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

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.

1. Introduction

The double pendulum, as one of the simplest nonlinear systems [1], has long attracted extensive attention in mechanics and physics due to its rich dynamical behaviors, such as chaotic motion and bifurcation phenomena [2,3,4]. In-depth research on double pendulum systems not only holds theoretical significance but also provides important insights for real-world controlled multibody system applications, including crane load transportation, robotic arm manipulation, and spacecraft attitude adjustment [5,6,7]. However, practical engineering systems are often subject to complex disturbances, such as time delay effects and stochastic noise [8], making traditional deterministic models insufficient for fully capturing their true dynamical characteristics [9,10,11].

Random bifurcation theory, as an extension of deterministic bifurcations under stochastic disturbances, investigates the qualitative changes in system behavior induced by parameter variations—such as the transition of a steady-state probability density function from unimodal to bimodal [12,13]. This theory holds significant applied value in fields such as mechanical vibration control and biological neurodynamics. For instance, in bridge wind-resistant design, random wind loads may trigger unexpected bifurcation-induced vibrations in structures. Similarly, in gene regulatory networks, noise can even induce bistable switching between cellular states [14,15].

In this study, we employ differential equations to model and analyze control systems. Compared with other approaches such as adaptive control and neural network methods, our method demonstrates distinct advantages. It excels in data efficiency as it relies on mathematical modeling and theoretical analysis of the system rather than extensive experimental data, which is particularly beneficial when data is scarce or costly to obtain. Furthermore, this method offers a clear mathematical framework that facilitates understanding of the working principles and dynamic behaviors of control strategies. The transparency of the model aids in analyzing the system’s stability, robustness, and sensitivity [16,17].

This study is developed based on the double pendulum system framework illustrated in Figure 1, advancing the research through several critical developments. First, we incorporate a time delay term into the lower pendulum’s driving mechanism to better reflect practical control scenarios, particularly addressing the signal transmission delays encountered in applications such as teleoperated robotic systems [18,19]. Second, environmental disturbances are systematically accounted for through the addition of stochastic noise to the cart’s motion dynamics, realistically modeling phenomena like the random vibrations induced by uneven terrain in crane operations. Finally, we develop an active delayed feedback control strategy to investigate the suppression of chaotic oscillations and bifurcation behaviors, bridging theoretical analyses with practical stabilization techniques. These extensions significantly enhance the model’s applicability to real-world engineering systems where time delays, stochastic excitations, and active control play crucial roles [20].

In terms of methodology, we first derive the system’s dynamic model from the Lagrange equations, then transform the delay differential equations into stochastic differential equations via stochastic averaging [21]. This approach enables systematic analysis of bifurcation characteristics under the coupled effects of noise and time delay. The study focuses on revealing how critical parameters (e.g., time delay, noise intensity, feedback gain) influence system stability and bifurcation thresholds, thereby providing theoretical foundations for robust control in practical engineering applications [22,23].

The structure of this paper is organized as follows: Section 2 contains a thorough model analysis, where numerical and theoretical solutions are mutually verified to identify the existence conditions for Hopf bifurcation and perform stability analysis. Section 3 primarily focuses on applying the center manifold theorem to reduce the dimensionality of the system equations, and then derives stochastic Itô equations through stochastic averaging and verifying the occurrence of stochastic bifurcation. Section 4 describes numerical simulations carried out to analyze the effects of time delay, white noise, and time-delayed feedback gain coefficients on the double pendulum system. Finally, Section 5 concludes with a summary of the main results obtained in this study.

2. The Analysis of the Model

The lower pendulum of this model represents a nonlinear system based on differential equations, which can be described by the following equation:

$$\begin{array}{c}{T}_1={\displaystyle \frac{1}{2}}{J}_1{{\dot{\theta }}_1}^2+{\displaystyle \frac{1}{2}}{m}_1[{(\dot{x}-l_1{\dot{\theta }}_{1}cos{\theta }_1)}^2+{(l_1{\dot{\theta }}_{1}sin{\theta }_1)}^2]\end{array}$$

(1)

$$\begin{array}{c}{V}_1=m_{1}g{l}_{1}cos{\theta }_1\end{array}$$

(2)

where $m_1$ is the mass of the lower pendulum, $l_1$ represents the distance between the rotation center and the centroid of the lower pendulum, $J_1$ denotes the moment of inertia of the lower pendulum, $\dot{x}$ represents the velocity of the cart, and $L_1$ is the length of the lower pendulum arm.

The kinetic energy and gravitational potential energy of the upper pendulum can be similarly expressed as follows:

$$\begin{array}{c}{T}_2={\displaystyle \frac{1}{2}}{J}_2{{\dot{\theta }}_2}^2+{\displaystyle \frac{1}{2}}{m}_2[{(\dot{x}-L_1{\dot{\theta }}_{1}cos{\theta }_1-l_2{\dot{\theta }}_{2}cos{\theta }_2)}^2+{(L_1{\dot{\theta }}_{1}sin{\theta }_1+l_2{\dot{\theta }}_{2}sin{\theta }_2)}^2]\end{array}$$

(3)

$$\begin{array}{c}{V}_2=m_{2}g(L_{1}cos{\theta }_1+l_{2}cos{\theta }_2)\end{array}$$

(4)

Based on the above formulations, the Lagrangian function can be constructed as follows:

$$\begin{array}{cc}\hfill L& =T_1+T_2-V_1-V_2\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{J}_1{\dot{\theta }}_1^2+{\displaystyle \frac{1}{2}}{m}_1\left[{(\dot{x}-l_1{\dot{\theta }}_{1}cos{\theta }_1)}^2+{\left(l_1{\dot{\theta }}_{1}sin{\theta }_1\right)}^2\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{J}_2{\dot{\theta }}_2^2+{\displaystyle \frac{1}{2}}{m}_2\left[{(\dot{x}-L_1{\dot{\theta }}_{1}cos{\theta }_1-l_2{\dot{\theta }}_{2}cos{\theta }_2)}^2\right.\hfill \\ \hfill & \left.+{(L_1{\dot{\theta }}_{1}sin{\theta }_1+l_2{\dot{\theta }}_{2}sin{\theta }_2)}^2\right]-m_{1}g{l}_{1}cos{\theta }_1-m_{2}g(L_{1}cos{\theta }_1+l_{2}cos{\theta }_2)\hfill \end{array}$$

(5)

where T represents the system’s kinetic energy and V denotes the system’s potential energy. The energy dissipation in the system is evaluated using the Rayleigh dissipation function, which also incorporates non-conservative forces as lumped parameters in the Lagrangian equations. The assumed form of the Rayleigh dissipation function is given below:

$$F_i={\displaystyle \frac{1}{2}}{f}_i{\dot{\theta }}_i^2(i=1,2)$$

(6)

We substitute the harmonic excitation $\mathrm{x}=Acos\omega t$ into the full system model and apply the following Lagrangian equation:

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\displaystyle \frac{\partial L}{\partial {\dot{\theta }}_i}}\right)-{\displaystyle \frac{\partial L}{\partial {\theta }_i}}=Q_i(i=1,2)$$

(7)

This study primarily investigates the time delay effects on the lower pendulum. The conservative force $Q_{\mathrm{i}}$ is characterized by the following expression:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial {\dot{\theta }}_1}}\right)-{\displaystyle \frac{\partial L}{\partial {\theta }_1}}& =-K_{p_1}\left({\theta }_1(t-\tau )\right)-K_{d_1}\left({\dot{\theta }}_1(t-\tau )\right)-F_1+{\eta }_1\left(t\right)\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial {\dot{\theta }}_2}}\right)-{\displaystyle \frac{\partial L}{\partial {\theta }_2}}& =-F_2+{\eta }_2\left(t\right)\hfill \end{array}$$

