Bifurcation Analysis of Driver's Characteristics in Car-Following Model

Yadav, Sunita; Redhu, Poonam

doi:10.1115/1.4063338

Abstract

According to traffic flow theory, traffic is affected not only by road conditions such as bottlenecks, the environment, interruptions, and so on but also by the driver's behavior. To control and manage increasingly complex traffic networks, it also becomes necessary to study the effects of driver characteristics significantly. In this research, a novel car-following model is proposed which considers both the driver's cautious and aggressive instincts for optimal and relative velocity integrals. To analyze the stability of the new model, a small perturbation method was used. Further, the modified Korteweg–de-Vries equations were established with the help of a reductive perturbation method. In bifurcation analysis, we examine the existence and stability of Hopf bifurcation in various systems. This helps to gain deeper insight into the behavior of these dynamical systems and can be used to develop more efficient control strategies. Numerical simulations and theoretical analyses both show that the aspects of the enhanced model related to driver characteristics have a major affect on traffic flow stability. Additionally, the model can adeptly handle traffic congestion and quickly return to its normal state if any disruption occurs.

1 Introduction

In developing cities, traffic congestion, accidents, and environmental pollution are major hindrances to people's regular travel and daily lives. Due to motorization, traffic problems had brought a serious impact on urban social and economic development. In recent years, the intricacy and nonlinearities involved in the formation of traffic jams have drawn the attention of scientists and engineers. To resolve these problems, numerous traffic flow models have been developed during the last few decades, including the car-following model [1–5], the cellular automaton model [6,7], the gas kinetic model [8,9], the continuum model [10–13], the lattice hydrodynamic model [14–19], etc. The car-following model is one of the common microscopic traffic models to study the effect of individuals in terms of headway, driver behavior, road infrastructure, energy consumption, etc.

First, in 1953, Pipes [20] introduced a car-following model to interpret the dynamics of automobiles interaction with each other as well as with the surrounding conditions. To resolve Pipe's model challenges, Newell [21] proposed a nonlinear model by taking speed into account in 1961. After Newell's model, the optimal velocity model (OVM), was put forth by Bando et al. [22], in which traffic congestion has been described in terms of stop-and-go traffic waves. Furthermore, many efforts have been done in the OVM model to investigate the effects of various realistic factors [23–30].

As we know, in describing any physical phenomena, the equations which appear in the formulation of mathematical models are nonlinear due to the complex nature of the problem. Therefore, nonlinear analysis becomes necessary in order to investigate the nonlinear behavior of the system. When a parameter is altered and reaches a critical value in the model, the qualitative status of the traffic system varies inherently. From a conceptual point of view, this is known as bifurcation, which leads to changes in traffic conditions like unrestricted flow and stop-and-go. Therefore, the bifurcation phenomenon is one of the primary causes of nonlinear behavior in traffic flow. Recently, Orosz and Stépán [31] investigated the stability and periodic bifurcation of the uniform flow equilibrium under various wave numbers in an optimal velocity model. Ren et al. [32] used the principles of bifurcation to determine the appropriate set of parameters for double time-delay feedback control to manage traffic flow. On the basis of vehicle's cautious and careless driving behavior, Zhai and Wu [33] suggested a novel optimal speed-following model and provided the model's linear stability criteria. From existing literature [34–38], we can say that the formation of bifurcation can result in the conversion of free flow to jammed flow. Therefore, investigating nonlinear patterns in the development and resolution of traffic congestion are also essential.

In actual traffic, there is an important role of driver's behavior on traffic flow at individual as well as aggregate level in predicting and estimating the traffic flow and further this behavior can be divided into two categories: aggressive and cautious. Aggressive and cautious drivers have distinct characteristics and adjust their driving behaviors in accordance with traffic situations. In cautious behavior, less experienced and cautious drivers will delay in estimating the best velocity due to timid and sluggish driving, but on the other hand, more experienced and aggressive drivers constantly adjust their estimates of optimal speed by predicting traffic circumstances. In this view, Peng et al. [39] and Cheng et al. [40] presented a traffic model by taking into account timid or aggressive behaviors in the optimal velocity model and continuum model, respectively. In the earlier studies, the impact of the driver's behavior is examined separately for both optimal velocity and relative local velocity [41–45]. However, no one has investigated the effect of a driver's behavior in the sense of being aggressive and cautious with bifurcation analysis, up to our knowledge. Therefore, in this study, a new car-following model has been explored by taking into account the optimal and relative velocity integrals for driver's aggressive and cautious behavior. Additionally, bifurcation analysis has been carried out for a better understanding of the nonlinear behavior of traffic flow.

The following is a description of the paper's structure. In Sec. 2, the fundamental models and proposed novel car-following model are analyzed. Sections 3 and 4 separately address linear and nonlinear stability. Stability analysis via bifurcation is studied in Sec. 5. Numerical simulation is carried out in Sec. 6. Finally, Sec. 7 is devoted to the conclusion.

