Rich dynamics in some discrete-time car-following models

Physica A 536 (2019) 120926

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Rich dynamics in some discrete-time car-following models Xiujuan Wang a,b , Mingshu Peng a , a b

∗

Department of Mathematics, Beijing Jiao Tong University, Beijing 100044, China Weifang University, Weifang, Shandong, 261061, China

highlights • • • •

Fewer works related to discrete-time car-following models. Propose discrete-time car-following models. Study local stability analysis in details. Find fractal properties by the computation of Lyapunov exponents and Lyapunov dimensions.

article

info

Article history: Received 28 June 2018 Available online 27 April 2019 MSC: 39A12 39A30 39A33 39A60 Keywords: Car-following models Chaotic maps Traffic flow

a b s t r a c t There are a great number of works to study delay differential equations modeling road traffic, but fewer related to discrete-time car-following models. In this paper, we propose two classes of discrete-time car-following models, which can be viewed as leader–follower models or discretization version of classic continuous-time car-following models. Local stability analysis is established in details. Rich dynamical behavior is to be explored, including local stability analysis, chaotic behavior etc. Fractal properties are discovered by the computation of Lyapunov exponents and Lyapunov dimensions. High codimensional bifurcations can be expected. We find that one of the proposed models can admit infinite nontrivial fixed points in its equivalent form but the other cannot do. Moreover, if the leading vehicle presents a regular (steady states or periodic) or irregular (chaotic) oscillation pattern, the following is to do the same likely. In a sense, a synchronous/heredity property can be exhibited in the underlying model. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In recent decades, accurate microscopic mathematical models for traffic engineering (called by car-following models) has been introduced to study the interaction of vehicle pairs and their individual regular/irregular motions in a single lane without overtaking. For example, the traditional well-known Gazis–Herman–Rothery (GHR) model was first brought forward by Gazis, DC et al. [1]. In the underlying model, the no-overtaking and inflow conditions restrict the vehicle to move in one spatial dimension. The vehicles are numbered from the front of the platoon, with the leading vehicle as vehicle 0 and the next as vehicle 1, 2, . . . , n. Specifying the stimulus as the relative velocity of vehicle and headway, the GHR model is expressed as follows: x¨ i (t + τ ) =

ax˙ m xi−1 (t) − x˙ i (t)] i (t)[˙

[xi−1 (t) − xi (t)]l

, i = 1, . . . , n,

∗ Corresponding author. E-mail address: [email protected] (M. Peng). https://doi.org/10.1016/j.physa.2019.04.162 0378-4371/© 2019 Elsevier B.V. All rights reserved.

(1.1)

2

X. Wang and M. Peng / Physica A 536 (2019) 120926

where xi (t) denotes the position of the ith vehicle at time t, di = xi−1 (t) − xi (t) is the headway of the ith vehicle, l and m are non-negative real numbers, and τ represents the driver reaction time and b > 0 is the sensitive coefficient, x˙ i (t) is denoted as the velocity and x¨ i (t) as the acceleration of vehicle i in the platoon at time t. Since there exists no allowance for the effect of the inter-car spacing independently of the relative velocity and vehicles are allowed to travel arbitrarily close together provided their velocities are identical, an attempt was made to achieve a ‘safe’ intervehicle separation and a modified GHR model of the following type was discussed by [2]: x¨ i (t + τ ) =

ax˙ m xi−1 (t) − x˙ i (t)] i (t)[˙

[xi−1 (t) − xi (t)]l

+ b[xi−1 (t) − xi (t) − Li ]3

(1.2)