(9)

where $K_{{\mathrm{p}}_1}$ is the position feedback gain coefficient, $K_{{\mathrm{d}}_1}$ denotes the velocity feedback gain coefficient, and ${\eta }_{\mathrm{i}}\left(t\right)$ represents white noise. Simplifying the equations yields the following expression:

$$\begin{array}{cc}\hfill & {\ddot{\theta }}_1\left(J_1+l_1^2{m}_1+L_1^2{m}_2\right)-A\omega sin\omega t\left(l_1{m}_1+L_1{m}_2\right)sin{\theta }_1\hfill \\ \hfill & +A{\omega }^{2}cos\omega t\left(l_1{m}_1+L_1{m}_2\right)cos{\theta }_1+{\ddot{\theta }}_2{L}_1{l}_2{m}_{2}cos\left({\theta }_1-{\theta }_2\right)\hfill \\ \hfill & -{\dot{\theta }}_2\left({\dot{\theta }}_1-{\dot{\theta }}_2\right)L_1{l}_2{m}_{2}sin\left({\theta }_1-{\theta }_2\right)\hfill \\ \hfill & -(m_1{l}_1+m_2{L}_1)\left(g-A\omega sin\omega t\right)sin{\theta }_1\hfill \\ \hfill & +{\dot{\theta }}_2{\dot{\theta }}_1{L}_1{l}_2{m}_{2}sin\left({\theta }_1-{\theta }_2\right)\hfill \\ \hfill & =-K_{p_1}\left({\theta }_1(t-\tau )\right)-K_{d_1}\left({\dot{\theta }}_1(t-\tau )\right)-{\displaystyle \frac{1}{2}}{f}_1{\dot{\theta }}_1^2+{\eta }_1\left(t\right)\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & {\ddot{\theta }}_2\left(J_2+l_2^2{m}_2\right)+{\ddot{\theta }}_1{L}_1{l}_2{m}_{2}cos\left({\theta }_1-{\theta }_2\right)\hfill \\ \hfill & -{\dot{\theta }}_1\left({\dot{\theta }}_1-{\dot{\theta }}_2\right)L_1{l}_2{m}_{2}sin\left({\theta }_1-{\theta }_2\right)\hfill \\ \hfill & -A\omega l_2{m}_{2}sin{\theta }_{2}sin\omega t+A{\omega }^2{l}_2{m}_{2}cos{\theta }_{2}cos\omega t\hfill \\ \hfill & -m_2{l}_2\left({\dot{\theta }}_2{\dot{\theta }}_1{L}_{1}sin\left({\theta }_1-{\theta }_2\right)+\left(g-A{\dot{\theta }}_2\omega sin\omega t\right)sin{\theta }_2\right)\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}{f}_2{\dot{\theta }}_2^2+{\eta }_2\left(t\right)\hfill \end{array}$$

(11)

The specific parameters involved in this study are listed in

Table 1. Description of designed system parameters.

Parameter	Description of Parameters	Value
$m_1$	Mass of the bottom bob	0.0158 (kg)
$m_2$	Mass of the top bob	0.01265 (kg)
$L_1$	Length of the bottom bob	0.084 (m)
g	Acceleration of gravity	9.81 (m/${\mathrm{s}}^2$)
$J_1$	Moment of inertia of the bottom bob	0.481 (kg ${\mathrm{m}}^2$)
$J_2$	Moment of inertia of the top bob	0.048 (kg ${\mathrm{m}}^2$)
$f_1$	The damping coefficient of the relative motion between the cart and bottom bob	0.08 (N m s)
$f_2$	The damping coefficient of the relative motion between the two bobs	0.085 (N m s)
$l_1$	Distance between the rotational center and the mass center of the bottom bob	0.0122 (m)
$l_2$	Distance between the rotational center and the mass center of the top bob	0.023 (m)
${\theta }_1$	Angle between bottom bob and the vertical	– (rad)
${\theta }_2$	Angle between top bob and the vertical	– (rad)

.

To determine the equilibrium points, we set both the first and second derivatives of angular velocity to zero.

$$\begin{array}{cc}\hfill & -A\omega sin\omega t\left(l_1{m}_1+L_1{m}_2\right)sin{\theta }_1+A{\omega }^{2}sin\omega t\left(l_1{m}_1+L_1{m}_2\right)cos{\theta }_1\hfill \\ \hfill & -(m_1{l}_1+m_2{L}_1)\left(g-A\omega sin\omega t\right)sin{\theta }_1=-K_{p_1}\left({\theta }_1(t-\tau )\right)+{\eta }_1\left(t\right)\hfill \end{array}$$

(12)

$$-A\omega l_2{\mathrm{m}}_2\sin {\theta }_2\sin \omega t-A{\omega }^2{l}_2{\mathrm{m}}_2\cos {\theta }_2\cos \omega t-{\mathrm{m}}_2{l}_{2}gsin{\theta }_2={\eta }_2\left(t\right)$$

(13)

We assume A to be sufficiently small to avoid destabilizing the system. Thus, we first set $A=0$ and let the noise term average to zero. For steady-state analysis, the time delay $\tau$ is set to 0 in the delayed terms. This yields point $\left(0,0\right)$ as the equilibrium and, at the same time, performs a Taylor expansion on the trigonometric function part at that point, which, upon linearization, can be expressed in the following vector form:

$${\displaystyle \frac{\mathrm{d}\theta \left(t\right)}{\mathrm{d}t}}=L\theta \left(t\right)+R\theta \left(t-\tau \right)+F$$

(14)

$$L=\left(\begin{array}{cccc}0& 1& 0& 0\\ L_{21}& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& L_{43}& 0\end{array}\right)$$

$$R=\left(\begin{array}{cccc}0& 0& 0& 0\\ R_{21}& R_{22}& 0& 0\\ 0& 0& 0& 0\\ R_{41}& R_{42}& 0& 0\end{array}\right)$$

where

$$\begin{array}{cc}\hfill L_{21}& ={\displaystyle \frac{\left(J_2+l_2^2{m}_2\right)g(m_1{l}_1+m_2{L}_1)}{\left(J_1+l_1^2{m}_1+L_1^2{m}_2\right)\left(J_2+l_2^2{m}_2\right)-{\left(L_1{l}_2{m}_2\right)}^2}}\hfill \\ \hfill L_{43}& ={\displaystyle \frac{\left(J_1+l_1^2{m}_1+L_1^2{m}_2\right)m_2{l}_{2}g}{\left(J_1+l_1^2{m}_1+L_1^2{m}_2\right)\left(J_2+l_2^2{m}_2\right)-{\left(L_1{l}_2{m}_2\right)}^2}}\hfill \\ \hfill R_{21}& ={\displaystyle \frac{-\left(J_2+l_2^2{m}_2\right)K_{p_1}}{\left(J_1+l_1^2{m}_1+L_1^2{m}_2\right)\left(J_2+l_2^2{m}_2\right)-{\left(L_1{l}_2{m}_2\right)}^2}}\hfill \\ \hfill R_{22}& ={\displaystyle \frac{-\left(J_2+l_2^2{m}_2\right)K_{d_1}}{\left(J_1+l_1^2{m}_1+L_1^2{m}_2\right)\left(J_2+l_2^2{m}_2\right)-{\left(L_1{l}_2{m}_2\right)}^2}}\hfill \\ \hfill R_{41}& ={\displaystyle \frac{\left(L_1{l}_2{m}_2\right)K_{p_1}}{\left(J_1+l_1^2{m}_1+L_1^2{m}_2\right)\left(J_2+l_2^2{m}_2\right)-{\left(L_1{l}_2{m}_2\right)}^2}}\hfill \\ \hfill R_{42}& ={\displaystyle \frac{\left(L_1{l}_2{m}_2\right)K_{{\mathrm{d}}_1}}{\left(J_1+l_1^2{m}_1+L_1^2{m}_2\right)\left(J_2+l_2^2{m}_2\right)-{\left(L_1{l}_2{m}_2\right)}^2}}\hfill \end{array}$$

(15)

The characteristic equation of the system can be expressed in determinant form as follows: $\left|\lambda E-L-R{\mathrm{e}}^{-\lambda \tau }\right|=0$.