2 Proposed Model

Jiang et al. [24] developed a “full velocity difference (FVD) model” and the model equation are given as

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

(1)

where $Δ x_{n} = x_{n + 1} - x_{n}$ denotes the headway between leading and following vehicle, $x_{n}$ denotes the position of the $n th$ vehicle, $Δ v_{n} (t) = v_{n + 1} - v_{n}$ is the relative speed, $v_{n}$ is the speed of the $n th$ vehicle, and $V (Δ x_{n})$ denotes the optimal velocity function, $a$ and $λ$ are the sensitivity coefficient of driver and relative speed, respectively.

Driving styles can be divided into two distinct categories: the aggressive and the cautious. Aggressive drivers are quick to respond to changes in their environment, often making lane changes and overtaking. On the other hand, cautious drivers tend to start slower, proceeding with caution and braking more often. Both driving styles have their own merits, but it is important for drivers to understand when each style should be used for optimal safety on the road. As we know, in actual traffic flow, each motorist behaves differently while driving; either they drive slowly owing to reckless driving, which causes delays, or they drive quickly due to skillful driving, which enables them to reach their anticipative optimal velocity. The above model is improved by including the velocity integral in

[t, t + τ_{0}]

and

[t, t - τ_{0}]

⁠, respectively, for cautious and aggressive effects on single lane highways. The model equation can be written as

\begin{matrix} \frac{d v_{n} (t)}{d t} = a [p V (\frac{1}{τ_{o}} \int_{t}^{t + τ_{o}} Δ x_{n} (u) d u) + (1 - p) V (\frac{1}{τ_{o}} \int_{t - τ_{o}}^{t} Δ x_{n} (u) d u) - v_{n} (t)] + λ [p (\frac{1}{τ_{o}} \int_{t}^{t + τ_{o}} Δ v_{n} (u) d u) + (1 - p) (\frac{1}{τ_{o}} \int_{t - τ_{o}}^{t} Δ v_{n} (u) d u)] \end{matrix}

(2)

where the weight parameter associated with two types of driver behavior characteristics is represented by $p (0 \leq p \leq 1)$ ⁠. It can be depicted that when $0.5 < p < 1$ ⁠, the aggressive behavior is dominated while cautious is dominated when $0 < p < 0.5$ ⁠. For $p = 1$ and $p = 0$ ⁠, the driver behavior is completely aggressive and cautious, respectively, but the contribution of the driver characteristics becomes equal for $p = 0.5$ ⁠.

By mean value theorem of integrals, there must be a

τ_{1} \in [t - τ_{o}, t + τ_{o}]

⁠, so we have

Δ x_{n} (t - τ_{1}) = \frac{1}{τ_{o}} \int_{t - τ_{o}}^{t} Δ x_{n} (u) d u, Δ x_{n} (t + τ_{1}) = \frac{1}{τ_{o}} \int_{t}^{t + τ_{o}} Δ x_{n} (u) d u

(3)

Δ v_{n} (t - τ_{1}) = \frac{1}{τ_{o}} \int_{t - τ_{o}}^{t} Δ v_{n} (u) d u, Δ v_{n} (t + τ_{1}) = \frac{1}{τ_{o}} \int_{t}^{t + τ_{o}} Δ v_{n} (u) d u

(4)

So, Eq. becomes

\frac{d v_{n} (t)}{d t} = a [p V (Δ x_{n} (t + τ_{1})) + (1 - p) V (Δ x_{n} (t - τ_{1})) - v_{n} (t)] + λ [p (Δ v_{n} (t + τ_{1})) + (1 - p) (Δ v_{n} (t - τ_{1}))]

(5)

Let $τ_{1} = α τ,$ where $α$ measures the different responses of aggressive and cautious drivers to road traffic conditions.

Then, Eq. yields

\frac{d v_{n} (t)}{d t} = a [p V (Δ x_{n} (t + α τ)) + (1 - p) V (Δ x_{n} (t - α τ)) - v_{n} (t)] + λ [p (Δ v_{n} (t + α τ)) + (1 - p) (Δ v_{n} (t - α τ))]

(6)

When

α = 0

and

p = 1

⁠, the new model reduces to the FVDM [24]. For the sake of simplicity, the nonlinear factors are ignored while determining the Taylor expansion of the variable

Δ x_{n} (t + α τ) = Δ x_{n} (t) + α τ Δ v_{n} (t), Δ x_{n} (t - α τ) = Δ x_{n} (t) - α τ Δ v_{n} (t)

(7)

Δ v_{n} (t + α τ) = Δ v_{n} (t) + α τ Δ a_{n} (t), Δ v_{n} (t - α τ) = Δ v_{n} (t) - α τ Δ a_{n} (t)

(8)

\begin{matrix} V (Δ x_{n} (t + α τ)) = V (Δ x_{n} (t)) + α τ Δ v_{n} (t) V^{'} (Δ x_{n} (t)) \\ V (Δ x_{n} (t - α τ)) = V (Δ x_{n} (t)) - α τ Δ v_{n} (t) V^{'} (Δ x_{n} (t)) \end{matrix}

(9)

where

V (Δ x_{n}) = \frac{v_{max}}{2} [tanh (Δ x_{n} (t) - h_{c}) + tanh (h_{c})]

(10)

where

v_{\max}

is the maximum velocity and

h_{c}

is the safe distance. Then, Eq. can be written as

\frac{d v_{n} (t)}{d t} = a [V (Δ x_{n} (t)) - v_{n} (t)] + (2 p - 1) α V^{'} (h) Δ v_{n} (t) + λ Δ v_{n} (t) + (2 p - 1) α τ Δ a_{n} (t)

(11)

This form illustrates how the proposed model can help address these challenges, allowing for more accurate predictions of traffic flow. It can also provide valuable insights into driver behavior and the potential consequences of different driving styles.

