Bifurcation Analysis and Control of Traffic Flow Model Considering the Impact of Smart Devices for Drivers

Ai, Wenhuan; Zeng, Jingming; Jianli, Fu; Lei, Zhengqing

doi:10.1115/1.4067887

Abstract

In order to comply with the development of intelligent transportation systems, many automotive suppliers upgrade motor vehicles, carrying a variety of intelligent vehicle equipment, which facilitates life but also has a lot of problems. Nowadays, intelligent devices have gradually become a new factor affecting traffic flow, but most traditional models rarely consider this factor and rarely use bifurcation theory methods to analyze traffic system state changes and control traffic flow state abrupt behavior. Without a full understanding of how smart devices affect traffic flow, the problem of traffic congestion cannot be solved well. In this paper, a macrotraffic flow model considering the influence of intelligent vehicle-borne communication devices is improved, which takes into account the change of drivers’ behavior under the influence of intelligent devices and thus the change of traffic flow state. A linear feedback controller is designed to analyze and control hopf bifurcation in the traffic flow system, so as to prevent or alleviate traffic congestion. First, a traffic flow stability model suitable for bifurcation analysis is established to transform the sudden change of traffic state into a system stability problem. The sudden change of stability, such as traffic congestion, is reflected from a macroperspective. Second, the bifurcation analysis of the traffic flow stability model is carried out to study the sudden change behavior of congestion and stability near the equilibrium point and bifurcation point of the expressway traffic system, and the change of actual traffic state is analyzed and predicted. Aiming at the unstable bifurcation points, a control scheme is designed by Chebyshev polynomial approximation and random feedback control to make the unstable bifurcation points delay or disappear and relieve traffic congestion. Finally, the simulation of hopf bifurcation control set on the model is carried out. The density space–time diagram and phase plane are used to verify the introduction of bifurcation control theory into traffic flow state control. It not only helps to improve the stability of traffic flow and avoid traffic jams but also provides a theoretical basis for the prevention of traffic jams. Numerical simulation results show that the improved model can well explain the formation and evolution mechanism of various congestion modes in real traffic, providing a scientific theoretical basis for preventing traffic congestion. And the addition of feedback controller to the model effectively inhibits traffic congestion, providing certain theoretical support for the implementation of effective traffic strategies and alleviating traffic congestion.

1 Introduction

Since the beginning of the 21st century, with the gradual improvement of the level of intelligence and urbanization in China, more and more traffic problems have also been generated while promoting the growth of intelligent traffic behavior and traffic activities, which troubling city managers and traffic engineers. In order to more reasonably understand, simulate, and predict various traffic phenomena (such as lane change, following, congestion, and phase change) that may occur in the urban traffic network, and effectively deal with various traffic problems in the process of urban development, multidisciplinary researchers, including physics and mathematics, try to conduct in-depth research and analysis of traffic flow through modeling.

In the study of traffic flow model, scholars have discussed various factors that may affect traffic flow operation. Considering the influence of curves, Wang established a lattice fluid mechanics model adapted to the special road conditions of curves on the basis of the classical Nagatani model [1]. Concerning weather reasons, Ali et al. [2] considered the impact of adverse weather conditions such as rain, snow, and ice on traffic flow. Zhang et al. [3] studied the macrotraffic flow model considering the speed difference of adjacent vehicles on uphill and downhill slopes and analyzed the stability of the model. They simulated the spatiotemporal evolution law of uphill and downhill traffic flows, and came to the conclusion that with the increase of slope, the unstable area on downhill slopes expands. Based on the Payne-Whitham model, Kühne considered the role of viscous terms [4]. Aw and Rascle considered substituting the convective derivative of “pressure” for its spatial derivative in the acceleration equation [5]. The researchers analyzed the dynamic characteristics of the model from various angles and proposed various traffic control schemes to effectively restrain traffic jams and improve the stability of traffic flow. However, with the development of science and technology, especially the Internet, Communication, and Automatic control technology, motor vehicles are equipped with more and more technical auxiliary functions and related equipment, such as driving computers, cruise control, adaptive cruise control, automatic braking equipment, and so on. The application of these new technologies will affect the operation of traffic flow to a certain extent, resulting in changes in traffic flow characteristics, as shown in Fig. 1. There are few related researches on the newly emerged factors affecting traffic flow. As we all know, the driver can receive the traffic operation status of the car in front of him in real-time with the help of intelligent communication devices and adjust the driving behavior accordingly, which will inevitably lead to changes in the driver’s driving behavior. In view of this fact, we consider the impact of smart devices on traffic conditions and improve a new macroscopic traffic flow model.

Fig. 1

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The intelligent vehicle system diagram

In the complex traffic system, there are often a variety of nonlinear traffic phenomena. These phenomena often change alternately. In theory, the nature of such changes is a bifurcation behavior. In order to study the impact of these changes on the stability of traffic flow, the bifurcation theory is introduced into traffic flow to reproduce sudden changes in traffic system. In 1999, Igarashi et al. [6] and Orosz et al. [7] studied the bifurcation of traffic flow in the optimal velocity model proposed by Bando et al. [8]. Igarashi proved the existence of hopf bifurcation by rigorous mathematical derivation; Orosz determined the type of hopf bifurcation and described its stability by drawing a bifurcation simulation diagram. Zhang et al. [9] conducted bifurcation analysis on a follower model considering time delay and speed difference, and concluded that the critical point of the hopf bifurcation of the model was located at the boundary of the unstable region, and found that the hopf bifurcation causes uniform traffic destabilization. Li et al. [10] used dynamic cooperation theory to study the traffic flow phenomenon of nonlinear following model, analyzed the stability of traffic flow by considering the dynamic cooperation with the nearest preceding vehicle and the nearest following vehicle, and analyzed the phenomenon of traffic congestion. In 2013, Carrillo et al. [11] studied the macroscopic second-order traffic flow model and proved the existence of the Take–Bogdanov bifurcation. In 2015, Delgado and Saavedra [12] used the traveling wave solution of traffic flow to prove the existence of Take–Bogdanov bifurcation, thus explaining the existence of hopf bifurcation and generalized Hopf bifurcation. At present, the bifurcation phenomenon of traffic system is mainly studied from the following model, but the bifurcation analysis of macrotraffic flow model is not very common. In particular, there are few researches on analyzing traffic flow phenomena through macroscopic traffic flow model and controlling traffic system bifurcation behavior. In this paper, an improved macroscopic traffic flow model is established, considering the influence of intelligent devices on the traffic flow system. Based on this model, bifurcation analysis is carried out to study the causes of traffic phenomena and the mechanism of traffic congestion dissipation. A feedback controller is designed to control the bifurcation behavior of the traffic system and thus provide theoretical support for alleviating traffic congestion.

In order to study the bifurcation phenomenon in nonlinear traffic system, this paper uses the bifurcation analysis method to analyze the improved macrotraffic flow model mentioned above to study the nonlinear dynamic behavior of traffic flow under bifurcation thresholds and bifurcation conditions. By setting different parameters for the model, the equilibrium point type and stability of the traffic system under different conditions can be obtained. The existence conditions and bifurcation types of hopf bifurcation and saddle bifurcation can be obtained through theoretical analysis of the model. Finally, the sudden change of the stability of the traffic system near the hopf bifurcation and saddle bifurcation can be reproduced through simulation diagram. Through the bifurcation analysis of the model, we can clearly explain how the qualitative state of the traffic flow system will change when the traffic parameters change and exceed a certain critical value, which will also reveal that the bifurcation will lead to the sudden change of the stability behavior of the traffic flow. It can be seen that the bifurcation analysis theory of traffic flow provides theoretical and scientific basis for preventing and alleviating traffic congestion and explains the formation mechanism of traffic congestion.

The structure of this paper is as follows: Sec. 2 considers the impact of intelligent vehicle equipment on traffic flow and proposes an improved macroscopic traffic flow model; in Sec. 3, the linear stability of the model is analyzed, and the neutral stability condition of the model is obtained. In Sec. 4, the KdV–Burger equation is obtained based on nonlinear stability analysis. Section 5 conducts bifurcation analysis on the model to study nonlinear traffic phenomena in the traffic system, deduces the type of equilibrium point of the model and the existence conditions of each bifurcation, and simulates the influence of intelligent equipment factors on the new model through numerical simulation to verify the correctness of the model. Section 6 describes the feedback controller of the new model. The feedback controller of the linear system is designed to delay the occurrence of hopf bifurcation. For example, the bifurcation appearing at the equilibrium point in the control system move forward, backward or disappear, and the hopf bifurcation are completely eliminated without changing the equilibrium point of the system, so as to achieve the purpose of controlling the abrupt behavior of the traffic system. Section 7 summarizes the work of the full text.

2 Improvement of Macrotraffic Flow Model Considering Intelligent Vehicle Systems Factor

In this section, a vehicle following model is established based on the traditional traffic flow model considering the driver’s reaction delay under the influence of the intelligent vehicle system, and the improved traffic flow model is analyzed by using bifurcation analysis to study the nonlinear dynamics of the traffic flow under bifurcation thresholds and bifurcation conditions. The model proposed in this section can better describe the phenomena such as traffic jam, stop-and-go, shock wave, and traffic flow in real traffic.

The full velocity difference model proposed by Jiang et al. [13]. The model expression is as follows:

\frac{d v_{n} (t)}{d t} = a [V (Δ x_{n} (t)) - v_{n} (t)] + λ Δ v_{n} (t)

(1)

where $λ$ is the influence factor of relative velocity.

According to Eq. (1), after the driver gets the traffic flow data such as the distance $Δ x_{n} (t)$ between the vehicle and the vehicle in front and the speed difference $Δ v_{n} (t)$ at the time $t$ ⁠, he will make the vehicle reach the optimal traveling speed $V (Δ x_{n} (t))$ by adjusting the $T$ time. Previous studies have shown that the size of $T$ will affect the operational stability of traffic flow. In general, the inverse of $T$ ⁠, $a = 1 / T$ is regarded as the sensitivity of the traffic system: the larger a is, the more sensitive the driver’s response to traffic changes is, and the operation of the traffic flow tends to be stable; the opposite also holds. In the following model, the main factors affecting a are the driver’s reaction speed and the driver’s error in estimating various parameters of the traffic flow, respectively. In the actual traffic scenario, there is a certain gap between the driver of the nth vehicle’s estimation of the traffic data (speed $v_{n + 1}$ ⁠, position $x_{n + 1}$ ⁠, acceleration/deceleration, etc.) of the vehicle in front of him, which inevitably leads to the difference in the estimated values of $Δ x_{n} (t)'$ and $Δ v_{n} (t)'$ from the accurate values $Δ x_{n} (t), Δ v_{n} (t)$ used in the follow-up model. In order to better simulate the actual traffic flow, when the values of traffic parameters such as $Δ x_{n} (t), Δ v_{n} (t)$ in the follow-up model are larger than the actual driver’s estimates, the value of $T$ is appropriately reduced during the calibration process of the follow-up model, and vice versa, the value of $T$ is increased.