Here, we denote the aforementioned characteristic determinant as $D\left(\lambda ,\tau \right)$:

$$\begin{array}{cc}\hfill D(\lambda ,\tau )=(& e^{-\lambda \tau }\left({\lambda }^2{J}_2(l_1^2{m}_1+L_1^2{m}_2)+l_2{m}_2{l}_1^2(-g+{\lambda }^2{l}_2)m_1-g{L}_1^2{m}_2\right)+J_1({\lambda }^2{J}_2\hfill \\ \hfill & +l_2(-g+{\lambda }^2{l}_2)m_2)-e^{\lambda \tau }g{J}_2{l}_1{m}_1+e^{\lambda \tau }{\lambda }^2{J}_2{l}_1^2{m}_1-e^{\lambda \tau }g{L}_1{J}_2{m}_2\hfill \\ \hfill & +e^{\lambda \tau }{\lambda }^2{L}_1^2{J}_2{m}_2-e^{\lambda \tau }g{l}_1{l}_2^2{m}_1{m}_2+e^{\lambda \tau }{\lambda }^2{l}_1^2{l}_2^2{m}_1{m}_2-e^{\lambda \tau }g{L}_1{l}_2^2{m}_2^2\hfill \\ \hfill & +\lambda K_{d_1}\left(J_2+l_2^2{m}_2\right)+e^{\lambda \tau }{\lambda }^2{J}_1\left(J_2+l_2^2{m}_2\right)+J_2{K}_{p_1}+l_2^2{m}_2{K}_{p_1})\hfill \\ \hfill & /{\left(l_1^2{l}_2^2{m}_1{m}_2+J_2\left(l_1^2{m}_1+L_1^2{m}_2\right)+J_1\left(J_2+l_2^2{m}_2\right)\right)}^2\hfill \end{array}$$

(16)

Separating the real part from the imaginary part, we obtain the following:

$$\begin{array}{cc}\hfill sin\omega \tau =& \omega K_{d_1}(l_2^2{m}_2(g{l}_1{m}_1+{\omega }^2{l}_1^2{m}_1+g{L}_1{m}_2)\hfill \\ \hfill & +J_2(g{l}_1{m}_1+{\omega }^2{l}_1^2{m}_1+L_1(g+{\omega }^2{L}_1)m_2)\hfill \\ \hfill & +{\omega }^2{J}_1(J_2+l_2^2{m}_2))/\hfill \\ \hfill & \left((J_2+l_2^2{m}_2)({\omega }^2{K}_{d_1}^2+K_{p_1}^2)\right)\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill cos\omega \tau =& \left(\right({\omega }^2{J}_1{J}_2+g{J}_2{l}_1{m}_1+{\omega }^2{J}_2{l}_1^2{m}_1+g{L}_1{J}_2{m}_2\hfill \\ \hfill & +{\omega }^2{L}_1^2{J}_2{m}_2+{\omega }^2{J}_1{l}_2^2{m}_2+g{l}_1{l}_2^2{m}_1{m}_2\hfill \\ \hfill & +{\omega }^2{l}_1^2{l}_2^2{m}_1{m}_2+g{L}_1{l}_2^2{m}_2^2\left)K_{p_1}\right)\hfill \\ \hfill & /\left((J_2+l_2^2{m}_2)({\omega }^2{K}_{d_1}^2+K_{p_1}^2)\right)\hfill \end{array}$$

(18)

Squaring and adding together the two equations, we have the following:

$$L\left(\mathrm{z}\right)={\omega }^4+{\mathrm{d}}_1{\omega }^3+d_2{\omega }^2+d_3\omega +{\mathrm{d}}_4=0$$

(19)

where

$$d_1=0$$

$${\mathrm{d}}_2={\displaystyle \frac{-K_{d_1}^2(J_2+{l_2}^2{m}_2)+2g(l_1{m}_1+L_1{m}_2)(J_2{l}_1^2{m}_1+L_1^2{J}_2{m}_2+l_1^2{l}_2^2{m}_1{m}_2+J_1(J_2+l_2^2{m}_2))}{(J_2+{l_2}^2{m}_2)(g^2{\left(l_1{m}_1+L_1{m}_2\right)}^2-K_{p_1}^2)}}$$

$$d_3=0$$

$$d_4={\displaystyle \frac{(J_2+{l_2}^2{m}_2)(g^2{\left(l_1{m}_1+L_1{m}_2\right)}^2-K_{p_1}^2)}{{(J_2{l}_1^2{m}_1+L_1^2{J}_2{m}_2+l_1^2{l}_2^2{m}_1{m}_2+J_1(J_2+l_2^2{m}_2))}^2}}$$

Since ${\displaystyle \underset{\omega \to \infty }{lim}}L\left(\omega \right)=\infty$, if condition (H1) $\exists {\omega }^{\ast }>0,L\left({\omega }^{\ast }\right)<0$ holds, Equation (19) has at least one positive real root ${\omega }_{\mathrm{i}}(1\le i\le 4)$. We denote the right-hand side of expression $sin\omega \tau$ as Y, and Equations (16) and (17) can be transformed as follows:

(1)

If $cos{\omega }_i\tau <0$

${\tau }_{\mathrm{ij}}={\displaystyle \frac{1}{{\omega }_i}}arcsin\left(Y\right)+{\displaystyle \frac{(2j+1)\pi }{{\omega }_i}}$$(j=0,1,2,\dots )$;

(2)

If $cos{\omega }_i\tau >0$ and $sin{\omega }_i\tau >0$

${\tau }_{\mathrm{ij}}=-{\displaystyle \frac{1}{{\omega }_{\mathrm{i}}}}arcsin\left(Y\right)+{\displaystyle \frac{2j\pi }{{\omega }_i}}$$(j=0,1,2,\dots )$;

(3)

If $cos{\omega }_i\tau >0$ and $sin{\omega }_i\tau <0$

${\tau }_{\mathrm{ij}}=-{\displaystyle \frac{1}{{\omega }_i}}arcsin\left(Y\right)+{\displaystyle \frac{(2j+2)\pi }{{\omega }_i}}$$(j=0,1,2,\dots )$.

Defining ${\tau }_0=min\left\{{\tau }_{i0},i=1,2,3,4,5\right\}$, when $\tau ={\tau }_0$, $\lambda =\mathrm{i}{\omega }_0({\omega }_0>0)$ denotes a pure imaginary root.