3 Linear Stability Analysis

In this section, a linear stability analysis of the proposed model is carried out in order to examine the impact of aggressive and cautious behavior on the transition of traffic jams. Assume that all vehicles travel at the optimal speed of

V (h)

with a constant headway of

h

⁠. The steady-state of traffic flow is as follows:

x_{n}^{0} (t) = h n + V (h) t, h = \frac{L}{N}

(12)

where N and L stand for the number of cars and the length of the road, respectively. In the steady-state

x_{n}^{0} (t),

we introduce a small deviation

y_{n} (t)

which is given below

x_{n} (t) = x_{n}^{0} (t) + y_{n} (t)

(13)

Using Eqs. and into Eq. and linearizing the derived equation, we get

\frac{d^{2} y_{n} (t)}{d t^{2}} = a [V^{'} (h) (Δ y_{n} (t)) - y_{n}^{'} (t)] + (2 p - 1) α V^{'} (h) Δ y_{n}^{'} (t) + λ Δ y_{n}^{'} (t) + (2 p - 1) α τ Δ y_{n}^{''} (t)

(14)

Expanding

y_{n} (t) = e^{ikn + z t}

in Eq. by using Fourier series, we get

z^{2} = a [V^{'} (h) (e^{i k} - 1) - z] + [λ + (2 p - 1) α V^{'} (h)] z (e^{i k} - 1) + (2 p - 1) α τ z^{2} (e^{i k} - 1)

(15)

Putting

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

and

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

and ignoring terms of order greater than

2

into Eq. , we get

\begin{matrix} {[z_{1} (i k) + z_{2} {(i k)}^{2}]}^{2} = a [V^{'} (h) ((i k) + \frac{{(i k)}^{2}}{2}) - z_{1} (i k) - z_{2} {(i k)}^{2}] \\ + [λ + (2 p - 1) α V^{'} (h)] [z_{1} (i k) + z_{2} {(i k)}^{2})] [(i k) + \frac{{(i k)}^{2}}{2}] \end{matrix}

(16)

Equating, the first- and second-order terms of

i k

⁠, we obtain

z_{1} = V^{'} (h)

(17)

z_{2} = \frac{a V' (h) - 2 z_{1}^{2} + 2 [λ + (2 p - 1) α V^{'} (h)] z_{1}}{2 a}

(18)

The condition of stability is written as

a = 2 V^{'} (h) [1 - (2 p - 1) α] - 2 λ

(19)

As a result, the condition satisfies the following formula which ensures that the uniform traffic flow maintains stability under small disturbances:

a > 2 V^{'} (h) [1 - (2 p - 1) α] - 2 λ

(20)

From condition (20), we can conclude that the parameters $p$ and $α$ ⁠, both have a measurable influence on stability of traffic flow. Figure 1 shows the key similarity of the new, FVD, and OV models for cautious $(0 < p < 0.5)$ and aggressive $(0.5 < p < 1)$ characteristics. On comparing the results of the OV and FVD models with the new model, it is obvious that the new model represents a more stable zone, revealing that the proposed model is an improvement over the existing literature. In the parameter space $(h_{c}, a_{c})$ ⁠, we replicate the neutral stability curves (solid lines) with coexisting curves (dotted lines) as depicted in Fig. 2. The apex of the neutral stability line is denoted by the critical point and the zone of stability is located above the neutral curve, whereas traffic remains uniform when a small perturbation is added, while the unstable region is under the neutral stability lines in which traffic jams will appear in the form of density waves.

Fig. 1

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Comparison between OVM, FVDM, and new model

Fig. 2

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Phase diagram in headway-sensitivity with fixed λ = 0.3 (a) for different value of p at fixed α, (b) when p = 0.2 for different α, and (c) when p = 0.8 for different α

Figure 2(a) shows that an increase in parameter $p$ leads to a decrease in critical points and an increase in the stable area. This indicates that aggressive driving strategies can improve traffic flow stability better than cautious driving, as it allows for more efficient navigation and faster clearance of congested areas. This is because, aggressive drivers are able to react quickly to the surrounding vehicle information, allowing them to reduce reaction time, better predictions, and smarter decisions while operating a vehicle.

When the parameter $p$ is set to $0 < p < 0.5$ ⁠, it implies that there is a delay in sensing the information. So, with increase in the response coefficient $α$ of cautious driving, the amplitude of the neutral stability curves rises which implies that optimal and relative velocities will ultimately reduce traffic flow stability as illustrated in Fig. 2(b).

From Fig. 2(c) as the response coefficient $α$ of aggressive driving increases, the amplitude of neutral stability curves gradually falls and enhances the stability region of traffic flow, when $0.5 < p < 1$ ⁠. This means that drivers are able to sense information in advance, which allows them to adjust their actions quickly. This can help in reducing accidents and improving road safety by allowing drivers to react more quickly and accurately to changing conditions on the road. In an aggressive driving style, the stability area enhances as $α$ rises while in a cautious driving style, it reduces as $α$ rises. Therefore, the aggressive behavior of the driver plays an important role in reducing traffic congestion while it enhances with an increase in the value of the anticipation effect in the case of cautious behavior. Thus, by altering their driving behavior, it can help relieve traffic congestion and improve the stability of the transportation system.