With the introduction of intelligent vehicle devices, drivers can accurately obtain various traffic operation statuses of the vehicle in front of them, such as the real-time speed of the vehicle in front of them, acceleration and deceleration, and the distance between the vehicle and the vehicle in front of it. Since it is not necessary to obtain the operation status of the front vehicle and its changes through manual estimation, the driver can focus more on the operation of the vehicle. In such a case, there are two kinds of changes in the operation of the traffic flow: (1) the introduction of the intelligent system can improve the sensitivity of the traffic system, and the degree of sensitivity improvement is related to the driver’s adaptation to the intelligent system; (2) the driver will anticipate the changes in the traffic flow in advance to a certain extent and change the state of the vehicle accordingly. When there is no intelligent vehicle system, the driver’s operation of the vehicle is based on the change of the distance between the vehicles, such as accelerating when $Δ x_{n} (t)$ becomes larger, and decelerating when it is vice versa. With the help of intelligent vehicle devices, the driver can accurately obtain the changes in the traffic status of the vehicle in front of him in real-time (when the vehicle in front of him starts to accelerate but $Δ x_{n} (t)$ does not change), and then make acceleration and deceleration operations.

We propose a follow-along model that considers the variation of optimal speed with the influence of smart vehicle devices

\frac{d v_{n} (t)}{d t} = [V (Δ x_{n} (t + ω T)) - v_{n} (t)] / T + λ Δ v_{n} (t)

(2)

where $ω = | [(2 D_{0} - D_{1}) - (({(v_{0} - v_{1})}^{2}) / 2 a_{\max})] | \cdot δ^{- γ}$ represents the delay time of the driver’s response; $D_{0}$ is the safety threshold distance; $D_{1}$ is for the intelligent device to detect the distance between the current vehicle and the front car; $a_{\max}$ is the maximum acceleration of the vehicle under normal conditions; and $v_{0}$ and $v_{1}$ are the normal driving speed of the vehicle and the driving speed detected by the intelligent device, respectively. $δ$ is the sensitive recognition degree of intelligent devices to the surrounding environment of the car: the more sensitive, the greater the value; $γ$ is the driver’s degree of adaptation to smart devices, $0 < γ < 1$ ⁠. It is assumed that each vehicle has an optimal following speed according to the change of workshop distance, and the driver’s following goal is to adjust his own speed to keep up with the optimal speed.

According to Newton’s theorem of motion, the distance

Δ x_{n} (t + ω T)

between the two vehicles in Eq. can be rewritten as

Δ x_{n} (t + ω T) = Δ x_{n} (t) + ω T \frac{d Δ x_{n} (t)}{d t}

(3)

Substituting Eq. into the right-hand side of Eq. and doing a Taylor expansion at

Δ x_{n} (t)

⁠, retaining up to the second term (i.e., the acceleration term), we can get

\begin{matrix} V (Δ x_{n} (t + ω T)) \\ = V (Δ x_{n} (t) + ω T \frac{d Δ x_{n} (t)}{d t}) \\ = V (Δ x_{n} (t)) + ω T V' (Δ x_{n} (t)) \frac{d Δ x_{n} (t)}{d t} \end{matrix}

(4)

where $V^{'} (Δ x_{n} (t))$ is the derivative of the optimal velocity function $V (x)$ at $x = Δ x_{n} (t)$ ⁠.

Substituting Eq. into Eq. , it can deduce that

\frac{d Δ v_{n} (t)}{d t} = [V (Δ x_{n} (t)) - v_{n} (t)] / T + (ω V' (Δ x_{n} (t)) + λ) Δ v_{n} (t)

(5)

In order to transform the above microtraffic flow model into a macrotraffic flow model [14], the following variables are introduced:

v_{n} (t) \to v (x, t) v_{n + 1} (t) \to v (x + Δ, t)

V (Δ x_{n} (t)) \to V_{e} (ρ) V' (Δ x_{n} (t)) \to - ρ^{2} V_{e}' (ρ)

where $Δ$ represents the distance of successive vehicles, $ρ (x, t)$ and $v (x, t)$ represent the macrodensity and macrovelocity, respectively, and $V_{e} (ρ)$ is the equilibrium velocity.

The optimal velocity function

V_{e} (ρ)

is chosen in the following form [15]:

V_{e} (ρ) = v_{f} {{[1 + exp (\frac{ρ / ρ_{m} - 0.25}{0.06})]}^{- 1} - 3.72 \times 10^{- 6}}

Expanding the

v (x + Δ, t)

Taylor series and neglecting higher order terms, we can obtain

Δ v_{n} (t) = v (x + Δ, t) - v (x, t) = v' (x, t) Δ - \frac{1}{2} v ¨ (x, t) Δ^{2}

(6)

Based on the above transformation, Eq. is rewritten as follows:

\frac{\partial v}{\partial t} + v \frac{\partial v}{\partial x} = [V_{e} (ρ) - v] / T + (ω V_{e}^{'} (ρ) + λ) Δ \frac{\partial v}{\partial x} + \frac{Δ^{2}}{2} (ω V_{e}^{'} (ρ) + λ) \frac{\partial^{2} v}{\partial x^{2}}

(7)

Combining the flow conservation equation with Eq. will lead to the derivation of a macroscopic traffic flow model in which the optimal speed varies with the influence of the intelligent vehicle device on the driver’s response

{\begin{matrix} ρ \frac{\partial v}{\partial x} + v \frac{\partial ρ}{\partial x} + \frac{\partial ρ}{\partial t} = 0 \\ \frac{\partial v}{\partial t} + (v - ω V_{e}^{'} (ρ) Δ - λ Δ) \frac{\partial v}{\partial x} = [V_{e} (ρ) - v] / T + \frac{Δ^{2}}{2} (ω V_{e}^{'} (ρ) + λ) \frac{\partial^{2} v}{\partial x^{2}} \end{matrix}

(8)

3 Linear Stability Analysis Considering Intelligent Devices Effects

For subsequent analysis, Eq. is rewritten in vector form

\frac{\partial U}{\partial t} + A \frac{\partial U}{\partial x} = E

(9)

included among these

U = [\begin{matrix} ρ \\ v \end{matrix}]; A = [\begin{matrix} v & ρ \\ 0 & v - ω V_{e}^{'} (ρ) Δ - λ Δ \end{matrix}]

(10)

E = [\begin{matrix} 0 \\ [V_{e} (ρ) - v] / T + \frac{Δ^{2}}{2} (ω V_{e}^{'} (ρ) + λ) \frac{\partial^{2} v}{\partial x^{2}} \end{matrix}]

(11)

By solving, the eigenvalues of A can be found as

λ_{1} = v, λ_{2} = v - ω V_{e}^{'} (ρ) Δ - λ Δ

(12)

The above characteristic speed $λ_{i} (i = 1, 2)$ is not greater than the macroscopic traffic flow speed, which indicates the anisotropic nature of the model.

We take the density and velocity of the initial state traffic flow as constants

ρ_{0}

and

v_{0}

⁠, and then add a small perturbation to the density and velocity of the initial traffic flow. The form of the small perturbation is as follows:

(\begin{matrix} ρ (x, t) \\ v (x, t) \end{matrix}) = (\begin{matrix} ρ_{0} \\ v_{0} \end{matrix}) + (\begin{matrix} \hat{ρ_{k}} \\ \hat{v_{k}} \end{matrix}) exp (ikx + σ_{k} t)

(13)

where k and $σ_{k}$ denote the wave number and frequency, respectively.

Substituting Eq. into Eq. , we get the following equation:

{\begin{matrix} (σ_{k} + v_{0} i k) \hat{ρ_{k}} + ρ_{0} i k \hat{v_{k}} = 0 \\ V_{e}^{'} (ρ_{0}) \hat{ρ_{k}} / T - [σ_{k} + (v_{0} - ω V_{e}^{'} (ρ) Δ - λ Δ)] i k + \frac{1}{T} - \frac{Δ^{2}}{2} (ω V_{e}^{'} (ρ) + λ) {(i k)}^{2}] \hat{v_{k}} = 0 \end{matrix}

(14)

If there exists a nonzero solution to Eq. , then the determinant of its coefficients must be equal to zero

[\begin{matrix} σ_{k} + v_{0} i k ρ_{0} i k \\ V_{e}^{'} (ρ_{0}) / T - [σ_{k} + (v_{0} - ω V_{e}^{'} (ρ) Δ - λ Δ)] i k + \frac{1}{T} - \frac{Δ^{2}}{2} (ω V_{e}^{'} (ρ) + λ) {(i k)}^{2}] \end{matrix}] = 0

(15)

the reason why

{(σ_{k} + v_{0} i k)}^{2} + V_{e}^{'} (ρ_{0}) ρ_{0} i k / T + (σ_{k} + v_{0} i k) [(ω V_{e}^{'} (ρ) Δ - λ Δ) i k + \frac{1}{T} - \frac{Δ^{2}}{2} (ω V_{e}^{'} (ρ) + λ) {(i k)}^{2}]

(16)

To determine the value of

σ_{k}

⁠, we perform the

σ_{k}

to the power series expansion of

i k

⁠, that is,

σ_{k} = σ_{1} i k + σ_{2} {(i k)}^{2} + \dots

⁠, and combining the terms into Eq. , we can obtain an estimate of the value of the

i k

⁠. The second-order expression is shown below:

[V_{e}^{'} (ρ_{0}) ρ_{0} + (σ_{1} + v_{0})] i k / T + [{(σ_{1} + v_{0})}^{2} + \frac{σ_{2}}{T} - (σ_{1} + v_{0}) (ω V_{e}^{'} (ρ) Δ - λ Δ)] {(i k)}^{2} = 0

(17)

To ensure that the above equation holds, the first-order term and the second-order term

{(i k)}^{2}

of the power series must be equal to zero. Thus, it can be obtained

[V_{e}^{'} (ρ_{0}) ρ_{0} + (σ_{1} + v_{0})] / T = 0

(18)

{(σ_{1} + v_{0})}^{2} + \frac{σ_{2}}{T} - (σ_{1} + v_{0}) (ω V_{e}^{'} (ρ) Δ - λ Δ) = 0

(19)

Combining and solving Eqs. and , we obtain

σ_{1} = - v_{0} - V_{e}^{'} (ρ_{0}) ρ_{0}

(20)

σ_{2} = - T V_{e}^{'} (ρ_{0}) ρ_{0} [V_{e}^{'} (ρ_{0}) ρ_{0} + (ω V_{e}^{'} (ρ) Δ - λ Δ)]

(21)

The traffic flow is stable when

σ_{2} > 0

is present, so the neutral stability condition satisfies the following equation:

T_{s} = {V_{e}^{'} (ρ_{0}) ρ_{0} [V_{e}^{'} (ρ_{0}) ρ + (ω V_{e}^{'} (ρ) Δ - λ Δ)]}^{- 1}

(22)

By using Taylor expansion, Eq. is rewritten as follows:

Im (σ_{k}) = - k [v_{0} + V_{e}^{'} (ρ_{0}) ρ_{0}] + O (k^{3})

(23)

According to Eq. , it can be obtained

c (ρ_{0}) = v_{0} + V_{e}^{'} (ρ_{0}) ρ_{0}

(24)

4 Nonlinear Stability Analysis Considering Intelligent Devices Factors

In order to analyze the evolutionary behavior of small perturbations in the vicinity of the neutral stability condition of the traffic system, a new coordinate system is introduced to transform it into the following form:

z = x - c t

(25)