where b, k > 0 and Li denotes the desired separation that the driver of vehicle i attempts to achieve from the vehicle ahead. The specific case, where the time delay τ is taken to be zero in System (1.2), has been discussed in [2]. In Ref. [3,4], the authors find that delay τ can play an import role in rich dynamic of system (1.1) (including stability analysis and periodic oscillation by Hopf bifurcations) with assumption that the leading vehicle remains constant speed. In a looped car-following model, there are a great number of works to explore rich dynamic and get the properties of traffic flows (please see [5,6] etc.). However, there is no leading vehicle as it is itself which follows the last vehicle in the stream. Until now, there are fewer papers to report nonzero delay results in an un-looped car-following model of the type (1.1) or (1.2), which motivate us to write this paper. In this paper, we are to propose discrete-time car-following models, which can be viewed as leader–follower models and discretization version of the continuous-time car-following model (1.1) or (1.1). Rich dynamical behavior is to be explored. We are interest in the motion of the following vehicles when the velocity of leading vehicle can present a regular (steady states or periodic) or irregular (chaotic) motion. By the way, stability analysis is given for some simple cases. For the leading vehicle presents period-2 motions in a logistic-like car following model, please see Ref. [7] and as to the dynamic gravity model, we refer the reader to Ref. [8]. The reminder of this paper is organized in what follows: In Section 2, our discrete-time car-following model is proposed. Further discussion is given in Section 3. Local stability analysis for the given system is established in Section 4. Numerical results related to the given system are shown in Section 5. The conclusions are drawn in Section 6. 2. System description In order to specify the velocity of leading vehicle taking a regular (steady states or periodic) or irregular (chaotic) form, we introduce car-following models as follows

⎧ ⎪ ⎪ x(t + 1) = 1 + µx(t − 1) − γ x(t)2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V1 (t + 1) = v 0 + Ax(t + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ S1 (t + 1) = S1 (t) + 0.5τ (V1 (t + 1) + V1 (t)) ⎪ ⎪ ⎪ ⎨ v m (t + 1)(vi (t) − vi+1 (t)) αi+1 (t + 1) = b i+1 ⎪ (Si (t) − Si+1 (t))l ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Vi+1 (t + 1) = Vi+1 (t) + 0.5τ (αi+1 (t + 1) + αi+1 (t)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Si+1 (t + 1) = Si+1 (t) + 0.5τ (Vi+1 (t + 1) + Vi+1 (t)) ⎪ ⎪ ⎪ ⎪ ⎩ i ∈ N = {1, . . . , n − 1}

(2.1)

and

= 1 + µx(t − 1) − γ x(t)2 ; = v 0 + Ax(t + 1) = S1 (t) + 0.5τ (V1 (t + 1) + V1 (t)) v m (t + 1)(vi (t) − vi+1 (t)) αi+1 (t + 1) = b i+1 + k(Si (t) − Si+1 (t) − Li )3 (Si (t) − Si+1 (t))l Vi+1 (t + 1) = Vi+1 (t) + 0.5τ (αi+1 (t + 1) + αi+1 (t)) Si+1 (t + 1) = Si+1 (t) + 0.5τ (Vi+1 (t + 1) + Vi+1 (t)),

⎧ x(t + 1) ⎪ ⎪ ⎪ ⎪ V1 (t + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ S1 (t + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(2.2)

where αi+1 (t + 1) = x¨ i (t + τ ), Vi+1 (t + 1) = x˙ i (t + τ ) ≈ x˙ i (t) + 0.5τ (x¨ i (t + τ ) + x¨ i (t)), Si+1 (t + 1) = x(t + τ ) ≈ x(t) + 0.5τ (x˙ i (t + τ ) + x˙ i (t)). Then system (2.1) (or (2.2)) can be viewed as discretization version of the continuous-time car-following model (1.1) (or (1.2)).

X. Wang and M. Peng / Physica A 536 (2019) 120926

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3. Further discussion For simplicity, set m = 0. Model (2.1) can be rewritten in another form

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

x(t + 1)

= 1 + µy(t) − γ x(t)2

y(t + 1)

= x(t)

S1 (t + 1)

= S1 (t) + v0 τ + 0.5Aτ + 0.5Aτ µy(t)

+ 0.5Aτ x(t) − 0.5Aτ γ x(t)2 ⎪ ⎪ ⎪ Vi (t) − Vi+1 (t) ⎪ ⎪ + k(Si (t) − Si+1 (t) − Li )3 αi+1 (t + 1) =b ⎪ l ⎪ (S ⎪ i (t) − Si+1 (t)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Vi+1 (t + 1) − 0.5τ αi+1 (t + 1) = Vi+1 (t) + 0.5τ αi+1 (t) ⎪ ⎪ ⎪ ⎪ ⎩ Si+1 (t + 1) − 0.5τ Vi+1 (t + 1) = Si+1 (t) + 0.5τ Vi+1 (t),

(3.1)

where V1 = v0 + Ax(t). Let us introduce U = (x, y, S1 , α2 , V2 , S2 , . . . , αn , Vn , Sn )T and F = (1 + µy(t) − γ x(t)2 , x, S1 (t) + v (t)−v (t) v0 τ + 0.5Aτ + 0.5Aτ µy(t) + 0.5Aτ x(t) − 0.5Aτ γ x(t)2 , b (S i(t)−S i+1(t))l , Vi+1 (t) + 0.5τ αi+1 (t), Si+1 (t) + 0.5τ Vi+1 (t), i ∈ N)T . i i+1 Then system (3.1) can be rewritten in a vector form

BU(t + 1) = F (U(t)),

(3.2)

and

⎡

I ⎢0

0 N

. B=⎢ ⎢ ..