4 Nonlinear Analysis

The nonlinear wave equation is inferred in this section to describe the nonlinear flow characteristics of traffic flow. On a coarse-grained scale, long wavelength modes are used to describe traffic flow, and the slow-changing feature of long waves are investigated near the critical stability point. For convenience of nonlinear analysis, Eq. is written as

\begin{matrix} \frac{d^{2} Δ x_{n} (t)}{d t^{2}} = a [V (Δ x_{n + 1} (t)) - V (Δ x_{n} (t)) - \frac{d Δ x_{n} (t)}{d t}] + [λ + (2 p - 1) α V^{'} (Δ x_{n} (t))] [\frac{d Δ x_{n + 1} (t)}{d t} - \frac{d Δ x_{n} (t)}{d t}] \\ + λ α τ (2 p - 1) [\frac{d^{2} Δ x_{n + 1} (t)}{d t^{2}} - \frac{d^{2} Δ x_{n} (t)}{d t^{2}}] \end{matrix}

(21)

Using the reductive perturbation method, we solve Eq. by introducing a small-scale positive parameter

ϵ

near the critical point

(h_{c}, a_{c})

⁠. For a small positive scaling parameter

ϵ

⁠, the slow variables

X

and

T

are defined as

X = ϵ (n + b t), T = ϵ^{3} t, 0 < ϵ \leq 1

(22)

where the unspecified constant is

b

⁠. The headway

Δ x_{n} (t)

is defined as

Δ x_{n} (t) = h_{c} + ϵ R (X, T)

(23)

From Eqs. – and expanding using Taylor's series expansion up to the fifth power of

ϵ

⁠, we get the following nonlinear evolution problem:

\begin{matrix} ϵ^{2} (V^{'} (h) - b) \partial_{X} R + ϵ^{3} [\frac{V^{'} (h)}{2} - τ b^{2} + [λ + (2 p - 1) α V^{'} (h)] b τ] \\ \partial_{X}^{2} R + ϵ^{4} [(\frac{V^{'} (h)}{6} + [λ + (2 p - 1) α V^{'} (h)] \frac{b τ}{2} + λ α τ^{2} (2 p - 1) b^{2}) \\ \partial_{X}^{3} R + (\frac{V^{‴} (h)}{6}) \partial_{X} R^{3} - \partial_{T} R] + ϵ^{5} [(\frac{V^{'} (h)}{24} + [λ + (2 p - 1) α V^{'} (h)] \\ \frac{b τ}{6} + λ α τ^{2} (2 p - 1) (\frac{b^{2}}{2})) \partial_{X}^{4} R + (\frac{V^{‴} (h)}{12}) \partial_{X}^{2} R^{3} \\ + (τ [λ + (2 p - 1) α V^{'} (h)] - 2 b τ) \partial_{X} \partial_{T} R] = 0 \end{matrix}

(24)

where

V^{'} (h_{c}) = \frac{d V (Δ x_{n})}{d Δ x_{n}} |_{Δ x_{n} = h_{c}}, V^{‴} (h_{c}) = \frac{d^{3} V (Δ x_{n})}{d Δ x_{n}^{3}} |_{Δ x_{n} = h_{c}}

On introducing

a_{c} / a = (1 + ϵ^{2})

⁠,

a_{c} = 2 V' (h_{c}) [1 - (2 p - 1) α] - 2 λ

⁠, and

b = V' (h)

near the critical point in Eq. and neglecting the terms of second and third orders of

ϵ

⁠, we get

ϵ^{4} (\partial_{T} R - q_{1} \partial_{X}^{3} R + q_{2} \partial_{X} R^{3}) + ϵ^{5} (q_{3} \partial_{X}^{2} R + q_{4} \partial_{X}^{4} R + q_{5} \partial_{X}^{2} R^{3}) = 0

(25)

where

\begin{matrix} q_{1} = \frac{V^{'} (h)}{6} + [λ + (2 p - 1) α V^{'} (h)] \frac{b τ_{c}}{2} + λ α τ^{2} (2 p - 1) b^{2} \\ q_{2} = - \frac{V^{‴} (h)}{6} \\ q_{3} = - b^{2} τ_{c} + [λ + (2 p - 1) α V^{'} (h)] b τ_{c} \\ q_{4} = - \frac{V^{'} (h)}{24} - [λ + (2 p - 1) α V^{'} (h)] \frac{b τ_{c}}{6} - λ α τ_{c}^{2} (2 p - 1) \frac{b^{2}}{2} + (2 b τ_{c} - τ_{c} [λ + (2 p - 1) α V^{'} (h)]) \\ (\frac{V^{'} (h)}{6} + [λ + (2 p - 1) α V^{'} (h)] + λ α τ_{c}^{2} (2 p - 1) b^{2}) \frac{b τ_{c}}{2} \\ q_{5} = \frac{V^{‴} (h)}{6} (2 b τ_{c} - τ_{c} [λ + (2 p - 1) α V^{'} (h)]) - \frac{V^{‴} (h)}{12} \end{matrix}