Substituting the above equation into Eq. yields

{\begin{matrix} - c ρ_{z} + q_{z} = 0 \\ - c v_{z} + (v - ω V_{e}^{'} (ρ) Δ - λ Δ) v_{z} = [V_{e} (ρ) - v], T + \frac{Δ^{2}}{2} (ω V_{e}^{'} (ρ) + λ) v_{z z} \end{matrix}

(26)

where the traffic flow q is the product of density and speed, i.e.,

q = ρ v

⁠, substituting the traffic flow formula into Eq. gives

v_{z} = \frac{c ρ_{z}}{ρ} - \frac{q ρ_{z}}{ρ^{2}}

(27)

v_{z z} = (\frac{c}{ρ} - \frac{q}{ρ^{2}}) ρ_{z z} - 2 (\frac{c}{ρ^{2}} - \frac{q}{ρ^{3}}) ρ_{z}^{2}

(28)

The second-order Taylor expansion for the flow rate A second-order Taylor expansion yields

q = ρ V_{e} (ρ_{0}) + b_{1} ρ_{z} + b_{2} ρ_{z z}

(29)

Substituting Eqs. and into the second equation of Eq. , we obtain

\begin{matrix} - c (\frac{c ρ_{z}}{ρ} - \frac{q ρ_{z}}{ρ^{2}}) + (\frac{q}{ρ} - ω V_{e}^{'} (ρ) Δ - λ Δ) (\frac{c ρ_{z}}{ρ} - \frac{q ρ_{z}}{ρ^{2}}) = [V_{e} (ρ) - \frac{q}{ρ}] / T \\ + \frac{Δ^{2}}{2} (ω V_{e}^{'} (ρ) + λ) (\frac{c ρ_{z z}}{ρ} - \frac{q ρ_{z z}}{ρ^{2}} - \frac{2 c ρ_{z}^{2}}{ρ^{2}} + \frac{2 q ρ_{z}^{2}}{ρ^{3}}) \end{matrix}

(30)

Since both

ρ_{z}

and

ρ_{z z}

are nonzero, the coefficients of

ρ_{z}

and

ρ_{z z}

in Eq. must be zero, thus giving

{\begin{matrix} b_{1} = T (c - V_{e} (ρ)) (c + \frac{Δ^{2}}{2} (ω V_{e}^{'} (ρ) + λ)) - c T V_{e} (ρ) \\ b_{2} = \frac{Δ^{2} T}{2} (ω V_{e}^{'} (ρ) + λ) (c - V_{e} (ρ)) \end{matrix}

(31)

In the neighborhood of the neutral stability condition, taking

ρ = ρ_{h} + \hat{ρ} (x, t)

⁠, a second-order Taylor expansion of the A second-order Taylor expansion yields

ρ V_{e} (ρ) \approx ρ_{h} V_{e} (ρ_{h}) + {(ρ V_{e})}_{ρ} |_{ρ = ρ_{h}} \hat{ρ} + \frac{1}{2} {(ρ V_{e})}_{ρ ρ} |_{ρ = ρ_{h}} {\hat{ρ}}^{2}

(32)

Substituting the above equation into

q = ρ V_{e} (ρ_{0}) + b_{1} ρ_{z} + b_{2} ρ_{z z}

and combining it with Eq. yields

- c ρ_{z} + [{(ρ V_{e} (ρ))}_{ρ} + {(ρ V_{e} (ρ))}_{ρ ρ} ρ] ρ_{z} + b_{1} ρ_{z z} + b_{2} ρ_{zzz} = 0

(33)

In order to transform Eq. into the standard KdV–Burgers equation, the following transformations need to be introduced:

U = - [{(ρ V_{e})}_{ρ} + {(ρ V_{e})}_{ρ ρ} ρ], X = m x, T = - m t

(34)

Then, substituting Eq. into Eq. yields the following standard KdV–Burgers equation:

U_{T} + U U_{X} - m b_{1} U_{X X} - m^{2} b_{2} U_{XXX} = 0

(35)

Solving the above equation yields the following solution:

U = - \frac{3 {(- m b_{1})}^{2}}{25 (- m^{2} b_{2})} [1 + 2 tanh (\pm \frac{- m b_{1}}{10 m^{2}}) (X + \frac{6 {(- m b_{1})}^{2}}{25 (- m^{2} b_{2})} T + ς_{0}) + {tanh}^{2} (\pm \frac{- m b_{1}}{10 m^{2}}) (X + \frac{6 {(- m b_{1})}^{2}}{25 (- m^{2} b_{2})} T + ς_{0})]

5 Bifurcation Analysis of Traffic Flow Models Improved by Considering the Characteristics of Intelligent Vehicle Systems

5.1 Type of Equilibrium and Stability Analysis of the Model.

In this paper, the main roadway is assumed to be an open boundary condition

ρ (1, t) = ρ (2, t), ρ (L, t) = ρ (L - 1, t), v (1, t) = v (2, t), v (L, t) = v (L - 1, t)

Assume that the model has traveling wave solutions

ρ (z)

and

v (z)

⁠, where

z = x - c t

and the traveling wave velocity

c < 0

⁠. Based on the theory above, this is obtained by substituting into Eq.

- c ρ_{z} + q_{z} = 0

(36)

- c v_{z} + (v - ω V_{e}^{'} (ρ) Δ - λ Δ) v_{z} = [V_{e} (ρ) - v] / T + \frac{Δ^{2}}{2} (ω V_{e}^{'} (ρ)) + λ) v_{z z}

(37)

from Eqs. and

v_{z} = \frac{c ρ_{z}}{ρ} - \frac{q ρ_{z}}{ρ^{2}}

(38)

v_{z z} = (\frac{c}{ρ} - \frac{q}{ρ^{2}}) ρ_{z z} - 2 (\frac{c}{ρ^{2}} - \frac{q}{ρ^{3}}) ρ_{z}^{2}

(39)

Substituting Eqs. and into Eq. yields

- \frac{ρ_{z}}{ρ^{3}} q^{2} + [\frac{(2 c + ω V_{e}^{'} (ρ) Δ + λ Δ) ρ_{z}}{ρ^{2}} + \frac{1}{T ρ} + \frac{(ω V_{e}^{'} (ρ) + λ) Δ^{2} ρ_{z z}}{2 ρ^{2}}] q = \frac{V_{e} (ρ)}{T} + \frac{c^{2} + c (ω V_{e}^{'} (ρ) + λ) Δ}{ρ} ρ_{z} + \frac{c (ω V_{e}^{'} (ρ) + λ) Δ^{2}}{2 ρ} ρ_{z z}

(40)

Equation is obtained by integrating over z

- c p + q = const = q_{*}

(41)

To wit

q_{*} = - c ρ + q

(42)

Substituting Eq. into Eq. yields

\begin{matrix} (\frac{(ω V_{e}^{'} (ρ) + λ) Δ^{2} (q_{*} + c ρ)}{2 ρ} - \frac{1}{2} c (ω V_{e}^{'} (ρ) + λ) Δ^{2}) ρ_{z z} \\ = (\frac{{(q_{*} + c ρ)}^{2}}{ρ^{2}} - \frac{(2 c + ω V_{e}^{'} (ρ) Δ + λ Δ) (q_{*} + c ρ)}{ρ} + (c^{2} + c (ω V_{e}^{'} (ρ) + λ) Δ)) ρ_{z} - \frac{q_{*} + c ρ}{T} + \frac{V_{e} (ρ) ρ}{T} \end{matrix}

(43)

Simplification leads to a second-order ordinary differential equation on

ρ (z)

second-order ordinary differential equation

ρ_{z z} - G (ρ, q_{*}) ρ_{z} - F (ρ, c, q_{*}) = 0

(44)

Among them

G (ρ, q_{*}) = [\frac{2}{Δ} - \frac{2 q_{*}}{(ω V_{e}^{'} (ρ) + λ) Δ^{2} ρ}]

(45)

F (ρ, c, q_{*}) = \frac{2 ρ}{(ω V_{e}^{'} (ρ) + λ) Δ^{2} T q_{*}} . (ρ V_{e} (ρ) - q_{*} - c ρ)

(46)

such that

y = d ρ / d z

⁠, Eq. can be transformed into a system of first-order ordinary differential equations

{\begin{matrix} \frac{d ρ}{d z} = y \\ \frac{d y}{d z} = G (ρ, q_{*}) y - F (ρ, c, q_{*}) \end{matrix}

(47)

When the right end of the system of Eq. is made to be zero, it leads to

y = 0

and

F = 0

⁠, from which its equilibrium point can be determined to be

(ρ_{i}, 0)

⁠. Taylor expansion of Eq. at the equilibrium point gives the following linear system:

{\begin{matrix} ρ' = y \\ y' = G (ρ_{i}, q_{*}) y - F' (ρ_{i}, c, q_{*}) (ρ - ρ_{i}) \end{matrix}

(48)

It can be derived that the system is at the equilibrium point

(ρ_{i}, 0)

⁠. The Jacobian matrix at the equilibrium point is given by

L = [\begin{matrix} 0 & 1 \\ F'_{i} & G_{i} \end{matrix}]

(49)

So the Jacobian characteristic equation can be obtained as

λ^{2} - G_{i} λ - F'_{i} = 0

(50)

When

G_{i} = G (ρ_{i}, q_{*})

and

F'_{i} = F' (ρ_{i}, c, q_{*})

⁠, the following equation can be obtained from Eqs. and :

F'_{i} = \frac{2 ρ_{i}}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} T q_{*}} . (ρ_{i} V_{e}^{'} (ρ_{i}) + V_{e} (ρ_{i}) - c)

(51)

G_{i} = [\frac{2}{Δ} - \frac{2 q_{*}}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} ρ_{i}}]

(52)

Since the equilibrium point is at

F = 0

⁠, it follows that

ρ_{i} V_{e}^{'} (ρ_{i}) + V_{e} (ρ_{i}) - c = 0

⁠, so

F'_{i}

can be written as

F'_{i} = \frac{2 (q_{*} + ρ_{i}^{2} V_{e} (ρ_{i}))}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} T q_{*}}

(53)

According to the qualitative theory of differential equations, the type of equilibrium point of linear system (48) can be determined as follows: (a) when $F'_{i} > 0$ ⁠, the equilibrium point is a saddle point; (b) the equilibrium point is a saddle point when $G_{i}^{2} + 4 F'_{i} > 0$ and $F'_{i} < 0$ ⁠, the equilibrium point is a node; (c) when $G_{i}^{2} + 4 F'_{i} < 0$ and $G_{i}^{2} \neq 0$ ⁠, the equilibrium point is the focus; and (d) when $F'_{i} < 0$ and $G_{i}^{2} \neq 0$ ⁠, the equilibrium point is the center. When $z \to \pm \infty$ ⁠, the linear system is unstable at both saddle points; when $G_{i}^{2} < 0$ (or $G_{i}^{2} > 0$ ⁠), the linear system is unstable at the nodes or foci along with the $z \to \pm \infty$ (or $z \to - \infty$ ⁠) is stabilized.