Based on the aforementioned theoretical derivation and numerical solution of Equations (17) and (18) (see Figure 2 for the numerical results), we conclusively determine that ${\tau }_0=0.292$ and ${\omega }_0=4.22$.

Here, we propose the following hypotheses:

$$\begin{array}{cc}\hfill H\left(2\right):I& =\left({\omega }_0{K}_{d_1}-K_{p_1}tan\left(\omega {\tau }_0\right)\right){\omega }_0[{\omega }_0^2{J}_2(l_1^2{m}_1+L_1^2{m}_2)\hfill \\ \hfill & +l_2{m}_2\left(l_1^2(g+{\omega }_0^2{l}_2)m_1+g{L}_1^2{m}_2\right)+J_1\left({\omega }_0^2{J}_2+l_2(g+{\omega }_0^2{l}_2)m_2\right)]\ne 0\hfill \end{array}$$

$$\begin{array}{cc}\hfill H\left(3\right):O& =\left[K_{d_1}\right(3{\omega }_0^2{J}_2(l_1^2{m}_1+F_1^2{m}_2)+l_2{m}_2\left(l_1^2(g+3{\omega }_0^2{l}_2)m_1+g{L}_1^2{m}_2\right)\hfill \\ \hfill & +J_1\left(3{\omega }_0^2{J}_2+l_2(g+3{\omega }_0^2{l}_2)m_2\right)]\hfill \\ \hfill & -[{\omega }_0^2{J}_2(l_1^2{m}_1+L_1^2{m}_2)+l_2{m}_2\left(l_1^2(g+{\omega }_0^2{l}_2)m_1+g{L}_1^2{m}_2\right)\hfill \\ \hfill & +J_1\left({\omega }_0^2{J}_2+l_2(g+{\omega }_0^2{l}_2)m_2\right)K_{p_1}{\tau }_0]\ne 0\hfill \end{array}$$

Theorem 1. 

If $H\left(1\right)$, $H\left(2\right)$, and $H\left(3\right)$ are satisfied, the system will generate Hopf bifurcation at the endemic equilibrium point (the specific meanings of H1, H2, and H3 can be found in Appendix B).

Proof. 

Taking the partial derivatives of Equation (16) with respect to both $\lambda$ and $\tau$ yields $Re\left({\displaystyle \frac{\mathrm{d}\lambda }{d\tau }}\right){|}_{\tau ={\tau }_0}^{\lambda =\mathrm{i}{\beta }_0}$. From Theorem 1, the following can be readily observed:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & Re\left({\displaystyle \frac{\mathrm{d}\lambda }{d\tau }}\right){|}_{\tau ={\tau }_0}^{\lambda =\mathrm{i}{\omega }_0}=-\left({\omega }_0{K}_{d_1}-K_{p_1}tan\left({\omega }_0{\tau }_0\right)\right){\omega }_0({{\omega }_0}^2{J}_2\left(l_1^2{m}_1+L_1^2{m}_2\right)\hfill \\ \hfill & +l_2{m}_2\left(l_1^2\left(g+{{\omega }_0}^2{l}_2\right)m_1+g{L}_1^2{m}_2\right)+J_1\left({{\omega }_0}^2{J}_2+l_2\left(g+{{\omega }_0}^2{l}_2\right)m_2\right))/\hfill \\ \hfill & (K_{d_1}3{{\omega }_0}^2{J}_2\left(l_1^2{m}_1+F_1^2{m}_2\right)+l_2{m}_2\left(l_1^2\left(g+3{{\omega }_0}^2{l}_2\right)m_1+g{L}_1^2{m}_2\right)\hfill \\ \hfill & +J_1\left(3{{\omega }_0}^2{J}_2+l_2\left(g+3{{\omega }_0}^2{l}_2\right)m_2\right))\hfill \\ \hfill & -({{\omega }_0}^2{J}_2\left(l_1^2{m}_1+L_1^2{m}_2\right)+l_2{m}_2\left(l_1^2\left(g+{{\omega }_0}^2{l}_2\right)m_1+g{L}_1^2{m}_2\right)\hfill \\ \hfill & +J_1\left({{\omega }_0}^2{J}_2+l_2\left(g+{{\omega }_0}^2{l}_2\right)m_2\right)K_{p_1}{\tau }_0)\ne 0\hfill \end{array}\end{array}$$

(20)

According to Hopf bifurcation theory, if condition $Re\left({\displaystyle \frac{\mathrm{d}\lambda }{d\tau }}\right){|}_{\tau ={\tau }_0}^{\lambda =\mathrm{i}{\omega }_0}\ne 0$ is satisfied (the transversality condition holds), then the system undergoes a Hopf bifurcation. Hopf bifurcation occurs at $\tau ={\tau }_0$ and the theorem is proved. □

3. Reduction and Stochastic Bifurcation of Systems

In this section, we employ the stochastic center manifold theorem to transform the time-delayed stochastic differential equation into a stochastic differential equation.

The first step is to perform the center manifold reduction, projecting the system from the infinite-dimensional state space of the DDE onto a two-dimensional invariant subspace tangent to the plane spanned by the critical eigenvectors. First, we need to define the state variable ${\mathrm{x}}_t$ as a functional that contains information about the system state at time t and its history over the past duration $\tau$. Specifically, it can be defined as follows:

$${\mathrm{x}}_t\left(y\right)=x\left(t+y\right),y\in \left[-\tau ,0\right]$$

(21)

Additionally, we introduce the following two operators:

$$Z\phi \left(y\right)=\left\{\begin{array}{ccc}{\displaystyle \frac{d}{dy}}\phi \left(y\right),\hfill & \hfill & \mathrm{if}-\tau \le y<0,\hfill \\ L\phi \left(0\right)+R\phi (-\tau ),\hfill & \hfill & \mathrm{if}y=0.\hfill \end{array}\right.$$

(22)

$$Q\phi \left(y\right)=\left\{\begin{array}{c}0\begin{array}{cc}& if-\tau \le y<0\end{array}\\ Q\begin{array}{cc}& ify=0\end{array}\end{array}\right.$$

(23)

Combining operator Z and the nonlinear term Q, we obtain the complete system equation in the following form:

$$\dot{x_t}=Z{x}_t+Q\left(x_t\right)$$

(24)

For the center manifold reduction, we require that the right eigenvectors of operator Z correspond to the critical eigenvalues ${\lambda }_{\mathrm{cr}}=\pm i{\omega }_0$. Then, the real eigenvectors $s_{1,2}\in X_{{\mathbb{R}}_4}$ satisfy the following equation:

$$Z{\mathrm{s}}_1=-\omega s_2\begin{array}{cc}& Z{s}_2=\omega s_1\end{array}$$

(25)

Based on the definition of the linear operator Z in the equation, this is a boundary value problem for $s_{1,2}\left(\theta \right)$. The solution takes the following form:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\mathrm{s}}_1\left(\theta \right)=S_{1}cos\left(\omega \theta \right)-S_{2}sin\left(\omega \theta \right)\hfill \\ \hfill & s_2\left(\theta \right)=S_{2}cos\left(\omega \theta \right)+S_{1}sin\left(\omega \theta \right)\hfill \end{array}\end{array}$$

(26)

where $S_{1,2}\in X_{{\mathbb{R}}_4}$ satisfies the linear homogeneous algebraic Equation (27):

$$\left[\begin{array}{cc}L+Rcos\omega & \omega I+Rsin\omega \\ -\left(\omega I+Rsin\omega \right)& L+Rcos\omega \end{array}\right]\left[\begin{array}{c}{S}_1\\ S_2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$$

(27)

From the above equations, $S_1$ and $S_2$ can be determined:

$$S_1=\left[\begin{array}{c}1\\ 0\\ {\displaystyle \frac{{\alpha }^2}{{\alpha }^2-{\omega }^2}}\\ 0\end{array}\right]\begin{array}{cc}& S_2=\left[\begin{array}{c}0\\ \omega \\ 0\\ {\displaystyle \frac{\omega {\alpha }^2}{{\alpha }^2-{\omega }^2}}\end{array}\right]\end{array}$$

(28)

For the projection of the system onto the plane generated by eigenvectors $s_1$ and $s_2$, it is necessary to use the left eigenvector $\mp \mathrm{i}{\omega }_0$ of the linear operator Z corresponding to the critical eigenvalue $n_{1,2}$.