⎢

.. .

··· ··· .. .

0

.

..

⎣

0

⎤

0 0⎥

⎥ .. ⎥ , .⎥ ⎦

(3.3)

N

where I is the identify matrix of order 3, and 1 −0.5τ 0

0 1 −0.5τ

[ N =

0 0 . 1

]

Moreover we have U(t + 1) = B−1 F (U(t)),

(3.4)

where

⎡

I ⎢0

. B−1 = ⎢ ⎢ ..

0 N −1

⎢

.. .

··· ··· .. .

0

.

⎣

0

..

⎤

0 0 ⎥

⎥ .. ⎥ , . ⎥ ⎦

(3.5)

N −1

and 1 0.5τ 0

[ N

−1

=

0 1 0.5τ

0 0 . 1

]

Similarly, system (2.2) can be rewritten in a vector form U(t + 1) = B−1 F1 (U(t)),

(3.6)

where F1 = (1 + µy(t) − γ x(t) , x, S1 (t) + v0 τ + 0.5Aτ + 0.5Aτ µy(t) + 0.5Aτ x(t) − 0.5Aτ γ x(t) , 2

Si+1 (t) − Li )3 , Vi+1 (t) + 0.5τ αi+1 (t), Si+1 (t) + 0.5τ Vi+1 (t), i ∈ N)T . 4. Local stability analysis Let us first consider system (2.1) with n = 2.

2

v (t)−v (t) b (S i(t)−S i+1(t))l i i +1

+ k(Si (t) −

4

X. Wang and M. Peng / Physica A 536 (2019) 120926

4.1. n = 2 Suppose that the first leading vehicle maintains a constant velocity, V1 (t) ≡ C . By introducing the headway of the ith vehicle as di (t) = Si (t) − Si+1 (t), one can transform system (2.1) into another equivalent form

⎧ C − V2 (t) ⎪ , α2 (t + 1) = b ⎪ ⎪ ⎨ dl1 (t) C −V (t) V2 (t + 1) = 0.5τ α2 (t) + V2 (t) + 0.5τ b l 2 , ⎪ d1 (t) ⎪ ⎪ ⎩ d1 (t + 1) = d1 (t) + τ (C − V2 (t)) − 0.52 τ 2 α2 (t).

(4.1)

The linearized system of (4.1) evaluated around the equilibrium (0, C , D1 ) (the nontrivial steady solution) leads to

⎧ α2 (t + 1) = ⎪ ⎪ ⎪ ⎨

−b Dl1

V2 (t), Dl −0.5τ b

V2 (t + 1) = 0.5τ α2 (t) + 1 l V2 (t), ⎪ D1 ⎪ ⎪ ⎩ d1 (t + 1) = −0.52 τ 2 α2 (t) − τ V2 (t) + d1 (t),

(4.2)

where D1 is the desired/optimal headway. It is obvious that D1 can be any nonzero constant, which implies that there exist infinite positive equilibria in system (4.1). Furthermore the corresponding characteristic matrix of (4.2) is

⎛

λ

⎜ ⎜ ⎝

Q1 (λ) = ⎜ −0.5τ

b Dl1

λ−

Dl1 −0.5τ b

0.25τ 2

Dl1

τ

0

⎞

0

⎟ ⎟ ⎟, ⎠

(4.3)

λ−1

and the characteristic equation becomes

(

detQ1 (λ) = (λ − 1) λ2 − λ

Dl1 − 0.5τ b Dl1

+

bτ 2Dl1

)

.