From the Hartman–Gorban linearization theorem, it follows that the nonlinear system (47) has the same equilibrium point as the linear system. For an equilibrium point that is not the center, the stability situation at the equilibrium point is the same for the nonlinear system (47) and the linear system (48). When given any set of values for the traveling wave velocity c and the traveling wave parameter, the equilibrium point of the linear system (42) $ρ_{i} (i = 1, 2, 3)$ can be solved for.

The values of the parameters in the model of this paper are as follows:

T = 10 s, V_{f} = 30 m / s, ρ_{m} = 0.2 veh / m, Δ = 30 m, ω = 0.2

When $ρ = 0$ time, this is a mundane equilibrium point of no practical significance, so only the other equilibrium points need to be discussed. From the discussion above and Eqs. (51) and (52), the type of equilibrium point and its stability can be determined, as shown in Table 1, where the equilibrium point is denoted by p (i = 1, 2, 3).

Table 1

Types of equilibrium points and their stability, given the model parameter $Δ_{i} - G_{i}^{2} + 4 F_{i}^{'}, i = 1, 2, 3$

$(c_{0}, q_{*})$	$ρ_{1}$	$ρ_{2}$	$ρ_{3}$
(−0.38, 0.058)	0.0065	0.1197	0.1557
	$F_{i}^{'} > 0$ ⁠, saddle	$Δ_{i} < 0$ ⁠, $G_{i} < 0$ ⁠, focus	$F_{i}^{'} > 0$ ⁠, saddle
	when	when	when
	$z \to \pm \infty$ ⁠, unstable	$z \to + \infty$ ⁠,	$z \to \pm \infty$ ⁠, unstable
		stabilized;
		when
		$z \to - \infty$ ⁠, unstable

(−0.32, 0.059)	0.0062	0.0971
	$F_{i}^{'} > 0$ ⁠, saddle	$Δ_{i} < 0$ ⁠, $G_{i} < 0$ ⁠, focus
	when	when
	$z \to \pm \infty$ ⁠, unstable	$z \to + \infty$ ⁠,
		stabilized;
		when
		$z \to - \infty$ ⁠, unstable

(−0.07, 0.3)	0.0059	0.0918
	$F_{i}^{'} > 0$ ⁠, saddle	$Δ_{i} < 0$ ⁠, $G_{i} < 0$ ⁠, focus
	when $z \to \pm \infty$ ⁠,	When
	unstable	$z \to + \infty$ ⁠, stabilized;
		when
		$z \to - \infty$ ⁠, unstable
(−1.371, 0.01)	0.0048
	$F_{i}^{'} > 0$ ⁠, saddle
	when
	$z \to \pm \infty$ ⁠, unstable

$(c_{0}, q_{*})$	$ρ_{1}$	$ρ_{2}$	$ρ_{3}$
(−0.38, 0.058)	0.0065	0.1197	0.1557
	$F_{i}^{'} > 0$ ⁠, saddle	$Δ_{i} < 0$ ⁠, $G_{i} < 0$ ⁠, focus	$F_{i}^{'} > 0$ ⁠, saddle
	when	when	when
	$z \to \pm \infty$ ⁠, unstable	$z \to + \infty$ ⁠,	$z \to \pm \infty$ ⁠, unstable
		stabilized;
		when
		$z \to - \infty$ ⁠, unstable

(−0.32, 0.059)	0.0062	0.0971
	$F_{i}^{'} > 0$ ⁠, saddle	$Δ_{i} < 0$ ⁠, $G_{i} < 0$ ⁠, focus
	when	when
	$z \to \pm \infty$ ⁠, unstable	$z \to + \infty$ ⁠,
		stabilized;
		when
		$z \to - \infty$ ⁠, unstable

(−0.07, 0.3)	0.0059	0.0918
	$F_{i}^{'} > 0$ ⁠, saddle	$Δ_{i} < 0$ ⁠, $G_{i} < 0$ ⁠, focus
	when $z \to \pm \infty$ ⁠,	When
	unstable	$z \to + \infty$ ⁠, stabilized;
		when
		$z \to - \infty$ ⁠, unstable
(−1.371, 0.01)	0.0048
	$F_{i}^{'} > 0$ ⁠, saddle
	when
	$z \to \pm \infty$ ⁠, unstable

Since the analytical solution of the nonlinear system (2) is not easy to be obtained, we selected the four sets of parameters in Table 1, respectively, and numerically simulated the stability of the nonlinear system (2) at the equilibrium point. The phase plane near the equilibrium point is shown in Fig. 2, where the equilibrium point $(ρ_{i}, 0)$ ⁠, and $i = 1, 2, 3$ ⁠, are shown as the converge curves in Fig. 2.

Fig. 2

View large Download slide

Phase plane diagram of traveling wave speed c and traveling wave parameter $q_{*}$ under different value conditions: (a) traveling wave speed c = −0.38, traveling wave parameter $q_{*}$ = 0.058, (b) traveling wave speed c = −0.32, traveling wave parameter $q_{*}$ = 0.059, (c) traveling wave speed c = −0.07, traveling wave parameter $q_{*}$ = 0.3, and (d) traveling wave speed c = −1.371, traveling wave parameter $q_{*}$ = 0.01

Figure 2(a) corresponds to the first set of data in Table 1. As can be seen from Fig. 2, when $z \to \pm \infty$ ⁠, the system is unstable at the equilibrium points $(ρ_{1}, 0)$ and $(ρ_{3}, 0)$ ⁠, and the nearby rails are all far away from this point. When $z \to + \infty$ ⁠, there are several spirals close to the saddle point $(ρ_{3}, 0)$ converging to the focal point $(ρ_{2}, 0)$ ⁠; when $z \to - \infty$ ⁠, they move away from the focal point and eventually converge to infinity. This shows that when $z \to + \infty$ ⁠, the system is stable at $(ρ_{2}, 0)$ ⁠; when $z \to - \infty$ ⁠, the system is unstable at $(ρ_{2}, 0)$ ⁠, which can be regarded as the saddle point–focus–saddle point solution of the system.

Figure 2(b) corresponds to the second set of data in Table 1. Figure 2 shows that when $z \to \pm \infty$ ⁠, the system is destabilized at the equilibrium point $(ρ_{1}, 0)$ ⁠, with nearby trajectories moving away from that point. The spiral trajectory from $(0.05, 0)$ converges to the focal point $(ρ_{2}, 0)$ when $z \to + \infty$ and the system is stable at that point; it moves away from the focal point when $(ρ_{2}, 0)$ $z \to - \infty$ and the system is unstable at that point.

Figure 2(c) corresponds to the third set of data in Table 1. Figure 2 shows that when $z \to \pm \infty$ ⁠, the system is unstable at the equilibrium point $(ρ_{1}, 0)$ ⁠, and the nearby rails are all moving away from that point. At $z \to - \infty$ ⁠, the spiraling trajectories close to the saddle point $(ρ_{1}, 0)$ converge to the focal point $(ρ_{2}, 0)$ ⁠; when $z \to + \infty$ ⁠, these trajectories move away from the focal point and eventually converge to infinity. It follows that the system is stable at $(ρ_{2}, 0)$ when $z \to - \infty$ and unstable at $(ρ_{2}, 0)$ when $z \to + \infty$ ⁠.

Figure 2(d) corresponds to the fourth set of data in Table 1; when $z \to \pm \infty$ ⁠, the system is unstable at $(ρ_{1}, 0)$ ⁠, and all the nearby tracks are far away from this point, and the value of the variable $ρ$ eventually tends to infinity. This indicates that the density of vehicles corresponding to this set of parameters will continue to increase, the traffic system becomes unstable, and the traffic flow will eventually become congested.

5.2 Derivation of Existence Conditions for Saddle-Node Bifurcation.

Lemma 5.3 [15]. Consider the system in which

x^{'} = f (x, λ), x \in R^{n}, λ \in R, λ

is a variable parameter. If

(x_{0,} λ)

satisfies the condition

{f (x, λ) |}_{(x_{0,} λ_{0})} = 0_{n \times 1}

at the equilibrium point, then we have

L = D_{x} {f (x, λ) |}_{(x_{0,} λ_{0})}

such that

α

and

β

are the left and right unit eigenvectors of

L

, i.e.,

α L = 0

and

L β = 0

, respectively, then

λ = λ_{0}

is a saddle-node type bifurcation of the system when the following condition is satisfied:

a = {α \cdot \frac{\partial}{\partial λ} f (x, λ) |}_{(x_{0}, λ_{0})} \neq 0

(54)

b = α \sum_{i = 1}^{n} e_{i} [β^{T} \frac{\partial^{2}}{\partial x^{2}} f (x, λ) |_{(x_{0}, λ_{0})} β] \neq 0

(55)

For an arbitrarily small

ε > 0

⁠, the approximate expression for the solution curve around

(x_{0}, λ_{0})

is

x = ε β + o (ε^{2}), λ = \frac{ε^{2} b}{2 a} + o (ε^{3})

(56)

For system (47), let $q_{*}$ be a variable parameter and the matrix of derivatives at the equilibrium point is shown in Eq. (60). When $q_{*}_{_{0}} = - ρ_{0}^{2} V_{e} (ρ_{0})$ ⁠, there exists $β = (\begin{matrix} 1 \\ 0 \end{matrix})$ satisfies $L β = 0$ ⁠.

Existence

α L = 0 \Rightarrow α = (\frac{2}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2}} (ω V_{e}^{'} (ρ_{i}) + λ) Δ - \frac{q_{*}_{_{0}}}{ρ_{0}} 1)

(57)

Substituting the variables into Eqs. and yields

\begin{matrix} a = \partial \frac{\partial}{\partial q_{*}} f (x, q_{*}_{_{0}}) |_{(x_{0, q_{*}_{_{0}}})} = (\frac{2}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2}} (ω V_{e}^{'} (ρ_{i}) + λ) Δ - \frac{q_{*}_{_{0}}}{ρ_{0}} 1) (\begin{matrix} 0 \\ - \frac{2 ρ_{0}}{T (ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} q_{*}_{_{0}}} \end{matrix}) \\ = - \frac{2 ρ_{0}}{T (ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} q_{*}_{_{0}}} \neq 0 \end{matrix}

(58)

\begin{matrix} b = α \sum_{i = 1}^{n} e_{i} [{β^{T} \frac{\partial^{2}}{\partial x^{2}} f (x, λ) |}_{(x_{0}, λ_{0})} β] = (\frac{2}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2}} (ω V_{e}^{'} (ρ_{i}) + λ) Δ - \frac{q_{*}_{_{0}}}{ρ_{0}} 1) \\ . (\begin{matrix} (1 0) (\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}) (\begin{matrix} 1 \\ 0 \end{matrix}) \\ (1 0) (\begin{matrix} \frac{4 V_{e} (ρ_{0}) + 8 ρ_{0} V_{e}^{'} (ρ_{0}) + 2 ρ_{0}^{2} V_{e} ″ (ρ_{0}) - 4 c}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} q_{*}_{_{0}}} & - \frac{2 q_{*}}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} ρ_{0}^{2}} \\ - \frac{2 q_{*}}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} ρ_{0}^{2}} & 0 \end{matrix}) (\begin{matrix} 1 \\ 0 \end{matrix}) \end{matrix}) \\ = \frac{4 V_{e} (ρ_{0}) + 8 ρ_{0} V_{e}^{'} (ρ_{0}) + 2 ρ_{0}^{2} V_{e} ″ (ρ_{0}) - 4 c}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} q_{*}_{_{0}}} \neq 0 \end{matrix}

(59)

So when $q_{*}_{_{0}} = - ρ_{0}^{2} {V^{'}}_{e} (ρ_{0})$ ⁠, system (41) has a saddle-node type bifurcation at $q_{*} = q_{*}_{_{0}}$ ⁠.