Consider the adjoint operator of $Z^{\ast }$ as follows:

$$Z^{\ast }\psi \left(\sigma \right)=\left\{\begin{array}{c}-{\displaystyle \frac{d}{d\sigma }}\psi \left(\sigma \right)\begin{array}{cc}& if0<\sigma <\tau \end{array}\\ L^{\ast }\psi \left(0\right)+R^{\ast }\psi \left(\tau \right)\begin{array}{cc}& if\sigma =0\end{array}\end{array}\right.$$

(29)

Let * represent simultaneously the conjugate transpose matrix and the adjoint operator in this context.

The real eigenvector $n_{1,2}\in X_{{\mathbb{R}}_4}^{\ast }$ satisfies Equation (30):

$$Z^{\ast }{\mathrm{n}}_1=\omega n_2\begin{array}{cc}& Z^{\ast }{\mathrm{n}}_2=-\omega n_1\end{array}$$

(30)

from which

$$\begin{array}{cc}\hfill & n_1\left(\sigma \right)=N_{1}cos\left(\omega \sigma \right)-N_{2}sin\left(\omega \sigma \right)\hfill \\ \hfill & n_2\left(\sigma \right)=N_{2}cos\left(\omega \sigma \right)+N_{1}sin\left(\omega \sigma \right)\hfill \end{array}$$

(31)

with $N_{1,2}\in {\mathbb{R}}_4$ satisfying

$$\left[\begin{array}{cc}{L}^{\ast }+R^{\ast }cos\omega & -(\omega I+R^{\ast }sin\omega )\\ \omega I+R^{\ast }sin\omega & L^{\ast }+R^{\ast }cos\omega \end{array}\right]\left[\begin{array}{c}{N}_1\\ N_2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$$

(32)

$$<n_1,s_1>=N_1^{\ast }{S}_1+{\int }_{\xi =-1}^0{n}_1^{\ast }(\xi +1)R{s}_1\left(\xi \right)d\xi =1$$

(33)

$$<n_1,s_2>=N_1^{\ast }{S}_2+{\int }_{\xi =-1}^0{n}_1^{\ast }(\xi +1)R{s}_2\left(\xi \right)d\xi =0$$

(34)

Based on the solution of the above equations, the following apply:

$$N_1=\left(\begin{array}{c}{N}_{11}\\ N_{12}\\ N_{13}\\ N_{14}\end{array}\right)\begin{array}{cc}& N_2=\left(\begin{array}{c}{N}_{21}\\ N_{22}\\ N_{23}\\ N_{24}\end{array}\right)\end{array}$$

(35)

where

$$\begin{array}{cc}\hfill N_{11}& =-\left(2\right(T{\omega }^4\left(T^2-2T(1+Y)\omega -Y^2{\omega }^2\right)\hfill \\ \hfill & +Y{\omega }^2\left(-T^2(-1+\omega )-2TY{\omega }^2+Y{\omega }^2(Y+2\omega +Y\omega )\right)\left)\right)/\hfill \\ \hfill & (-4T^3{\omega }^3+T^4(-1+2{\omega }^2)\hfill \\ \hfill & +Y^3{\omega }^4\left(2(-1+\omega )\omega +Y(-1+2{\omega }^2)\right)\hfill \\ \hfill & +2T^2{\omega }^2\left(2{\omega }^2+Y\omega (-1+3\omega )+Y^2(-1+2{\omega }^2)\right)\hfill \\ \hfill & -2Y\omega \left(-T^2(-1+\omega )-2TY{\omega }^2+Y{\omega }^2(Y+2\omega +Y\omega )\right))\hfill \end{array}$$

$$\begin{array}{cc}\hfill N_{12}& =\left(2{\omega }^2\right(T^3(-1+\omega )+T^{2}Y{\omega }^2\hfill \\ \hfill & +T{Y}^2(-1+\omega ){\omega }^2+Y^3{\omega }^4\left)\right)/\hfill \\ \hfill & (-4T^3{\omega }^3+T^4(-1+2{\omega }^2)\hfill \\ \hfill & +Y^3{\omega }^4\left(2(-1+\omega )\omega +Y(-1+2{\omega }^2)\right)\hfill \\ \hfill & +2T^2{\omega }^2\left(2{\omega }^2+Y\omega (-1+3\omega )+Y^2(-1+2{\omega }^2)\right)\hfill \\ \hfill & -2Y\omega \left(-T^2(-1+\omega )-2TY{\omega }^2+Y{\omega }^2(Y+2\omega +Y\omega )\right))\hfill \end{array}$$

$$N_{13}=0\begin{array}{cc}& N_{14}=0\end{array}$$

$$\begin{array}{cc}\hfill N_{21}& =\left(2{\omega }^3\right(-2T^{2}Y{\omega }^2+T{Y}^2(-1+\omega ){\omega }^2\hfill \\ \hfill & -T^3(1+\omega )+Y\left(-T^2+2TY\omega +Y^2{\omega }^2\right)\left)\right)/\hfill \\ \hfill & (-4T^3{\omega }^3+T^4(-1+2{\omega }^2)\hfill \\ \hfill & +Y^3{\omega }^4\left(2(-1+\omega )\omega +Y(-1+2{\omega }^2)\right)\hfill \\ \hfill & +2T^2{\omega }^2\left(2{\omega }^2+Y\omega (-1+3\omega )+Y^2(-1+2{\omega }^2)\right)\hfill \\ \hfill & -2Y\omega \left(-T^2(-1+\omega )-2TY{\omega }^2+Y{\omega }^2(Y+2\omega +Y\omega )\right))\hfill \end{array}$$

$$\begin{array}{cc}\hfill N_{22}& =\left(2{\omega }^3\right(-T^3-T{Y}^2{\omega }^2+Y^3(-1+\omega ){\omega }^2\hfill \\ \hfill & +T^2\left(Y(-1+\omega )+2\omega \right)-2Y^2\omega \left)\right)/\hfill \\ \hfill & (-4T^3{\omega }^3+T^4(-1+2{\omega }^2)\hfill \\ \hfill & +Y^3{\omega }^4\left(2(-1+\omega )\omega +Y(-1+2{\omega }^2)\right)\hfill \\ \hfill & +2T^2{\omega }^2\left(2{\omega }^2+Y\omega (-1+3\omega )+Y^2(-1+2{\omega }^2)\right)\hfill \\ \hfill & -2Y\omega \left(-T^2(-1+\omega )-2TY{\omega }^2+Y{\omega }^2(Y+2\omega +Y\omega )\right))\hfill \end{array}$$

$$N_{23}=0\begin{array}{cc}& N_{24}=0\end{array}$$

The above Q, W, T, and Y correspond to $L_{21}$, $L_{43}$, $R_{21}$, and $R_{22}$ in Equation (15), respectively.