To obtain the mKdV equation, the following transformation (change of scale variable) is applied to Eq. :

T = \frac{1}{q_{1}} T^{'}, R = \sqrt{\frac{q_{1}}{q_{2}}} R^{'}

(26)

Therefore, with such a

O (ϵ)

correction term, the conventional mKdV equation is given as

\partial_{T^{'}} R^{'} = \partial_{X}^{3} R^{'} - \partial_{X} {R^{'}}^{3} - ϵ (\frac{q_{3}}{q_{1}} \partial_{X}^{2} R^{'} + \frac{q_{4}}{q_{1}} \partial_{X}^{4} R^{'} + \frac{q_{5}}{q_{1}} \partial_{X}^{2} {R^{'}}^{3})

(27)

On ignoring the terms

O (ϵ)

in Eq. , the “kink-antikink” soliton is defined as

R_{0}^{'} (X, T^{'}) = \sqrt{c} tanh [\sqrt{\frac{c}{2}} (X - c T^{'})]

(28)

Here,

c

is the propagation speed at which the kink-antikink solution propagates and is determined by solving the requisite condition below

R_{0}^{'}, M ([R_{0}^{'}]) \equiv \int_{- \infty}^{\infty} R_{0}^{'} M [R_{0}^{'}] d X = 0

(29)

where

M [R_{0}^{'}] = (\frac{q_{3}}{q_{1}} \partial_{X}^{2} R^{'} + \frac{q_{4}}{q_{1}} \partial_{X}^{4} R^{'} + \frac{q_{5}}{q_{1}} \partial_{X}^{2} {R^{'}}^{3})

With the help of the method described in Ref. [46], we get propagation velocity

c

as

c = \frac{5 q_{2} q_{3}}{2 q_{2} q_{4} - 3 q_{1} q_{5}}

(30)

As a result, the generic kink-antikink solution is as follows:

Δ x_{n} (t) = h_{c} \pm \sqrt{\frac{q_{1} c}{q_{2}} (\frac{τ}{τ_{c}} - 1)} \times [(1 - c q_{1}) (\frac{τ}{τ_{c}} - 1) t + n]

(31)

The amplitude is

A = \sqrt{\frac{q_{1} c}{q_{2}} (\frac{τ}{τ_{c}} - 1)}

(32)

where $V ‴ < 0$ ⁠. The coexisting curves for the jammed and the free flow phase can be described as $Δ x_{n} = h_{c} \pm A$ ⁠. The coexisting and neutral stability curves that split the phase plane into three regions—stable, metastable, and unstable are depicted by the solid and dashed lines, respectively, in Fig. 2. In the cautious behavior of a driver, the associated neutral and coexisting curves rise together with the anticipation ability coefficient $α$ ⁠. On the other hand, for aggressive behavior, the region of stability enhances as $α$ increases. Additionally, the amplitude of neutral and coexisting curves drops as the value of $α$ increases. The nonlinear analysis reveals that the mKdV equation's solution, close to the critical point, may effectively represent the propagating characteristic of traffic jams.

5 Bifurcation Analysis

In order to investigate the appropriate relationship between the critical points of Hopf bifurcation and instability, we initially conducted a theoretical analysis of the new model's bifurcation behaviors in this part. The uniform traffic state is defined as a situation in which vehicles have an optimal velocity

V (h^{*})

and a constant vehicle distance

x_{n}^{0}

v_{n}^{0} (t) = V (h^{*}) + η_{n} (t), x_{n}^{0} (t) = n h^{*} + V (h^{*}) t + ξ_{n} (t)

(33)

where

ξ_{n}

and

η_{n}

are the small deviations in position and velocity. Equation can be written as

{\begin{matrix} \frac{d η_{n} (t)}{d t} = a [p V^{'} (ξ_{n + 1} (t + α τ) - ξ_{n} (t + α τ)) + (1 - p) V^{'} (ξ_{n + 1} (t - α τ) - ξ_{n} (t - α τ)) - η_{n} (t)] \\ + λ [p (η_{n + 1} (t + α τ) - η_{n} (t + α τ)) + (1 - p) (η_{n + 1} (t - α τ) - η_{n} (t - α τ))] \\ \frac{d ξ_{n} (t)}{d t} = η_{n} (t) \end{matrix}

(34)

Taking into account the periodic boundary, Eq. may be reduced to matrix form as follows:

(\begin{matrix} \frac{d η_{n} (t)}{d t} \\ \frac{d ξ_{n} (t)}{d t} \end{matrix}) = (\begin{matrix} - a I & O \\ I & O \end{matrix}) (\begin{matrix} η_{n} (t) \\ ξ_{n} (t) \end{matrix}) + (\begin{matrix} - ϵ_{1} A & - ϵ_{2} A \\ O & O \end{matrix}) (\begin{matrix} η_{n} (t + α τ) \\ ξ_{n} (t + α τ) \end{matrix}) + (\begin{matrix} - γ_{1} A & - γ_{2} A \\ O & O \end{matrix}) (\begin{matrix} η_{n} (t - α τ) \\ ξ_{n} (t - α τ) \end{matrix})

(35)

where

O

denotes the

N \times N

null matrix,

I

denotes the

N \times N

identity matrix

A = (\begin{matrix} 1 & - 1 & 0 & \dots & 0 & 0 \\ 0 & 1 & - 1 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & 1 & - 1 \\ - 1 & 0 & 0 & \dots & 0 & 1 \end{matrix})

and $ϵ_{1} = λ p, ϵ_{2} = a p V^{'}, γ_{1} = (1 - p) λ, and γ_{2} = (1 - p) a V^{'}$ ⁠.

