In this paper, we investigate the control and dynamic behavior of the inverted pendulum system with time delay under fractional Gaussian noise excitation. For H=1/2 and H, we analyze the stochastic dynamic characteristics of the system under Hopf bifurcation, utilizing time delay and noise intensity as bifurcation parameters, and validate the theoretical conclusions through numerical simulations. We also defined the engineering application range of angle and angular velocity under both asymptotically stable and periodic oscillation dynamic states. Furthermore, using the stochastic equation, we determined the values of time delay and noise intensity that satisfy the maximum engineering application range of angle and angular velocity, and verified their accuracy against the original equation. Additionally, we observed stochastic D-bifurcation and P-bifurcation arising from the combined effects of time delay and noise. Our results exhibit remarkable consistency between analytical and numerical findings, affirming the robustness of our approach and shedding light on the intricate dynamics of the system.

中文翻译：

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分数高斯噪声激励时滞倒立摆系统的控制与随机动态行为


在本文中，我们研究了分数高斯噪声激励下时滞倒立摆系统的控制和动态行为。对于H=1/2和H，利用时滞和噪声强度作为分岔参数，分析了Hopf分岔下系统的随机动态特性，并通过数值模拟验证了理论结论。我们还定义了渐近稳定和周期振荡动态下角度和角速度的工程应用范围。此外，利用随机方程确定了满足角度和角速度最大工程应用范围的时延和噪声强度值，并与原方程验证了其准确性。此外，我们还观察到由于时间延迟和噪声的综合影响而产生的随机 D 分岔和 P 分岔。我们的结果表现出分析结果和数值结果之间的显着一致性，证实了我们方法的稳健性并揭示了系统复杂的动力学。