To restrict the dynamics to the invariant center manifold, the infinite-dimensional state ${\mathrm{x}}_t$ is projected onto the right eigenvectors ${\mathrm{s}}_1$ and ${\mathrm{s}}_2$ via the corresponding inner product between the left eigenvectors ${\mathrm{n}}_{1,2}$ and ${\mathrm{x}}_t$. The infinite-dimensional part w of the state ${\mathrm{x}}_t$ is computed by simple subtraction. Thus, the new scalar coordinates ${\mathrm{z}}_{1,2}$ on the center manifold are introduced as follows:

$${\mathrm{z}}_1=<n_1,x_t>$$

(36)

$${\mathrm{z}}_2=<n_2,x_t>$$

(37)

$$\mathrm{w}=x_t-z_1{s}_1-z_2{s}_2$$

(38)

With these new variables, since w represents higher-order terms and can be neglected, Equation (21) can be rewritten in the following form:

$$\left[\begin{array}{c}{z}_1^{\prime }\\ z_2^{\prime }\end{array}\right]=\left[\begin{array}{cc}0& \omega \\ -\omega & 0\end{array}\right]\left[\begin{array}{c}{z}_1\\ z_2\end{array}\right]+\left[\begin{array}{c}{N}_1^{\ast }\times F\left({\mathrm{x}}_t\right)\left(0\right)\\ N_2^{\ast }\times F\left({\mathrm{x}}_t\right)\left(0\right)\end{array}\right]$$

(39)

where $N_{\mathrm{i}}^{\ast }$ represents the transpose of $N_{\mathrm{i}}$.

The equation can be rewritten in the following form:

$$\left\{\begin{array}{c}{\mathrm{z}}_1^{\prime }=\omega {\mathrm{z}}_2+N_{11}^{\ast }{F}_1+N_{12}^{\ast }{F}_2+N_{13}^{\ast }{F}_3+N_{14}^{\ast }{F}_4\\ {\mathrm{z}}_2^{\prime }=-\omega {\mathrm{z}}_1+N_{21}^{\ast }{F}_1+N_{22}^{\ast }{F}_2+N_{23}^{\ast }{F}_3+N_{24}^{\ast }{F}_4\end{array}\right.$$

(40)

where $F_1=0\begin{array}{cc}& F_3=0\end{array}$

$$\begin{array}{cc}\hfill F_2& =(\left(J_2+l_2^2{m}_2\right)\left(-A{\omega }^{2}cos\left(t\omega \right)cos\left({\displaystyle \frac{z_1}{\omega }}+{\displaystyle \frac{z_2{P}_1}{{\omega }^2{D}_1}}\right)\left(l_1{m}_1+L_1{m}_2\right)+t{n}_1\right)\hfill \\ \hfill & -cos\left({\displaystyle \frac{z_1}{\omega }}+{\displaystyle \frac{z_2{P}_1}{{\omega }^2{D}_1}}\right)L_1{l}_2{m}_2(-A{\omega }^{2}cos\left(t\omega \right)l_2{m}_2\hfill \\ \hfill & +t{n}_1+sin\left({\displaystyle \frac{z_1}{\omega }}+{\displaystyle \frac{z_2{P}_1}{{\omega }^2{D}_1}}\right)L_1{l}_2{m}_2{\left(z_2-{\displaystyle \frac{z_1{P}_1}{\omega D_1}}\right)}^2\left)\right)\hfill \\ \hfill & /\left(-L_1^2{l}_2^2{m}_2^2+\left(J_1+l_1^2{m}_1+L_1^2{m}_2\right)\left(J_2+l_2^2{m}_2\right)\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill F_4& =(-L_1{l}_2{m}_2\left(-A{\omega }^{2}cos\left(t\omega \right)cos\left({\displaystyle \frac{z_1}{\omega }}+{\displaystyle \frac{z_2{P}_1}{{\omega }^2{D}_1}}\right)\left(l_1{m}_1+L_1{m}_2\right)+t{n}_1\right)\hfill \\ \hfill & +\left(J_1+l_1^2{m}_1+L_1^2{m}_2\right)(-A{\omega }^{2}cos\left(t\omega \right)l_2{m}_2\hfill \\ \hfill & +t{n}_1+sin\left({\displaystyle \frac{z_1}{\omega }}+{\displaystyle \frac{z_2{P}_1}{{\omega }^2{D}_1}}\right)L_1{l}_2{m}_2{\left(z_2-{\displaystyle \frac{z_1{P}_1}{\omega D_1}}\right)}^2\left)\right)\hfill \\ \hfill & /\left(-L_1^2{l}_2^2{m}_2^2+\left(J_1+l_1^2{m}_1+L_1^2{m}_2\right)\left(J_2+l_2^2{m}_2\right)\right)\hfill \end{array}$$

The polar coordinate transformation is carried out by using the stochastic averaging method as follows:

$$\left\{\begin{array}{c}{\mathrm{z}}_1=R\left(t\right)cos\mathrm{k},\\ z_2=-R\left(t\right)sink,\\ k={\omega }_{0}t+\phi \left(t\right),\end{array}\right.$$

(41)

where $R\left(t\right)$ and $\phi \left(t\right)$ represent the amplitude and phase of the solution, respectively. We can derive the stochastic differential equations governing $R\left(t\right)$ and $\phi \left(t\right)$.

Since both $R\left(t\right)$ and $\phi \left(t\right)$ are slowly varying processes, the method of integral averaging is employed to transform the state variables of the vector field on the center manifold into amplitude–phase relationships. We transcribe the aforementioned expressions for ${\mathrm{z}}_1$ and ${\mathrm{z}}_2$ into the following forms:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}{z}_1}{dt}}=\stackrel{\cdot }{R}=f_1+g_{11}\eta \left(t\right)\hfill \\ \hfill & {\displaystyle \frac{\mathrm{d}{z}_2}{dt}}=\stackrel{\cdot }{\phi }=f_2+g_{21}\eta \left(t\right)\hfill \end{array}$$

(42)

Based on Equation (42), we can separately determine $g_{11}$ and $g_{21}$. Subsequently, by applying the knowledge of stochastic averaging, we can obtain the following results:

$$\mathrm{m}\left(R\right)={〈f_1〉}_t+\pi k_{11}{〈g_{11}{\displaystyle \frac{\partial }{\partial z_1}}{g}_{11}〉}_t+\pi k_{11}{〈g_{21}{\displaystyle \frac{\partial }{\partial z_2}}{g}_{11}〉}_t$$

(43)

$$\sigma \left(R\right)=\sqrt{2\pi k〈g_{11}{g}_{11}〉}$$

(44)

where ${〈\cdot 〉}_{\mathrm{t}}={\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }(\cdot )d\theta$.

After performing time averaging and the averaging operation from 0 to $2\pi$, we obtain the following Itô stochastic differential equations:

$$\begin{array}{cc}\hfill & dR=m\left(R\right)dt+\sigma \left(R\right)dB\left(t\right)\hfill \\ \hfill & m\left(R\right)={\mu }_{1}R+{\mu }_2{R}^3\hfill \\ \hfill & \sigma \left(R\right)=\sqrt{{\mu }_3{R}^2}\hfill \\ \hfill & {\mu }_1={\partial }_1\hfill \\ \hfill & {\mu }_2={\partial }_2\hfill \\ \hfill & {\mu }_3={\partial }_3\hfill \end{array}$$

(45)

The specific expressions of ${\partial }_1$, ${\partial }_2$, and ${\partial }_3$ are given in Appendix A.