(4.4)

Now, we give a detailed study of the zero distribution of a polynomial of the type

λ2 − λ

Dl1 − 0.5τ b Dl1

+

bτ 2Dl1

,

(4.5)

where b > 0, D1 > 0, l > 0 and τ > 0. By direct computation, one can find that

√ • if (3 − 2 2)τ b < Dl1 <

τb

, then there exist one pair of complex conjugate roots of (4.5) with modulus greater than 2 √ 1, and system (4.2) or system (4.1) is unstable; whereas if τ2b < Dl1 < (3 + 2 2)τ b, there exist one pair of complex conjugate eigenvalues with modulus less than 1, which implies that system (4.2) or system (4.1) is locally stable; √ • if Dl1 > (3 + 2 2)τ b, then there exists two real eigenvalues √ with absolute value less than 1, and system (4.2) or system (4.1) is locally stable; whereas if 0 < Dl1 < (3 − 2 2)τ b, there exists at least one real root with absolute value greater than 1, and system (4.2) or system (4.1) is unstable. • if Dl1 = τ2b , there can simultaneously occur λ1 = 1 and complex conjugate roots λ = ±i in (4.4), and high-codimensional bifurcations with strong resonances can be expected (related works please see [9]). 4.2. General cases Suppose that the first leading vehicle maintains a constant velocity, V1 (t) ≡ C . System (2.1) can be written as

⎧ C − V2 (t) ⎪ ⎪ α2 (t + 1) = b , ⎪ ⎪ ⎪ dl1 (t) ⎪ ⎪ ⎪ ⎪ C − V2 (t) ⎪ ⎪ , V2 (t + 1) = 0.5τ α2 (t) + V2 (t) + 0.5τ b ⎪ ⎪ ⎪ dl1 (t) ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ d1 (t + 1) = d1 (t) + τ (C − V2 (t)) − 0.5 τ α2 (t), V1+i (t) − V2+i (t) α2+i (t + 1) = b , ⎪ ⎪ dli+1 (t) ⎪ ⎪ ⎪ ⎪ ⎪ V1+i (t) − V2+i (t) ⎪ ⎪ , ⎪ ⎪ V2+i (t + 1) = 0.5τ α2+i (t) + V2+i (t) + 0.5τ b ⎪ dli+1 (t) ⎪ ⎪ ⎪ ⎪ ⎪ d1+i (t + 1) = d1+i (t) + τ (V1+i (t) − Vi+2 (t)) − 0.52 τ 2 α2+i (t), ⎪ ⎪ ⎩ i = 1, . . . , n − 2.

(4.6)

X. Wang and M. Peng / Physica A 536 (2019) 120926

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The linearized system of (4.6) evaluated around the equilibrium (0, C , D1 , . . . , 0, C , Dn−1 ) (the nontrivial steady solution) becomes

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

α2 (t + 1) =

−b Dl1

V2 (t),

V2 (t + 1) = 0.5τ α2 (t) +

Dl1 − 0.5τ b Dl1

V2 (t),

d1 (t + 1) = −0.52 τ 2 α2 (t) − τ V2 (t) + d1 (t),

V1+i (t) − V2+i (t) ⎪ ⎪ ⎪ α2+i (t + 1) = b , ⎪ ⎪ Dli+1 ⎪ ⎪ ⎪ ⎪ ⎪ V1+i (t) − V2+i (t) ⎪ ⎪ V2+i (t + 1) = 0.5τ α2+i (t) + V2+i (t) + 0.5τ b , ⎪ ⎪ Dli+1 ⎪ ⎪ ⎪ ⎩ d1+i (t + 1) = d1+i (t) − τ Vi+2 (t) − 0.52 τ 2 α2+i (t) + τ Vi+1 (t) + 0.52 τ 2 α1+i (t),

(4.7)

where Di is the desired/optimal headway, which can be any nonzero constant. This implies that there exist a practically infinite number of positive equilibria in system (4.6). Then one can yield the characteristic matrix of (4.7)

Q1 (λ) ⎜ B1

⎛

⎜ . Qn (λ) = ⎜ ⎜ .. ⎝ 0

0 Q2 (λ)

.. .

··· ··· .. .

0 0

··· ···

0

0 0

0 0

Q2 (λ) Bn−2

0 Qn−1 (λ)

.. .

.. .

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

where

⎛

λ

⎜ ⎜ Qi = ⎜ −0.5τ ⎝ 0.25τ 2

b Dli

λ−

0

Dli −0.5τ b Dli

τ

⎞

⎟ ⎟ 0 ⎟, ⎠ λ−1

−b

⎡

0

⎢

0

−0.5τ b

−0.25τ 2

−τ

Bi = ⎢ ⎣

Dli Dli

⎤

0

0⎥ ⎦.