The corresponding characteristic equation is

{[λ^{2} + a λ + ϵ_{1} e^{α τ λ} λ + γ_{1} e^{- α τ λ} λ + ϵ_{2} e^{α τ λ} + γ_{2} e^{- α τ λ}]}^{N} - {[ϵ_{1} e^{α τ λ} λ + γ_{1} e^{- α τ λ} λ + ϵ_{2} e^{α τ λ} + γ_{2} e^{- α τ λ}]}^{N} = 0

(36)

The following equation is equivalent to Eq. :

{[\frac{λ^{2} + a λ + ϵ_{1} e^{α τ λ} λ + γ_{1} e^{- α τ λ} λ + ϵ_{2} e^{α τ λ} + γ_{2} e^{- α τ λ}}{ϵ_{1} e^{α τ λ} λ + γ_{1} e^{- α τ λ} λ + ϵ_{2} e^{α τ λ} + γ_{2} e^{- α τ λ}}]}^{N} = 1

(37)

The solution of Eq. is

\begin{matrix} λ^{2} + a λ + ϵ_{1} e^{α τ λ} λ + γ_{1} e^{- α τ λ} λ + ϵ_{2} e^{α τ λ} + γ_{2} e^{- α τ λ} = (ϵ_{1} e^{α τ λ} λ + γ_{1} e^{- α τ λ} λ + ϵ_{2} e^{α τ λ} + γ_{2} e^{- α τ λ}) \times (cos \frac{2 k π}{N} + i sin \frac{2 k π}{N}) \end{matrix}

(38)

where $k \in {1, 2, \dots, N}$ is wave number associated with the oscillation mode.

Inserting

λ = μ + i ω

(where

μ

and

ω

are real and imaginary components of

λ

⁠, respectively) into Eq. and the real and imaginary parts are

{\begin{matrix} μ^{2} - ω^{2} + a μ + ϵ_{1} e^{α τ μ} (μ c_{1} + ω s_{1}) + γ_{1} e^{- α τ μ} (μ c_{1} - ω s_{1}) + ϵ_{2} e^{α τ μ} c_{1} + γ_{2} e^{- α τ μ} c_{1} \\ = (ϵ_{1} e^{α τ μ} (μ c_{1} + ω s_{1}) + γ_{1} e^{- α τ μ} (μ c_{1} - ω s_{1}) + ϵ_{2} e^{α τ μ} c_{1} + γ_{2} e^{- α τ μ} c_{1}) c_{k} - (ϵ_{1} e^{α τ μ} (ω c_{1} + μ s_{1}) \\ + γ_{1} e^{- α τ μ} (ω c_{1} - μ s_{1}) + ϵ_{2} e^{α τ μ} s_{1} + γ_{2} e^{- α τ μ} s_{1}) s_{k} \\ 2 μ ω + a ω ϵ_{1} e^{α τ μ} (ω c_{1} + μ s_{1}) + γ_{1} e^{- α τ μ} (ω c_{1} - μ s_{1}) + ϵ_{2} e^{α τ μ} s_{1} - γ_{2} e^{- α τ μ} s_{1} \\ = (ϵ_{1} e^{α τ μ} (μ c_{1} - ω s_{1}) + γ_{1} e^{- α τ μ} (μ c_{1} + ω s_{1}) + ϵ_{2} e^{α τ μ} c_{1} + γ_{2} e^{- α τ μ} c_{1}) s_{k} - (ϵ_{1} e^{α τ μ} (ω c_{1} + μ s_{1}) \\ + γ_{1} e^{- α τ μ} (ω c_{1} - μ s_{1}) + ϵ_{2} e^{α τ μ} s_{1} + γ_{2} e^{- α τ μ} s_{1}) c_{k} \end{matrix}

(39)

where $c_{k} = cos (2 k π / N)$ ⁠, $s_{k} = sin (2 k π / N)$ ⁠, $c_{1} = cos α τ μ$ ⁠, and $s_{1} = sin α τ μ$ ⁠. Notice that for $k = N$ ⁠, the point $(μ, ω) = (0, 0)$ and $(μ, ω) = (f, 0)$ are solution of the system, where $f$ depends on $α$ and $p$ ⁠. Figure 3 illustrates how the real and imaginary components of the eigenvalues of the above equation are distributed when $k \neq N$ for $N = 7$ ⁠. The distribution of eigenvalues for aggressive and cautious factors, when $α = 0$ ⁠, is shown in Fig. 3(a). When $V' = 0$ ⁠, eigenvalues are located on the real axis, each eigenvalue disperses from the real axis to both sides as $V'$ rises. The left side of the complex plane contains all of the eigenvalues, when $V'$ is small enough, demonstrating that the system is asymptotically stable. However, when $V'$ is large enough, the eigenvalues cross the imaginary axis, and the system suddenly loses stability.