The Lyapunov exponent serves as a crucial parameter for characterizing both deterministic and stochastic behaviors in nonlinear dynamical systems. Based on its definition, the following approximation can be derived:

$$\begin{array}{cc}\hfill \lambda & =\underset{t\to \infty }{lim}{\displaystyle \frac{1}{t}}lnV=\underset{t\to \infty }{lim}{\displaystyle \frac{1}{t}}{\int }_0^t\left\{m^{\prime }(R=0)-{\displaystyle \frac{1}{2}}{\left[{\sigma }^{\prime }(R=0)\right]}^2\right\}dt\hfill \\ \hfill & =m^{\prime }(R=0)-{\displaystyle \frac{1}{2}}{\left[{\sigma }^{\prime }(R=0)\right]}^2={\mu }_1-{\displaystyle \frac{1}{2}}{\mu }_3\hfill \end{array}$$

(46)

When ${\mu }_1<{\displaystyle \frac{1}{2}}{\mu }_3$ is satisfied, $\lambda$ is negative, which indicates that the equilibrium solution exhibits local asymptotic stability under such circumstances. Conversely, if ${\mu }_1<{\displaystyle \frac{1}{2}}{\mu }_3$ holds true, that is to say, $\lambda$ is positive, the equilibrium solution will become unstable. This instability served as a pivotal turning point for our subsequent investigation into the dynamic behavior of the system. To achieve a comprehensive understanding of the system’s global dynamic characteristics, this study employs a strategy that integrates boundary classification with the three-index method. Specifically, when the parameter ${\mu }_2$ assumes a negative value, the right-hand boundary $R\to \infty$ is transformed into an entry boundary. This transformation lays a solid foundation for the overall behavioral analysis of the system in this study.

Within the theoretical framework of boundary classification, we conduct an in-depth analysis of the left boundary $R=0$, and determine that it belongs to the first-class singular boundary. Further analysis reveals that, when ${\displaystyle \frac{2{\mu }_1}{{\mu }_3}}>1$ is satisfied, a non-trivial steady-state probability density $p\left(R\right)$ exists within the system, and its specific form is as follows:

$$\mathrm{p}\left(R\right)={\displaystyle \frac{C}{{\sigma }^2\left(R\right)}}exp\left[\int {\displaystyle \frac{2m\left(R\right)}{{\sigma }^2\left(R\right)}}dR\right]$$

(47)

When $\alpha -\beta =1$, the stationary probability density can be simplified as $p\left(R\right)=O\left(R^{c-a}\right)$. Through analysis, we can infer that D-bifurcation occurs at ${\displaystyle \frac{2{\mu }_1}{{\mu }_3}}=1$, and P-bifurcation occurs at ${\displaystyle \frac{2{\mu }_1}{{\mu }_3}}=2$, as shown in Figure 3.

4. Numerical Simulation

In this section, in order to better reveal the dynamic behavior of the hem, we primarily adopt the data from Table 1 for numerical analysis by controlling key variables such as the time delay $\left(\tau \right)$, noise intensity $\left(D\right)$, angular delayed feedback gain coefficient $\left(K_{{\mathrm{p}}_1}\right)$, and angular velocity delayed feedback gain coefficient $\left(K_{{\mathrm{d}}_1}\right)$.

This paper first examines the motion of the noise-free cart–pendulum system with $K_{{\mathrm{d}}_1}=K_{p_1}=0.2$, where theoretical analysis reveals a critical time delay ${\tau }_0$ such that the system maintains asymptotic stability when $\tau <{\tau }_0$ but becomes unstable and undergoes Hopf bifurcation, yielding periodic solutions, when $\tau >{\tau }_0$, as demonstrated through the time history plots of angular displacement and velocity for representative cases $\tau =0.15<{\tau }_0$ (showing stability convergence) and $\tau =0.75>{\tau }_0$ (exhibiting sustained oscillations), which confirm the theoretical predictions. Figure 4 and Figure 5 are presented below.

Subsequently, with the time delay fixed at $\tau =0$ and control parameters set to $K_{{\mathrm{d}}_1}=K_{p_1}=0.2$, we investigate the vibrational characteristics of the double pendulum system under varying noise conditions. Comparative analysis of the results under different noise intensities $D=0.001,D=0.1$ reveals that increasing noise strength significantly alters the system’s stability, leading to the emergence of stochastic bifurcation phenomena, as clearly demonstrated in Figure 6 and Figure 7.

Furthermore, we present the probability density distributions of both angular displacement and angular velocity under varying noise intensities in Figure 8. The results clearly demonstrate three distinct regimes: at low noise intensity $\left(D=0.000001\right)$, the distribution exhibits a unimodal profile; with moderate noise, it increases $\left(D=0.001\right)$ while maintaining unimodality, and the curve shows significant broadening; at high noise levels $\left(D=1\right)$, the distribution undergoes a fundamental transition to bimodality. This systematic evolution from a unimodal to bimodal distribution provides direct evidence for noise-induced bifurcation, unambiguously revealing the crucial role of stochastic fluctuations in driving phase transitions of the system.

Since excessive noise can induce stochastic bifurcation phenomena, to ensure the reliability of our research results, we control the noise intensity D in all experimental scenarios within the range of $D\le 1$.

Furthermore, this study conducts a comparative analysis of the relative influences of time delay $\tau$ and noise intensity D by systematically controlling both parameters (see Figure 9). Through numerical simulations examining two distinct dynamical regimes—the asymptotically stable state pre-bifurcation and the periodic oscillation state post-bifurcation—we generate time series plots for various combinations of $\tau$ and D. The results unequivocally demonstrate that variations in noise intensity D exert a substantially greater impact on the system’s behavior than changes in time delay $\tau$, as clearly evidenced by the more pronounced alterations in both the amplitude and frequency characteristics visible in the comparative time series plots. This quantitative comparison establishes noise intensity as the dominant factor governing the system’s stochastic dynamics across different operational regimes.

In the following section, we investigate how modifying key parameters $\left(\tau ,K_{p_1},K_{d_1}\right)$ can enhance system stability (see Figure 10 and Figure 11). First, we examine the effect of time delay $\tau$ by fixing $D=0.2$ and $K_{d_1}=K_{{\mathrm{p}}_1}=0.2$. To analyze pre-bifurcation behavior, we select $\tau =0.05$ and $\tau =0.25$. For post-bifurcation analysis, we choose $\tau =0.5$ and $\tau =0.7$, enabling comprehensive observation of the influence of $\tau$ on system dynamics across different stability regimes.

The simulation results clearly demonstrate that increasing the time delay $\tau$ consistently accelerates the variation of both angular displacement and angular velocity, regardless of whether the system is in the pre-bifurcation $\left(\tau <{\tau }_0\right)$ or post-bifurcation $\left(\tau >{\tau }_0\right)$ regime. To further investigate the influence of feedback gain coefficients on system dynamics, we perform numerical simulations with fixed parameters $\tau =0.05$ and $D=0.2$ while systematically varying the control gains through two representative cases, (i) $K_{d_1}=K_{p_1}=0.2$ and (ii) $K_{d_1}=K_{p_1}=0.4$, which enables the comparative analysis of gain effects under identical noise and delay conditions (see Figure 12 and Figure 13).