5.3 Derivation of Hopf Bifurcation Conditions for the Model.

The condition for the existence of hopf bifurcation follows from Lemma 4.1 [16]. For system , let

q_{*}

be a variable parameter which has an equilibrium point

(ρ_{0}, 0)

for everything

q_{*}

⁠, and the derivative matrix

L

at the equilibrium point is also the Jacobian matrix of the system at the equilibrium point as follows:

L (q_{*}) = (\begin{matrix} 0 & 1 \\ \frac{2 (q_{*} + ρ^{2} V_{e'} (ρ))}{T (ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} q_{*}} & \frac{2}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2}} (\frac{q_{*}}{ρ} - (ω V_{e}^{'} (ρ_{i}) + λ) Δ) \end{matrix}) |_{\begin{matrix} ρ = ρ_{0} \\ q_{*} = q_{*}_{_{0}} \end{matrix}} ≜ (\begin{matrix} b_{1} & b_{2} \\ b_{3} & b_{4} \end{matrix})

(60)

Let its eigenvalues be

λ

⁠,

λ = R (q_{*}) \pm j I (q_{*})

⁠, then its characteristic equation is

λ^{2} - (b_{1} + b_{2}) λ + (b_{1} b_{4} - b_{2} b_{3}) = 0

(61)

Let the pair of eigenvalues of this equation be

R (q_{*}) \pm j I (q_{*})

⁠. Then we can obtain

R (q_{*}) = \frac{b_{1} + b_{4}}{2} = \frac{q_{*} - (ω V_{e}^{'} (ρ_{i}) + λ) Δ ρ_{0}}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} ρ_{0}}

(62)

I (q_{*}) = \sqrt{(b_{1} b_{4} - b_{2} b_{3}) - \frac{{(b_{1} + b_{4})}^{2}}{4}} = \sqrt{\frac{- 2 q_{*} - 2 ρ^{2} V_{e}^{'} (ρ)}{T (ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} q_{*}} - \frac{{(q_{*} - (ω V_{e}^{'} (ρ_{i}) + λ) Δ ρ)}^{2}}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} ρ^{2}}}

(63)

c = R' (q_{*}) | q_{*}_{_{0}} = \frac{1}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} ρ_{0}} \neq 0

(64)

Let

R (ρ_{0}, q_{*}_{_{0}}) = 0

be

R (ρ_{0}, q_{*}_{_{0}}) = (\frac{q_{*}_{_{0}} - (ω V_{e}^{'} (ρ_{i}) + λ) Δ ρ_{0}}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} ρ_{0}}) |_{q_{*} = q_{*}_{_{0}}} ≜ 0

(65)

Thus, it follows that

q_{*}_{_{0}} = (ω V_{e}^{'} (ρ_{i}) + λ) Δ ρ_{0}

(66)

Also at the same time

I (ρ_{0}, q_{*}) = {\sqrt{\frac{- 2 q_{*} - 2 ρ^{2} V_{e}^{'} (ρ)}{T (ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} q_{*}} - \frac{{(q_{*} - (ω V_{e}^{'} (ρ_{i}) + λ) Δ ρ)}^{2}}{(ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} ρ^{2}}} |}_{\begin{matrix} ρ = ρ_{0} \\ q_{*} = q_{*}_{_{0}} \end{matrix}}

(67)

From Eq.

I (ρ_{0}, q_{*}) = \sqrt{\frac{- 2 q_{*} - 2 ρ^{2} V_{e}^{'} (ρ)}{T (ω V_{e}^{'} (ρ_{i}) + λ) Δ^{2} q_{*}}} |_{\begin{matrix} ρ = ρ_{0} \\ q_{*} = q_{*}_{_{0}} \end{matrix}}

(68)

Since ${V^{'}}_{e} (ρ) < 0$ ⁠, when $- ρ_{0}^{2} V_{e} (ρ_{0}) > q_{*}_{_{0}} > 0$ ⁠, $I (ρ_{0}, q_{*}_{_{0}}) > 0$ ⁠. At this point, the system has a hopf bifurcation at $q_{*} = q_{*}_{_{0}}$ ⁠.

5.4 Derivation of Hopf Bifurcation Types for Model.

For the hopf bifurcation problem, only the hopf bifurcation of 2D systems is considered since the n-dimensional systems can be restricted to 2D central manifolds by the central manifold method.

By Lemma 4.2 [16]: set up a two-dimensional system containing the parameter

x^{'} = f (x, φ) x = {(x_{1}, x_{2})}^{T}, φ \in R

(69)

The equilibrium point of the system is the origin

o (0, 0)

⁠, and there are eigenvalues at the origin

R (φ) \pm j I (φ)

⁠. Also when

φ = 0

⁠, the system has purely imaginary eigenvalues of the partial derivative matrix

j I_{0}

⁠, i.e.,

R (0) = 0, I (0) = I_{0} > 0

⁠. After the coordinate transformation, the system can be written as follows:

{\begin{matrix} x_{1}^{'} = R (φ) x_{1} - I (φ) x_{2} + \tilde{f_{1}} (x_{1}, x_{2}, φ) \\ x_{2}^{'} = R (φ) x_{1} + I (φ) x_{2} + \tilde{f_{2}} (x_{1}, x_{2}, φ) \end{matrix}

(70)

which is ${\tilde{f}}_{1}, {\tilde{f}}_{2} = O ({x^{'}}_{1} + {x^{'}}_{2})$ ⁠.

Theorem 5.1 [17] (hopf bifurcation theorem for models). Let the partial derivative matrix of systemat the origin

J (φ)

have eigenvalues

R (φ) + j I (φ)

and satisfy

R (φ) = 0, I (φ) = I_{0} > 0, c = R^{'} (φ) \neq 0

, then we have the following analytic function:

φ (ε) = \sum_{k = 2}^{\infty} φ_{k} ε^{k}

(71)

such that for

φ = φ (ε) \neq 0

the systemhas a unique closed orbit in a sufficiently small neighborhood at the origin with period

T_{ε} = \frac{2 π}{I_{0}} (1 + \sum_{k = 2}^{\infty} τ_{k} ε^{k})

(72)

When $ε \to 0$ ⁠, $φ (ε) \to 0$ ⁠, $Γ_{ε}$ converge to the origin. Denote the coefficients of the first nonzero term in the expansion(71)as $φ_{k 1}$ , then $Γ_{ε}$ is a stable limit ring when $φ_{k 1}$ is congruent with c, and $Γ_{ε}$ is an unstable limit ring when $φ_{k 1}$ is dissimilar with c.

The proof of Theorem 5.1 and the formula for the coefficient

φ_{k}, τ_{k}

are stated in the literature [18] when

k = 2

φ_{2} = - a / c, τ_{2} = - (b + φ_{2} I^{'} (0)) / I_{0}

(73)

included among these

\begin{matrix} a = (1 / 16) [{\tilde{f}}_{1 x x x} + {\tilde{f}}_{1 x x y} + {\tilde{f}}_{2 x x y} + {\tilde{f}}_{2 y y y}] + (1 / 16 I_{0}) [{\tilde{f}}_{1}_{x y} ({\tilde{f}}_{1 x x} + {\tilde{f}}_{1 y y}) - {\tilde{f}}_{2 x y} ({\tilde{f}}_{2 x x} + {\tilde{f}}_{2 y y}) - {\tilde{f}}_{1 x x} {\tilde{f}}_{2 x x} + {\tilde{f}}_{1 y y} {\tilde{f}}_{2 y y}] \end{matrix}

(74)

where the partial derivative of ${\tilde{f}}_{1}, {\tilde{f}}_{2}$ at $(0, 0, 0)$ is denoted as ${\tilde{f}}_{1}_{xxx}, {\tilde{f}}_{2}_{xxx}$ ⁠, etc. Here, the parameter a is used to determine the stability of the limit ring, assuming that the conditions of Theorem 5.1 [17] hold, and by hopf bifurcation theory, one can conclude the following explicit stability:

When $a < 0$ ⁠, the limit ring $Γ_{ε}$ is stable; when $a > 0$ ⁠, the limit ring $Γ_{ε}$ is unstable.
A hopf bifurcation is supercritical if $a$ is the same number as $c$ ⁠, and subcritical if $a$ is a different number from $c$ ⁠.
If $a = 0$ ⁠, then the hopf bifurcation is degenerate.

For the system , there is a variable parameter

q_{*}

⁠, which has an equilibrium point

(ρ_{0}, 0)

for everything

q_{*}

⁠, so that

\tilde{ρ} = ρ - ρ_{0}

performs a coordinate translation to move the equilibrium point to the origin, at which point the system can be expressed as follows:

{\begin{matrix} \tilde{ρ^{'}} = y \\ y^{'} = \frac{2}{(λ + N) Δ^{2}} (\frac{q_{*}}{\tilde{ρ} + ρ_{0}} - (λ + N) Δ) y - \frac{2 (\tilde{ρ} + ρ_{0})}{T (λ + N) Δ^{2} q_{*}} \end{matrix} [q_{*} + c (\tilde{ρ} + ρ_{0}) - (\tilde{ρ} + ρ_{0}) V_{e} (\tilde{ρ} + ρ_{0})]

(75)

A Taylor expansion of the system at the equilibrium point

(\tilde{ρ}, y) = (0, 0)

yields a linear form of

\tilde{x^{'}} = L (q_{*}) \tilde{x} + f

(76)

where the components of the smooth vector function

f

⁠,

f_{1, 2}

are a Taylor expansion of the least quadratic term about

\tilde{x}

⁠, which can be expressed in the following form:

f = [\begin{matrix} 0 \\ k_{11} {\tilde{ρ}}^{2} + k_{22} y^{2} + k_{12} \tilde{ρ} y + k_{111} {\tilde{ρ}}^{3} + k_{222} y^{3} + k_{112} {\tilde{ρ}}^{2} y + k_{122} \tilde{ρ} y^{2} + o {(\tilde{ρ}, y)}^{4} \end{matrix}]

(77)

The Jacobi matrix

L (q_{*})

has the following form:

L (q_{*}) = (\begin{matrix} 0 & 1 \\ \frac{2 (q_{*} + ρ_{0}^{2} {V^{'}}_{e} (ρ_{0}))}{T (λ + N) Δ^{2} q_{*}} & \frac{2}{(λ + N) Δ^{2}} (\frac{q_{*}}{ρ_{0}} - (λ + N) Δ) \end{matrix}) = (\begin{matrix} 0 & 1 \\ m (q_{*}) & n (q_{*}) \end{matrix})

(78)

Its characteristic equations and eigenvalues are as follows:

λ^{2} - σ λ + Δ = 0

(79)

where

σ = σ (q_{*}) = n (q_{*}) = tr L (q_{*}), Δ = Δ (q_{*}) = - m (q_{*}) = \det L (q_{*})

λ_{12} (q_{*}) = \frac{1}{2} (σ (q_{*}) \pm \sqrt{σ {(q_{*})}^{2} - 4 Δ (q_{*})})

(80)

By the hopf bifurcation condition

σ (0) = 0, Δ (0) = I_{0}^{2} > 0

(81)

Introducing variables for the smaller

| q_{*} |

R (q_{*}) = \frac{1}{2} σ (q_{*}), I (q_{*}) = \frac{1}{2} \sqrt{4 Δ - σ^{2} (q_{*})}

(82)

Its eigenvalue expression can be obtained as

λ_{1} (q_{*}) = λ (q_{*}), λ_{2} (q_{*}) = \bar{λ} (q_{*})

(83)

Among them

λ (q_{*}) = R (q_{*}) + j I (q_{*}), R (0) = 0, I (0) = I_{0} > 0

(84)

Let the eigenvector of

L (q_{*})

corresponding to the eigenvalue

λ (q_{*})

be

I_{re} + j I_{im}

⁠, then we have

L (I_{re} + j I_{im}) = j I_{0} (I_{re} + j I_{im})

(85)

Since the real and imaginary parts of both sides of the equation are equal, we get

{\begin{matrix} L I_{im} = I_{0} I_{re} \\ L I_{re} = - I_{0} I_{im} \end{matrix}

(86)

Organizing Eq. yields the following relationship:

L [I_{im} I_{re}] = [I_{im} I_{re}] [\begin{matrix} 0 & - I_{0} \\ I_{0} & 0 \end{matrix}]

(87)

It follows that

{[I_{im} I_{re}]}^{- 1} L [I_{im} I_{re}] = [\begin{matrix} 0 & - I_{0} \\ I_{0} & 0 \end{matrix}]

(88)

honorific title

\tilde{y} = {[I_{im} I_{re}]}^{- 1} \tilde{x}

(89)

assume (office)

\tilde{y^{'}} = {[I_{im} I_{re}]}^{- 1} {\tilde{x}}^{'}

(90)

Substituting Eq. into Eq. yields

\begin{matrix} \tilde{y^{'}} = {[\begin{matrix} I_{im} & I_{re} \end{matrix}]}^{- 1} L [\begin{matrix} I_{im} & I_{re} \end{matrix}] \tilde{y} + {[\begin{matrix} I_{im} & I_{re} \end{matrix}]}^{- 1} f \\ = [\begin{matrix} 0 & - I_{0} \\ I_{0} & 0 \end{matrix}] \tilde{y} + {[\begin{matrix} I_{im} & I_{re} \end{matrix}]}^{- 1} f \end{matrix}

(91)

and the feature vector of

L (q *)

can be calculated as follows:

I_{re} + j I_{im} = [\begin{matrix} 0 \\ 1 \end{matrix}] + j [\begin{matrix} - \frac{1}{\sqrt{- m (q_{*})}} \\ 0 \end{matrix}]

(92)

Substituting Eq. and the values of

I_{im}

and

I_{re}

into Eq. yields

\tilde{y^{'}} = [\begin{matrix} 0 & - I_{0} \\ I_{0} & 0 \end{matrix}] \tilde{y} + [\begin{matrix} 0 \\ k_{11} {\tilde{ρ}}^{2} + k_{22} y^{2} + k_{12} \tilde{ρ} y + k_{111} {\tilde{ρ}}^{3} + k_{222} y^{3} + k_{112} {\tilde{ρ}}^{2} y + k_{122} \tilde{ρ} y^{2} + O {(ρ, y)}^{4} \end{matrix}]

(93)

Since Eq. is of the same form as Eq. , the value of a in system can be calculated as follows according to Eq. :

a = (1 / 16) [{\tilde{f}}_{2 {\tilde{y}}_{1} {\tilde{y}}_{1} {\tilde{y}}_{2}} + {\tilde{f}}_{2 {\tilde{y}}_{2} {\tilde{y}}_{2} {\tilde{y}}_{2}}] + (1 / 16 I_{0}) [- {\tilde{f}}_{2 {\tilde{y}}_{1} {\tilde{y}}_{2}} ({\tilde{f}}_{2 {\tilde{y}}_{1} {\tilde{y}}_{1}} + {\tilde{f}}_{2 {\tilde{y}}_{2} {\tilde{y}}_{2}})]

(94)

So, the value of

c

can be calculated as follows:

c = R^{'} (0) = \frac{1}{(λ + N) Δ^{2} ρ_{0}} > 0

(95)

Summarizing the above analysis, for system (47), hopf bifurcation is supercritical when $a < 0$ ⁠; hopf bifurcation is subcritical when $a > 0$ ⁠.

5.5 Simulation of the Impact of Smart Device Models on Traffic Flow

5.5.1 Influence of Different Initial Densities on Traffic Flow.

For the traffic flow model , in order to verify the fit between the model and the reality, the influence of different initial densities on expressway traffic flow was considered, and a local small disturbance was added to the initial uniform traffic flow. The expression of initial density was given as follows:

ρ (x, 0) = ρ_{0} + Δ ρ_{0} {\cosh^{-}^{2} [\frac{160}{L} (x - \frac{5 L}{16})] - \frac{1}{4} \cosh^{- 2} [\frac{40}{L} (x - \frac{11 L}{32})]}, x \in [0, L]

(96)

v (x, 0) = V (ρ (x, 0)), x \in [0, L]

(97)

where

ρ_{0}

is the initial density,

Δ ρ_{0} = 0.01 veh / m

is the disturbance density, and

L = 32.2 km

is the section length. The dynamic critical conditions are given as follows:

ρ (1, t) = ρ (2, t), ρ (L, t) = ρ (L - 1, t), v (1, t) = v (2, t), v (L, t) = v (L - 1, t)

(98)

In order to facilitate the simulation, the spatial spacing is set to

100 m

⁠, and the time interval is set to

1 s

⁠. Other parameters in the model are described as follows:

T = 10 s, V_{f} = 30 m / s, ρ_{m} = 0.2 veh / m, β = 0.3, S_{d} = 100 m

(99)

When the parameter values are as above, the critical density of the model is $0.052 veh / m$ and $0.103 veh / m$ ⁠, that is, the traffic flow is linearly unstable when the initial density is in the range of $0.052 veh / m < ρ_{0} < 0.103 veh / m$ ⁠.

When the initial density is $0.052 veh / m$ the density is within the stable range of the model. When a small disturbance is applied to the traffic flow at this time, it can be seen from Fig. 3 that the traffic flow can become stable again with the passage of time. The initial density selected is $0.064 veh / m$ ⁠, which is within the instability range of the model. At this time, the fluctuation amplitude of traffic flow density caused by small perturbations increases significantly, and it evolves into a local clustering phenomenon, indicating that traffic congestion occurs and the traffic system is in an unstable state.

Fig. 3

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Space–time evolution of density waves at different initial densities $ρ_{0}$ ⁠: (a) $ρ_{0}$ = 0.052 veh/m, (b) $ρ_{0}$ = 0.064 veh/m, (c) $ρ_{0}$ = 0.081 veh/m, and (d) $ρ_{0}$ = 0.103 veh/m

As shown in Figs. 3(b) and 3(c), when the initial density is between $0.064 veh / m$ and $0.081 veh / m$ ⁠, the higher the initial density, the larger the unstable region, and the more obvious the traffic flow clustering phenomenon. In addition, the traffic flow will evolve into a local cluster, resulting in a stop-and-go phenomenon, when the traffic flow is in an unstable state. As shown in Fig. 3(d), when the density increases to $0.103 veh / m$ ⁠, the density wave will return to a stable state, at which time the traffic flow gradually tends to a uniform flow. In summary, the model can well reflect the relationship between traffic flow state and initial density, and the correctness of the model has been verified.

5.5.2 Impact of Smart Devices on Traffic Flow.

Next, the influence of intelligent devices on traffic flow is considered, and the evolution of the spatiotemporal density graph under different response degrees of drivers is used to verify whether the theoretical results of the model are consistent with the actual traffic conditions.

As can be seen from Fig. 4, as the value of $ω$ increases, the amplitude and range of density wave gradually increase, which indicates that when the vehicle is driving on a certain road section, the on-board equipment will make different prompts and responses according to the road conditions. The less familiar the driver is with the operation of the intelligent system, the slower the response will be, and the more obvious the change in vehicle speed will be. The driver’s response to stimuli (acceleration) is also significantly stronger than the situation after familiarity with the vehicle equipment, and the traffic situation will become unstable. And with the slower the response, the unstable area of traffic flow also gradually increases. This also verifies the consistency of theoretical analysis and practical situation.

Fig. 4

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Spatial and temporal evolution of density under different reaction degrees of drivers when the initial density $ρ_{0}$ is 0.45 veh/m: (a) $ω$ = 15.11 s, (b) $ω$ = 10.06 s, (c) $ω$ = 8.51 s, and (d) $ω$ = 5.36 s

6 Establishment and Control Simulation of Hopf Branch Feedback Controller for Intelligent Device Model

After studying and understanding the bifurcation phenomenon in traffic flow and the state changes of traffic flow through various bifurcation, sometimes in order to achieve a certain traffic state, it is necessary to change the different characteristics of various bifurcation of the traffic dynamics system through some control methods, and control the unstable bifurcation points in the system. The unstable bifurcation points can be controlled by designing corresponding control schemes to make the unstable bifurcation points become stable bifurcation points, or delay or even disappear, so as to achieve the ideal stable traffic state. Introducing the bifurcation control theory into the state control of traffic flow not only helps to improve the stability of the traffic flow, avoiding traffic congestion, but also provides some theoretical support for the guiding suggestions to alleviate traffic congestion.

Based on the microscopic traffic flow model considering the driver’s reaction time delay under the influence of intelligent vehicle control system, a linear state feedback method is used to study the control of hopf bifurcation in the traffic flow system, and a linear state feedback term is added to the nonlinear ordinary differential form suitable for bifurcation analysis generated from the model by traveling-wave substitution to generate the corresponding controlled system, and to prove the existence of the hopf bifurcation in the controlled system, and this method is employed to alter the characteristics of bifurcation phenomena in the microscopic traffic flow model, thereby achieving the desired dynamic behavior of the system.

The objective of bifurcation control is to design a controller to modify the bifurcation properties of the original nonlinear system so as to obtain the desired dynamical behavior. Consider a simple two-dimensional system that

{\begin{matrix} \dot{x} = f (x, y; μ) \\ \dot{y} = g (x, y; μ) \end{matrix}

(100)

where $μ$ is a real parameter, $f$ and $g$ are continuously differentiable functions and satisfy the condition $f (x_{*}, y_{*}; μ_{*}) = g (x_{*}, y_{*}; μ_{*}) = 0$ ⁠, then $(x_{*}, y_{*})$ is the equilibrium point of the system. Let $u$ be the designed feedback controller, and assume that the system is at point $(x_{0}, y_{0}; μ_{0})$ ⁠.