Fig. 3

View large Download slide

Eigen value of the system for N = 7 and λ = 0.3 when (a) α = 0 and (b) α = 0.2

The imaginary axis and the trajectory of the eigenvalue then cross a point known as the Hopf bifurcation point. In Fig. 3, we can see that the eigenvalues for

k = 1

and

k = 6

cross the imaginary axis first, followed by

k = 2

and

k = 5

⁠, and finally

k = 3

and

k = 4

⁠, which cross the axis last and require more time to do so. Figure 3(b) shows the distribution of eigenvalues when

α > 0

⁠. Two eigenvalues curves with

k = 1

and

k = 6

take longer to pass through the virtual axis once the

α

has a positive impact on it. Two eigenvalue curves with

k = 3

and

k = 4

have opposite trajectories from those in Fig. 3(a). There is no Hopf bifurcation point in this eigenvalue since all of its paths are on the left half of the complex plane. It is demonstrated that the aggressive and cautious characteristics can prevent or postpone the emergence of Hopf bifurcations. We can conclude that the right choice of

α

can reduce or completely eliminate the effects of Hopf bifurcation, increase system stability and prevent traffic jams. In other words, the Hopf bifurcation will happen when a pair of pure imaginary roots

λ_{1, 2} = \pm i ω

occur in the system. Thus, by adding

λ = i ω

into Eq. , the crucial condition for Hopf bifurcation, is obtained as

{\begin{matrix} - ω^{2} + ϵ_{1} ω s_{1} - γ_{1} ω s_{1} + ϵ_{2} c_{1} + γ_{2} c_{1} \\ = (ϵ_{1} ω s_{1} - γ_{1} ω s_{1} + ϵ_{2} c_{1} + γ_{2} c_{1}) c_{k} - (ϵ_{1} ω c_{1} + γ_{1} ω c_{1} + ϵ_{2} s_{1} + γ_{2} s_{1}) s_{k} \\ a ω + ϵ_{1} ω c_{1} + γ_{1} ω c_{1} + ϵ_{2} s_{1} - γ_{2} s_{1} \\ = (- ϵ_{1} ω s_{1} + γ_{1} ω s_{1} + ϵ_{2} c_{1} + γ_{2} c_{1}) s_{k} - (ϵ_{1} ω c_{1} + γ_{1} ω c_{1} + ϵ_{2} s_{1} + γ_{2} s_{1}) c_{k} \end{matrix}

(40)

The findings suggest that the driver behavior parameters play distinct roles in the critical point region of the Hopf bifurcation, but they can collectively contribute to improving the stability of the traffic system. By adjusting these factors, it is possible to influence the stability of the traffic system and potentially reduce the occurrence of chaotic or unstable behavior. Thus, in the Bifurcation analysis, we have found that the sensitivity and driver behavior parameters have a noticeable impact on the occurrence of Hopf bifurcation in the car-following model.

6 Numerical Simulation

The novel model represented by Eq. is numerically simulated under the periodic boundary condition to validate the findings of the theoretical outcomes and furthermore, the effect of the cautious and aggressive behaviors are studied under a small perturbation. On the length L = 400 of the road, all vehicles N = 100 are traveling at the same headway and also distributed evenly. A small perturbation is added in the steady-state flow of traffic, namely,

\begin{matrix} Δ x_{n} (0) = Δ x_{n} (1) = Δ x_{n} (2) = 4.0, (n \neq 50, 51) \\ Δ x_{n} (0) = Δ x_{n} (1) = Δ x_{n} (2) = 4.0 + A, (n = 50) \\ Δ x_{n} (0) = Δ x_{n} (1) = Δ x_{n} (2) = 4.0 - A, (n = 51) \end{matrix}

The numerical outcomes of the proposed model are discussed below:

6.1 Effect of Weight Parameter of Two Types of Driver Behavior Characteristics.

Figure 4(a) represents the headway profile of the vehicles for different values of $p$ at $t = 10, 300$ and it can be concluded that when a small perturbation is introduced to the conventional traffic flow, stop-and-go traffic congestion appears in the unstable region and expands in downstream with time. As there is an increase in the value $p$ ⁠, the amplitude, as well as the number of kink-antikink waves, decreases, and if we enter into the stable region, the perturbation dies out which leads to uniform flow for $p = 0.8$ ⁠. The spatiotemporal evolution of headway is shown in Fig. 5 in which the initial perturbation evolves into congested flow in the form of kink waves which propagates in the backward direction and oscillates near the critical headway as shown in Figs. 5(a)–5(c). As we reach into the stable region for $p = 0.8$ ⁠, it is clear from Fig. 5(d) that the congested flow converts into a uniform flow which indicates that the aggressive behavior of the driver reduces the traffic congestion effectively.