⎥

0

Hence, the characteristic equation is detQ(λ) = Q1 (λ) • · · · • Qn−1 (λ).

(4.8)

According to local stability theory, it is not easy to find that

• if Dli > τ2b for all (i ∈ N, then system (4.7) or system (4.6) is locally stable; • if Dl1 = τ2b , then there can simultaneously occur λ1 = 1 and complex conjugate roots λ = ±i in (4.8) which leads to high-codimensional bifurcations with strong resonances (which need a further study in the future);

• Otherwise, system (4.7) or system (4.6) is unstable. It is similar to give a detailed study of local stability analysis of Eq. (4.9) and it is omitted. Please note there exists a unique fixed point (0, C , L1 , . . . , 0, C , Ln−1 ) in the following equivalent form of (2.2):

⎧ C − V2 (t) ⎪ ⎪ α2 (t + 1) = b + k(d1 (t) − L1 )3 , ⎪ ⎪ ⎪ dl1 (t) ⎪ ⎪ ⎪ ⎪ C − V2 (t) ⎪ ⎪ , V2 (t + 1) = 0.5τ α2 (t) + V2 (t) + 0.5τ b ⎪ ⎪ ⎪ dl1 (t) ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ d1 (t + 1) = d1 (t) + τ (C − V2 (t)) − 0.5 τ α2 (t), V1+i (t) − V2+i (t) + k(di+1 (t) − Li+1 )3 , α2+i (t + 1) = b ⎪ l ⎪ (t) d ⎪ i + 1 ⎪ ⎪ ⎪ ⎪ V1+i (t) − V2+i (t) ⎪ ⎪ V2+i (t + 1) = 0.5τ α2+i (t) + V2+i (t) + 0.5τ b + 0.5τ k(di+1 (t) − Li+1 )3 , ⎪ ⎪ ⎪ dli+1 (t) ⎪ ⎪ ⎪ ⎪ ⎪ d1+i (t + 1) = d1+i (t) + τ (C − V2 (t)) − 0.52 τ 2 α2+i (t), ⎪ ⎪ ⎩ i = 1, . . . , n − 2.

(4.9)

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X. Wang and M. Peng / Physica A 536 (2019) 120926

5. Chaos or not and its determination 5.1. Lyapunov exponents and Lyapunov dimensions The computation of Lyapunov exponents (LE) is a fundamental problem for understanding rich dynamical behavior in nonlinear systems. We first briefly describe the well-known Eckmann–Ruelle algorithm [10] for calculating the LE of a discrete system. Given a Cr (r ≥ 1) map M : U → Rd on an open set U ⊂ Rd . Let x0 be the starting point ∈ U, and {xn = n



  M (x0 ), n = 1, 2, . . .} the corresponding orbit calculated by iteration of M, where M = M ◦ . . . ◦ M. The tangent map J(x0 ) = Dx M |x=x0 . is the matrix of partial derivatives of the d components of M with respect to the d components of x, n

n

then the sequence of tangent maps can be defined as: Dx M n (x)⏐x=x = J(M n−1 (xn−1 )) · · · J(M 1 (x1 ))J(x0 ).

⏐

0

(5.1)

The Lyapunov exponents (LE) can then be obtained from the QR factorization of tangent map expressed in Eq. (5.1), where Q is an orthogonal matrix and R is upper triangular with positive diagonal elements. Starting with J(x0 ) = Q1 R1 and writing subsequently J(M n−1 (xn−1 ))Qn−1 = Qn Rn , we have qr(J(M n−1 (xn−1 )) · · · J(M 1 (x1 ))J(x0 )) = Qn Rn · · · R1 . It is then shown in [10] that the d diagonal elements λii (i = 1, . . . , d) of upper triangular product Rn · · · R1 satisfy lim

ln λii

n→∞

n

= λi

where λi (i = 1, . . . , d) in the descending order are called by the LE. Furthermore the chaotic attractor is characterized by the existence of positive exponents. The Kaplan–Yorke conjecture connects the Lyapunov dimension [11] DL with the Lyapunov exponents spectrum by

∑j DL = j +

λi , |λj+1 | i=1

where the Lyapunov exponents are sorted in decreasing order (λ1 ≥ λ2 ≥ · · · ≥ λd ) and j is defined by the conditions j ∑ i=1