There is no hopf bifurcation at the point, but it is desired to be able to generate a hopf bifurcation at that point, then the design controlled system is

{\begin{matrix} \dot{x} = f (x, y; μ) \\ \dot{y} = g (x, y; μ) + u (x, y; μ) \end{matrix}

(101)

From the definition of bifurcation, it can be seen that if the controlled system wants bifurcation to occur at that point, it must first satisfy the equilibrium point. According to the control theory, the controller should be designed to be as simple and practical as possible without affecting the control purpose. Therefore, for the microscopic traffic flow model in this paper, the bifurcation control will be based on its nonlinear constant differential form generated by traveling wave substitution suitable for bifurcation analysis, and the state variables of the model

ρ

and

y

⁠, then the state feedback controller can be designed as follows:

u = k s_{1}^{2} \cdot y

(102)

where

k s_{1}^{2}

is the feedback controller gain. Applying a control quantity

u

to the system results in a controlled system as shown below:

{\begin{matrix} \dot{ρ} = y \\ \dot{y} = g (ρ, q_{*}) y + f (ρ, c, q_{*}) + u \end{matrix}

(103)

shows that at this point the linear state feedback control does not change the equilibrium point of the system.

According to hopf’s bifurcation theorem, for the controlled system , let

q_{*}

be a variable parameter and there exists an equilibrium

(ρ_{0}, 0)

L (q_{*}) = [\begin{matrix} 0 & 1 \\ f_{0}^{'} & g_{0} + k s_{1}^{2} \end{matrix}]

(104)

The characteristic equation is

λ^{2} - (g_{0} + k s_{1}^{2}) - f_{0}^{'} = 0

(105)

Let the eigenvalue be

λ = α (q_{*}) \pm j β (q_{*})

⁠, which gives

α (q_{*}) = \frac{g_{0} + k s_{1}^{2}}{2}

(106)

β (q_{*}) = \frac{\sqrt{- 4 f_{0}^{'} - (g_{0} + k s_{1}^{2})}}{2}

(107)

Then, it can be deduced

\dot{α} (q_{*}) = \frac{g_{0}'}{2} = \frac{1}{2 μ_{0} ρ_{0}} > 0

(108)

If

c = α (q_{*}_{_{0}}) \neq 0

is true, making

c = α (q_{*}_{_{0}}) \neq 0

can be obtained

g_{0} + k s_{1}^{2} = 0 \Rightarrow q_{*}_{_{0}} = ρ_{0} (c_{0} - k s_{1}^{2} μ_{0})

(109)

Substituting Eq. into Eq. yields

β (q_{*}_{_{0}}) = \sqrt{- f_{0}^{'}} = \sqrt{\frac{- q_{*}_{_{0}} - ρ_{0}^{2} V_{e}^{'} (ρ_{0})}{T μ_{0} q_{*}_{_{0}}}}

(110)

Therefore, it can be seen that when $0 < q_{*}_{_{0}} < ρ_{0}^{2} V_{e}^{'} (ρ_{0})$ ⁠, if $β (q_{*}_{_{0}}) > 0$ is true. Thus, the hopf bifurcation exists in the system when $q_{*} = q_{*}_{_{0}}$ ⁠.

6.1 Hopf Bifurcation Control Numerical Simulation.

By controlling the value of the parameter $k, s_{1}$ in the nonlinear feedback controller $u = k s_{1}^{2} \cdot y$ ⁠, the hopf bifurcation phenomenon can be advanced or delayed under the improved model.

When the system (103) is not added to the control equation, the system will be in the equilibrium point (p, y) = (0.109729, 0) at the f bifurcation occurs, substituting the relevant parameters to plot the phase plane diagram of the system at this bifurcation point, as shown in Fig. 5. At this time, the system does not join the feedback controller, when $z \to + \infty$ ⁠, the trajectory of the line in the graph is converging to the focal point (0.109729, 0), which indicates that at the equilibrium point, the state of the system is stable; when $z \to - \infty$ ⁠, the trajectory of the curve disperses to the far side and produces the same amplitude of the phenomenon of the oscillations, which indicates that the system at the point of the state is not stable.

Fig. 5

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Phase plane diagram of the system before control

When feedback control is added to the system, it can be seen in Fig. 6 that the state of the system changes differently when the control parameters take different values. The parameter k takes a constant value, so that when s₁= 0.5, the controlled system hopf bifurcation critical value is delayed. At this time, the trajectory of the system converges to the focus (0.096279, 0); when $z \to + \infty$ ⁠, the trajectory of the line in the figure is converging to the focus (0.096279, 0), which indicates that the state of the system at the equilibrium point is stable; when $z \to - \infty$ ⁠, the trajectory of the curve is dispersed to a far distance, and the same amplitude of the shock phenomenon will be produced, which indicates that the system at the point of the system state is not stable. Compared with Fig. 5, the case of hopf bifurcation advancement occurs, as shown in Fig. 6(a); the parameter k takes the same value, so that when $s_{1} = 0.4$ ⁠, the hopf bifurcation critical value of the controlled system is advanced. The corresponding system trajectory converges to the focal point (0.096043, 0), and when $z \to + \infty$ ⁠, the trajectory in the figure converges to the focal point (0.096043, 0), which indicates that at this equilibrium point, the state of the system is stable; when $z \to - \infty$ ⁠, the trajectory at the focal point (0.096043, 0) disperses to a far distance, and ultimately an oscillation is formed. This indicates that the system is destabilized at that equilibrium point. Compared to Fig. 5, there is a backward shift of the hopf bifurcation, as shown in Fig. 6(b). The equilibrium point (0.12506043, 0) continues to move backward when $k = 0.1, s_{1} = 0.4$ as shown in Fig. 6(c).

Fig. 6

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Phase plane diagram of controlled system change trajectories as the controlled system parameter $k, s_{1}$ is varied hop: (a) $k = 0.1, s_{1} = - 0.5$ ⁠, (b) $k = 0.1, s_{1} = - 1.2$ ⁠, and (c) $k = 0.1, s_{1} = 0.4$

The evolution of the stability of the traffic system can be analyzed by observing the changes in the density spatiotemporal map. It is found that when the values of some traffic parameters of the traffic flow in the uniform state reach the state quantity at the hopf bifurcation point, the stability of the system will be changed abruptly and there is an oscillation phenomenon. The density value at the bifurcation point is chosen as the initial average density for the spatiotemporal evolution of density, and the actual traffic phenomenon can be simulated along with the spatiotemporal displacement. In this section, the numerical simulation is carried out by considering the case of local perturbation under the condition of initial uniform density, which can show the traffic phenomenon in the system more clearly. The hopf bifurcation point state variable $ρ_{0} = 0.060061 veh / m$ is chosen as the initial uniform density value, and the density spatiotemporal evolution is plotted as shown in Fig. 7.

Fig. 7

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Spatiotemporal plot of density of the system with hopf bifurcation as initial value

As can be seen in Fig. 7, the amplitude of the initial small perturbation gradually increases with the space–time displacement, and the phenomenon of equal-amplitude periodic oscillations occurs in the system. This is exactly the nature of the limit loop solution. Figure 7 demonstrates the existence of the limit loop solution and reflects the phenomenon of oscillating congested traffic as opposed to stop-and-go traffic.

When the original system is added to the linear feedback controller $u$ ⁠, changing the values of the control parameters can see the changes in the system as shown in Fig. 8.

Fig. 8

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Spatiotemporal plot of density of the system for different parameter values: (a) $k = 0.12, s_{1} = - 0.5$ and (b) $k = 0.2, s_{1} = 0.4$

The change in the state of the system after the addition of bifurcation control nonlinear feedback to the system model is shown in Fig. 8. The change of the amplitude oscillation phenomenon in the figure, i.e., the change of hopf bifurcation, can be seen by improving the variable control parameters. When the control parameter $k = 0.12, s_{1} = - 0.5$ is added to the system after the small perturbation, the small perturbation gradually increases with the temporal displacement, and then the system appears to go and stop fluctuations. Compared with Fig. 7, the equal-amplitude oscillation phenomenon in the system is obviously advanced, i.e., the hopf bifurcation is advanced, as shown in Fig. 8(a); when the control parameter $k = 0.2, s_{1} = 0.4$ is used, the small perturbation added to the system dissipates with time, and there is no time-walking and time-stopping fluctuation, and the hopf bifurcation of the system disappears, as shown in Fig. 8(b).

For the uncontrolled system, i.e., when $k s_{1}^{2} = 0$ ⁠, the bifurcation diagram of the system is shown in Fig. 9, and the system has one hopf bifurcation and two LP bifurcations. Since the subcritical hopf bifurcation is unstable, the UHB is used to denote the subcritical hopf bifurcation and the SHB is used to denote the supercritical hopf bifurcation.

Fig. 9

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Global bifurcation diagram for an uncontrolled system when $k s_{1}^{2} = 0$

The original hopf bifurcation and the limit point (LP) bifurcation appear to be unchanged, and a new LP bifurcation is generated. When $q_{*}$ = 0.40424, there is a hopf bifurcation of the system at the point (0.10424, 0); at this time the eigenvalue is, which can be regarded as a pair of conjugate pure imaginary numbers, satisfying the hopf bifurcation condition. At this point, its first Lyapunov coefficient is 4.993256 × 10⁺⁰¹ which is greater than zero, which means that it is a subcritical hopf bifurcation, and when | $q_{*}$ | is small enough, there exists a unique unstable limit ring with period 198.930271. The LP bifurcation at (0.112377, 0) has an eigenvalue of 2.68412 × 10⁻⁰⁹, which can be considered approximately equal to zero, and the other eigenvalue of −0.0625065 is less than zero, which satisfies the condition of saddle-node bifurcation and indicates that the LP bifurcation point is stable.

A hopf bifurcation corresponds to a point where the system either maintains a periodic oscillation or develops an increasing oscillation and ultimately causes the system to collapse. The potentially threatening subcritical hopf bifurcation point greatly reduces the range of the stability domain, greatly affects the stability of the system, greatly reduces the load capacity of the system, and when the system is operated at this point and subjected to a small perturbation, an increase in oscillations occurs which ultimately leads to the collapse of the system.

The change of the response phase trajectory of the uncontrolled system before and after the subcritical hopf bifurcation is shown in Fig. 10. When $q_{*}$ < 0.41424, so that $q_{*}$ = 0.40424, the response phase-track diagram before the subcritical hopf bifurcation is shown in Fig. 10(a), and the phase trajectory gradually converges to a closed trajectory, generating an unstable limit ring with a period of 199, and the limit ring is shown in Fig. 10(b). Therefore, the uncontrolled system generates an unstable limit ring before the subcritical hopf bifurcation, and after the subcritical hopf bifurcation, the unstable limit ring gradually shrinks and merges with a stable equilibrium point at the bifurcation point to eventually disappear, and the equilibrium point becomes unstable.

Fig. 10

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Phase plane diagram plot of the response of the uncontrolled system before and after the subcritical hopf bifurcation: (a) $q_{*}$ = 0.10424 and (b) $q_{*}$ = 0.1048

For hopf bifurcation control, one can advance or delay the inherent hopf bifurcation, but the system still has a hopf bifurcation; change the type of hopf bifurcation in order to stabilize the unstable hopf bifurcation, i.e., control the criticality (i.e., subcritical or supercritical) of the hopf bifurcation; or eliminate hopf bifurcation so that the system has no hopf bifurcation.