Fig. 4

View large Download slide

Headway profile with fixed λ = 0.3 when (a) for different values of p, (b) with fixed p = 0.2, and (c) with fixed p = 0.8 for different parameter α

Fig. 5

View large Download slide

Space-time evolution of the headway at λ = 0.3 for (a) p = 0, (b) p = 0.2, (c) p = 0.5, and (d) p = 0.8

6.2 Effect of Response Coefficient for Both Aggressive and Cautious Behavior.

The headway profiles of the cars are shown in Figs. 4(b) and 4(c) for different values of $α$ ⁠, which correspond to $p = 0.2$ and $p = 0.8$ ⁠, respectively. From Fig. 4(b), we can conclude that when there is a delay in driver's anticipation, the fluctuating amplitude of the stop-and-go waves substantially maximizes with an increase in the value of $α$ for $p < 0.5$ ⁠, while the amplitude of these waves minimizes with the increase in the value of $α$ when there is advancement in the driver's anticipation effect as shown in Fig. 4(c) for $p > 0.5$ ⁠. Figures 6 and 7 are space–time evolution of the headway with different parameter $α$ in respect to Figs. 4(b) and 4(c) with cautious and aggressive behavior, respectively.

Fig. 6

View large Download slide

Space-time evolution of the headway for p = 0.2 at (a) α = 0.8, (b) α = 0.5, (c) α = 0.2, and (d) α = 0

Fig. 7

View large Download slide

Space-time evolution of the headway with different parameter α at p = 0.8 (a) α = 0, (b) α = 0.2, (c) α = 0.4, and (d) α = 0.6

When $a = 1.2$ ⁠, we fall in unstable region from $α = 0.2$ to $α = 0.8$ and so the initial perturbation evolves into a kink-antikink wave, which oscillates near the critical headway as shown in Figs. 6(a)–6(c) and in these figures, the number of stop and go waves as well as their amplitude increases with the increment in the value of $α$ due to delay decision. Figure 6(d) shows that once we approach in the stable zone for $α = 0$ ⁠, the congested flow transforms into a uniform flow. In the case of aggressive driving, for smaller values of $α$ the provided perturbation exhibits the form of stop-and-go waves which propagate in a backward direction in course of time as shown in Figs. 7(a)–7(d). A further increase in the value of $α = 0.6$ ⁠, the density of waves dies out and the amplitude of waves approaches to zero which indicates that the traffic jam is disappeared. On comparing the results of aggressive and cautious behavior of drivers, we can say that role of the driver's anticipation effect is more prominent in case of aggressive driving. Thus, it can be concluded that aggressive driving behavior is significantly more effective in reducing traffic congestion than cautious driving behavior. Therefore, it is reasonable from theoretical and simulation results that traffic jams can be effectively suppressed by including the driver's characteristics in the traffic system.

7 Conclusion

In a real-world traffic environment, different actions and reactions of drivers can be seen on roadways due to the diverse nature of human beings. But, the effect of driver anticipation in case of aggressive and cautious behavior is rarely studied in the car-following traffic flow model. Motivated by this, we invented a novel car-following model by taking the role of the driver's characteristics at optimal and relative velocity integrals to present a more realistic phenomenon. Through the linear and nonlinear stability analysis, the neutral and coexisting stability curves are obtained near the critical point. Also, the mKdV equation is derived using the reductive perturbation approach to explore the nonlinear behavior of traffic. Further, the bifurcation analysis is conducted to get the stability condition, which is also in accordance with the linear stability analysis. From theoretical and numerical observation, it is found that cautious driving worsens traffic stability while aggressive driving improves traffic stability. Furthermore, it can be easily observed that the driver's anticipation effect destabilizes the traffic flow in case of delay while it stabilizes the traffic flow during the aggressive driving of the driver, which is in good agreement with the theoretical analysis results. It validates that driving skills have a significant impact on traffic flow. Finally, we can say that the new model is found effective in stabilizing traffic flow and alleviating traffic congestion. In the future, this work has been extended by considering the implication of the heterogeneous behavior of drivers.

Acknowledgment

The first author gratefully acknowledges the Council of Scientific and Industrial Research (CSIR), New Delhi, India for providing financial support under file no. 09/382(0245)/2019-EMR-I.

Funding Data

The Council of Scientific and Industrial Research (CSIR), New Delhi, India for providing financial support under File No. 09/382(0245)/2019-EMR-I. Ministry of Human Resource Development (Funder ID: 10.13039/501100004541).

Data Availability Statement

No data, models, or code were generated or used for this paper.

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Bifurcation Analysis of Driver's Characteristics in Car-Following Model

Abstract

1 Introduction

2 Proposed Model

3 Linear Stability Analysis

4 Nonlinear Analysis

5 Bifurcation Analysis

6 Numerical Simulation

6.1 Effect of Weight Parameter of Two Types of Driver Behavior Characteristics.

6.2 Effect of Response Coefficient for Both Aggressive and Cautious Behavior.

7 Conclusion

Acknowledgment

Funding Data

Data Availability Statement

References

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Bifurcation Analysis of Driver's Characteristics in Car-Following Model

Abstract

1 Introduction

2 Proposed Model

3 Linear Stability Analysis

4 Nonlinear Analysis

5 Bifurcation Analysis

6 Numerical Simulation

6.1 Effect of Weight Parameter of Two Types of Driver Behavior Characteristics.

6.2 Effect of Response Coefficient for Both Aggressive and Cautious Behavior.

7 Conclusion

Acknowledgment

Funding Data

Data Availability Statement

References

Get Email Alerts

Cited By

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