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

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  • 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
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    References listed on IDEAS

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