By performing hopf bifurcation analysis on controlled and uncontrolled systems with state feedback, Table 2 lists the values of the parameter $q_{*}$ when the hopf bifurcation point occurs in the corresponding system when $k s_{1}^{2}$ takes different and increasing values.

Table 2

Changes in hopf bifurcation of the corresponding controlled system when goes to the same value and increases gradually

$k s_{1}^{2}$	Value of $q_{*}$ when UHB occurs	Value of $q_{*}$ when SHB occurs
0	0.41424	—
0.01	0.575627	—
0.02	0.529587	—
0.03	0.428383	—
0.036	0.405640	—
0.0368	0.401424	—
0.04	0.393950	—
0.05	0.373088	—
0.052	—	0.261705
0.058	—	0.280315
0.06	—	0.186007
0.0625	—	0.115198
0.0626	—	—
0.063	—	—
0.07	—	—
0.1	—	—

$k s_{1}^{2}$	Value of $q_{*}$ when UHB occurs	Value of $q_{*}$ when SHB occurs
0	0.41424	—
0.01	0.575627	—
0.02	0.529587	—
0.03	0.428383	—
0.036	0.405640	—
0.0368	0.401424	—
0.04	0.393950	—
0.05	0.373088	—
0.052	—	0.261705
0.058	—	0.280315
0.06	—	0.186007
0.0625	—	0.115198
0.0626	—	—
0.063	—	—
0.07	—	—
0.1	—	—

As can be seen from Table 2, when the system is not controlled, there is a subcritical hopf bifurcation point in the system at this time; when feedback control is added to the system and the value of $k s_{1}^{2}$ is gradually increased, the value of $q_{*}$ is gradually narrowed down, thus indicating that the hopf bifurcation of the system has been advanced and finally disappeared. The change of hopf bifurcation has gone through the subcritical, supercritical, and disappeared three processes, and when $k s_{1}^{2}$ is taken as different values in the three processes, the corresponding bifurcation diagrams of the controlled system are obtained. The corresponding bifurcation diagram of the controlled system, as shown in Fig. 11, can clearly see the changes of hopf bifurcation at different stages.

Fig. 11

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Corresponding bifurcation of the feedback controlled system when takes different values: (a) $k s_{1}^{2}$ = 0.01, (b) $k s_{1}^{2}$ = 0.03, (c) $k s_{1}^{2}$ = 0.05, (d) $k s_{1}^{2}$ = 0.058, (e) $k s_{1}^{2}$ = 0.06, and (f) $k s_{1}^{2}$ = 0.063

(1) When $0 < k s_{1}^{2} < 0.052$ ⁠, the subcritical hopf bifurcation in the controlled system is advanced and the value of $q_{*}$ decreases gradually, as shown in Figs. 11(a)–11(c). Taking $k s_{1}^{2}$ = 0.03 as an example, when $q_{*}$ = 0.428383, the hopf bifurcation appears at the point (0.10936, 0); obviously, the appearance of the hopf bifurcation is advanced, and the bifurcation diagram of the system at this time is shown in Fig. 11(b). The eigenvalue of the hopf bifurcation is 4.91178 × 10⁻⁰³ 0.6368i, which can still be regarded as a pair of conjugate pure imaginary numbers, and it satisfies the condition of the hopf bifurcation. The first Lyapunov exponent is 1.552665 × 10⁺⁰¹ which is greater than zero, so the hopf bifurcation is advanced but its type is not changed and still subcritical, and the system exists the only unstable limit ring with period 155. The phase plane diagram of the system at this point is shown in Fig. 12.

Fig. 12

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Phase plane diagram before and after subcritical hopf bifurcation in the controlled system when $k s_{1}^{2}$ = 0.03: (a) $q_{*}$ = 0.418383 and (b) $q_{*}$ = 0.438383

When $k s_{1}^{2}$ = 0.03, the response phase track change of the controlled system before and after the subcritical hopf bifurcation is shown in Fig. 12. When $q_{*}$ < 0.428383, let $q_{*}$ = 0.418383, the response phase track before the subcritical hopf bifurcation generates a closed curve, and the nearby trajectories are far away from this closed curve, thus forming an unstable limit loop as shown in Fig. 12(a). When $q_{*}$ > 0.428383, so that $q_{*}$ = 0.438383, the equilibrium point (0.10981, 0) is the focus of instability, and the response phase track spirals away from this point before the subcritical hopf bifurcation, as shown in Fig. 12(b), It can be seen that, since the hopf bifurcation point is still subcritical, the change of the phase track is in line with the previous description.

(2) Continue to increase the value of, when $k s_{1}^{2}$ = 0.058, the subcritical hopf bifurcation of the controlled system becomes supercritical, and the analysis shows that the hopf bifurcation of the system is not only advanced but also its type is changed to supercritical hopf bifurcation when 0.05 < $k s_{1}^{2}$ < 0.063, as shown in Figs. 11(d) and 11(e). Taking $k s_{1}^{2}$ = 0.06 as an example, the bifurcation diagram of the system is shown in Fig. 11(d); when $q_{*}$ = 0.186007, the hopf bifurcation is generated at the point (0.11208, 0) with the eigenvalue of 9.31687 × 10⁻⁰⁴ 0.51976i, and the first Lyapunov exponent of −2.98104 × 10⁻⁰¹ is less than zero; thus, the hopf bifurcation is supercritical, and when | $q_{*}$ | is enough hours, there exists a unique stable limit ring with period 170. The phase plane diagram of the system at this time is shown in Fig. 13.

Fig. 13

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Phase plane diagram before and after subcritical hopf bifurcation in the controlled system when $k s_{1}^{2}$ = 0.03: (a) $q_{*}$ = 0.270315 and (b) $q_{*}$ = 0.290315

When $k s_{1}^{2}$ = 0.058, the response phase trajectory change of the controlled system before and after the subcritical hopf bifurcation is shown in Fig. 13. When $q_{*}$ < 0.280315, let $q_{*}$ = 0.270315, the phase trajectory before the supercritical hopf bifurcation gradually spirals and converges to the stabilizing focus (0.113048, 0), as shown in Fig. 13(a). When $q_{*}$ > 0.280315, so that $q_{*}$ = 0.290315, the response phase trajectory after the supercritical hopf bifurcation gradually converges to a closed curve, forming a stabilizing limit ring, as shown in Fig. 12(b). Thus, for the supercritical hopf bifurcation, the phase trajectory converges to a stable equilibrium point before the bifurcation, while the equilibrium point becomes unstable after the bifurcation and generates a stable limit ring.

Based on the bifurcation analysis of macrotraffic flow model considering the influence of intelligent devices on drivers, the hopf bifurcation of the control system is analyzed and controlled by using the linear state feedback control method, and the corresponding controlled system is obtained by changing the feedback controller and analyzing the change of the hopf bifurcation.

7 Conclusion

In this paper, based on the full velocity difference (FVD) following model, a single-lane following model considering the influence of intelligent vehicle communication technology is established, and the influence of this technology on traffic flow operation is investigated. According to the feature that intelligent vehicle devices can provide traffic information in real-time, a parameter $ω$ reflecting the driver’s advance reaction degree after receiving the traffic information provided by the intelligent devices is introduced into the FVD following model, which is analyzed theoretically and simulated numerically. In the linear and nonlinear stability analysis, the neutral stability conditions and KdV–Burger equations of the model are derived, while the saddle-node bifurcation analysis of the model is carried out. In addition to this, the existence conditions and stability of hopf bifurcation of the improved model are theoretically analyzed, the hopf bifurcation of the control system is analyzed and controlled by using the linear state feedback control method, and the corresponding controlled system is obtained by changing the gain of the feedback controller and the change of the hopf bifurcation is analyzed. The feedback controller is used to advance or delay the inherent hopf bifurcation, eliminate the hopf bifurcation, change the type of hopf bifurcation to stabilize the unstable hopf bifurcation, and apply the phase plane method to study the system state changes and specific traffic phenomena caused by the different types of hopf bifurcations. The numerical simulation results show that the analysis and control of hopf bifurcation in the traffic system model are well realized by the bifurcation control method in the microscopic traffic flow theory, and the bifurcation control not only helps to control various nonlinear dynamics of traffic phenomena in the traffic flow but also provides a theoretical basis for the development of the control decision and induces and prevents traffic congestion in advance.

Funding Data

National Natural Science Foundation of China (Grant No. 72361031; Funder ID: 10.13039/501100001809).
Industrial Support Plan Project of Department of Education of Gansu Province (Grant No. 2024CYZC-09).
Gansu Province University Youth Doctoral Support Project (Grant No. 2023QB-049).

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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Bifurcation Analysis and Control of Traffic Flow Model Considering the Impact of Smart Devices for Drivers

Abstract

1 Introduction

2 Improvement of Macrotraffic Flow Model Considering Intelligent Vehicle Systems Factor

3 Linear Stability Analysis Considering Intelligent Devices Effects

4 Nonlinear Stability Analysis Considering Intelligent Devices Factors

5 Bifurcation Analysis of Traffic Flow Models Improved by Considering the Characteristics of Intelligent Vehicle Systems

5.1 Type of Equilibrium and Stability Analysis of the Model.

5.2 Derivation of Existence Conditions for Saddle-Node Bifurcation.

5.3 Derivation of Hopf Bifurcation Conditions for the Model.

5.4 Derivation of Hopf Bifurcation Types for Model.

5.5 Simulation of the Impact of Smart Device Models on Traffic Flow

5.5.1 Influence of Different Initial Densities on Traffic Flow.

5.5.2 Impact of Smart Devices on Traffic Flow.

6 Establishment and Control Simulation of Hopf Branch Feedback Controller for Intelligent Device Model

6.1 Hopf Bifurcation Control Numerical Simulation.

7 Conclusion

Funding Data

Data Availability Statement

References

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Bifurcation Analysis and Control of Traffic Flow Model Considering the Impact of Smart Devices for Drivers

Abstract

1 Introduction

2 Improvement of Macrotraffic Flow Model Considering Intelligent Vehicle Systems Factor

3 Linear Stability Analysis Considering Intelligent Devices Effects

4 Nonlinear Stability Analysis Considering Intelligent Devices Factors

5 Bifurcation Analysis of Traffic Flow Models Improved by Considering the Characteristics of Intelligent Vehicle Systems

5.1 Type of Equilibrium and Stability Analysis of the Model.

5.2 Derivation of Existence Conditions for Saddle-Node Bifurcation.

5.3 Derivation of Hopf Bifurcation Conditions for the Model.

5.4 Derivation of Hopf Bifurcation Types for Model.

5.5 Simulation of the Impact of Smart Device Models on Traffic Flow

5.5.1 Influence of Different Initial Densities on Traffic Flow.

5.5.2 Impact of Smart Devices on Traffic Flow.

6 Establishment and Control Simulation of Hopf Branch Feedback Controller for Intelligent Device Model

6.1 Hopf Bifurcation Control Numerical Simulation.

7 Conclusion

Funding Data

Data Availability Statement

References

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