In real life, the types of noise are complex and diverse. Gaussian noise is often the first choice for research and application due to its favorable mathematical properties and relatively simple model form. However, with the deepening of research and the development of technology, non-Gaussian distributed fluctuations can more accurately reflect the complexity and uncertainty of the market. Therefore, to more comprehensively and accurately understand and analyze practical problems, it is of great theoretical and practical significance to explore the impact of non-Gaussian noise. Therefore, we propose this noise driven by Gaussian noise. Although its driving source is Gaussian noise, after certain processing or transformation, it exhibits different characteristics from the original Gaussian noise, that is, it has autocorrelation. The specific formula is as follows:

$${\displaystyle \frac{\mathrm{d}\eta }{\mathrm{d}t}}=-{\displaystyle \frac{1}{{\tau }_2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\eta }}{M}_q\left(\eta \right)+{\displaystyle \frac{1}{{\tau }_2}}\epsilon \left(t\right)$$

(48)

$$M_{\mathrm{q}}\left(\eta \right)={\displaystyle \frac{p}{{\tau }_2(q-1)}}ln\left[1+{\displaystyle \frac{{\tau }_2}{p}}(q-1){\displaystyle \frac{{\eta }^2}{2}}\right]$$

(49)

where $\epsilon \left(t\right)$ is the mean zero Gaussian white noise term, and its statistical property is given by $〈\epsilon \left(t\right)\epsilon \left(t^{\prime }\right)〉=2P\delta (t-t^{\prime })$.

When $\mathrm{q}<1$, under Equations (48) and (49), the non-Gaussian noise n(t) can be written as follows:

$${\displaystyle \frac{\mathrm{d}\eta }{\mathrm{d}t}}=-{\displaystyle \frac{1}{{\tau }_1}}\eta +{\displaystyle \frac{1}{{\tau }_1}}{\epsilon }_1\left(t\right)$$

(50)

where ${\tau }_1={\displaystyle \frac{2(2-q)}{5-3q}}{\tau }_2$. We may as well assume that $q={\displaystyle \frac{1}{2}}$ and ${\tau }_2={\displaystyle \frac{1}{4}}$. We plot the non-Gaussian noise driven by different noise intensities. It is not difficult to see from Figure 14 that, as the noise intensity increases, there is a transition from a unimodal to a bimodal distribution.

Additionally, in real-world applications, the time delays encountered by each system may differ. To more precisely investigate the impact of multiple time delays on the performance of networked control systems, we reformulate Equations (8) and (9) into the following form to further analyze the effects of encountering various time delays:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial {\dot{\theta }}_1}}\right)-{\displaystyle \frac{\partial L}{\partial {\theta }_1}}& =-K_{p_1}\left({\theta }_1(t-{\tau }_1)\right)-K_{d_1}\left({\dot{\theta }}_1(t-{\tau }_2)\right)-F_1+{\eta }_1\left(t\right)\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial {\dot{\theta }}_2}}\right)-{\displaystyle \frac{\partial L}{\partial {\theta }_2}}& =-F_2+{\eta }_2\left(t\right)\hfill \end{array}$$

(52)

From Figure 15 and Figure 16 below, it can be clearly observed that, regardless of whether the system is subjected to Gaussian or non-Gaussian noise, a transition from unimodal to bimodal distribution occurs when the noise intensity becomes sufficiently large. However, compared to the probability density plots discussed earlier in this paper, the trend of this transition appears to be somewhat mitigated.

In summary, the results demonstrate that the dynamic response of the double pendulum system exhibits two distinct control paradigms: (1) decreasing the time delay $\tau$ or increasing the feedback gain coefficients $\left(K_{{\mathrm{p}}_1}/K_{d_1}\right)$ significantly reduces the rates of change in both angular displacement and velocity, leading to more gradual oscillations; (2) increasing $\tau$ or decreasing the gains conversely accelerates the angular dynamics, intensifying the oscillatory behavior. These quantitatively established relationships provide crucial design principles for practical applications of double pendulum systems, particularly in vibration control and stabilization scenarios where precise modulation of dynamic response through time delay and gain adjustments is essential. The anti-phase relationship between $\tau$ and $K_{{\mathrm{p}}_1}/K_{d_1}$ effects further suggests potential compensation strategies for robust control system design.

5. Conclusions

This paper investigated the nonlinear dynamic behavior of a cart–double pendulum system with single time delay feedback control. The stability analysis of the system is of crucial dynamic significance, as the system exhibits remarkable time delay-dependent dynamic characteristics. When the time delay $\tau$ is below the critical value ${\tau }_0$, the system maintains asymptotic stability; when $\tau$ exceeds ${\tau }_0$, the system undergoes a Hopf bifurcation and generates stable periodic oscillations (see Figure 4 and Figure 5). This phenomenon demonstrates the critical role of the time delay parameter as a key bifurcation control variable.

Furthermore, we extended our investigation to stochastic excitation environments. The study reveals that white noise intensity significantly affects system stability. Numerical simulations (see Figure 8) demonstrated that, as noise intensity increases, the system exhibits typical stochastic bifurcation phenomena. Through comparative analysis of system responses under different parameter combinations (see Figure 9), we found that noise intensity has a more pronounced effect on system stability compared to time delay variations.

Regarding control strategies, this study proposed a stability optimization method based on time delay regulation. Through coordinated adjustment of the time delay parameter and feedback gain coefficient, we obtained the following conclusions: reducing the time delay or increasing the feedback gain coefficient can effectively enhance system stability, while appropriately increasing the time delay or decreasing the feedback gain coefficient can inject greater driving force into the system. This dual-effect parameter regulation provides flexible control strategies for engineering applications (see Figure 10, Figure 11, Figure 12 and Figure 13).

The methodologies proposed in this study are capable of addressing practical challenges related to noise and latency, for example, in the context of modern wireless communication, when mobile phone calls in urban environments are affected by electromagnetic interference from automotive electronics and household appliances, which compromises call quality, and in satellite communication, where changes in atmospheric conditions lead to variations in signal propagation delay that necessitate precise tuning to ensure the accuracy of data transmission. These two real-world examples highlight the importance and challenge of accurately measuring noise and finely tuning latency parameters. The approaches to studying noise and latency variations presented in this paper provide a stronger theoretical foundation for resolving these practical issues [24,25].

In this study, we conducted an in-depth analysis of vehicle suspension systems using the theory of stochastic dynamics. Based on the aforementioned results, we concluded that precise adjustment of the time delay $\left(\tau \right)$ and the gain coefficient $\left(K\right)$ can effectively control the transition between the stable state and the dynamic response state of the vehicle suspension system. This finding holds significant practical implications for enhancing the driving stability and dynamic performance of vehicles [26,27].

This research establishes a theoretical framework for the stochastic stability analysis of time delay-controlled multibody systems. The core innovation lies in the revelation of the bifurcation mechanisms under the coupling effects of time delay and stochastic excitation, as well as the development of optimization control strategies for time delay feedback parameters. These findings not only contribute to theoretical advancement in nonlinear dynamics but also provide valuable insights for engineering applications involving vibration control systems with inherent time delay characteristics. We intend to conduct corresponding physical experiments in the future to further advance our research. Potential future research directions may explore more complex scenarios, including multiple coupled time delays and non-Gaussian stochastic excitations.