & Yang, Lijian & Fu, Ziying & Jia, Ya, 2023. "Neuronal morphology and network properties modulate signal propagation in multi-layer feedforward network," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    19. Wang, Luxuan & Niu, Ben & Wei, Junjie, 2016. "Dynamical analysis for a model of asset prices with two delays," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 447(C), pages 297-313.
    20. Ding, Qianming & Wu, Yong & Hu, Yipeng & Liu, Chaoyue & Hu, Xueyan & Jia, Ya, 2023. "Tracing the elimination of reentry spiral waves in defibrillation: Temperature effects," Chaos, Solitons & Fractals, Elsevier, vol. 174(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:gam:jmathe:v:13:y:2025:i:13:p:2161-:d:1692941. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.