λi ≥ 0,

j+1 ∑

λi < 0.

i=1

5.2. Numerical simulation Suppose that Li = L, i = 1, . . . , n − 1. Given A = 0.5, v0 = 19, and the initial condition x(0) =random, y(0) = random, Si (1) = (i − 1)L0 , Vi = v0 and αi = 0. Choose M = B−1 F or M = B−1 F1 . Then Lyapunov Characteristic exponents and Lyapunov dimensions related to the corresponding parameters in Model (2.1) or (2.2) can be directly computed. In what follows, the chaotic behavior of the leading vehicle is shown with the Lyapunov dimension DL = 1.22 (or 1.26) for µ = 0.38, γ = 1.098 (or 1.198). In Fig. 1, n = 2. There can occur rich dynamics in system (2.1), including the stable steady state (Fig. 1(i)), periodic oscillations with period 4 (Fig. 1(ii)), and chaotic behavior (Fig. 1(iii)). Complicate dynamic behavior for n vehicles in an un-looped road without overtaking and in-flows is depicted respectively for n = 4 (Fig. 2) and n = 10 (Fig. 3). We find that there can occur rich dynamics in both system (2.1) and (2.2). But traffic dynamic is more sensitive to the parameters in model (2.2) than that in (2.1), such as the action delay τ , sensitivity coefficient b, strong nonlinear term coefficients k, l, the desired separation Li and even the starting headway distance L0 etc. This illustrates that there exist a strong dependence on these parameters when we are interested in their oscillation patterns. Actually the additional spacing-dependent terms can play a great role in System (2.2). Fractal properties are discovered by the computation of Lyapunov dimensions in Figs. 1–3. Lyapunov dimensions are strongly correlated to the number of the vehicles in model (2.2) or (2.1). Moreover, different oscillation patterns of the leading vehicle can greatly influence the behavior of the following. For example, when there occur regular oscillation patterns for the leading vehicle, so does the following (Fig. 1(i–ii)), whereas there occur chaotic patterns for the leading vehicle, so does the following (Fig. 1(iii) and Figs. 2–3). It is natural to give a further study of synchronous traffic flows in car-following models. By the way, one may find that there are fewer fluctuations about the Lyapunov dimensions when compared those in model (2.1) with (2.2) for very small k(≪ 1). The latter can be viewed as a perturbed model of the former. Such a property may be called dynamic continuing.

X. Wang and M. Peng / Physica A 536 (2019) 120926

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Fig. 1. Rich dynamics in system (2.1) with A = 0.5, v0 = 19, L0 = 10, l = 1, b = 1.63 and τ = 0.2.

Fig. 2. Chaotic behavior in system (2.1) or (2.2)n = 4, τ = .2; v0 = 29, Li = 10 = L0 , l = 1, µ = 0.38, b = 1.63.

6. Conclusions In this paper, we propose two classes of discrete-time models of single-lane road traffic, which can be viewed as a leader–follower model or a discretization version of the traditionally continuous-time car-following model (1.1) or (1.2). One admits a series of nonzero fixed points in its equivalent form but the other cannot do. Fractal properties are discovered by the computation of Lyapunov exponents and Lyapunov dimensions. An attempt by numerical simulation is made to identify different types of dynamic behavior in the model. We find that traffic dynamic in model (2.2) is more sensitive to reaction time delay τ than that in (2.1). Local stability analysis is established for model (2.1) in details. Furthermore, limit cycles by Neimark–Sacker bifurcations [9] related to the parameter τ or b cannot easily be observed in the given systems. The secret lies on the properties related to the root distribution of the corresponding characteristic equations, which is detailly discussed in Section 4. The route to chaos by following the Ruelle–Takens–Newhouse scenario (stable

8

X. Wang and M. Peng / Physica A 536 (2019) 120926

Fig. 3. Chaotic behavior in system (2.1) or (2.2) for n = 10, τ = 0.2; v0 = 19, Li = 10 = L0 , l = 1, µ = 0.38, b = 1.063.

steady state-two-dimensional tori-three-dimensional tori-chaotic attractors) cannot be observed, which was the main topic in [5]. We find that the motion of leading vehicle can play a great role in that of the following vehicles. When the leading vehicle presents a regular (steady states or periodic) or irregular (chaotic) motion, the following is to do the same likely. In some sense they are likely synchronous but not exactly. Further study may focus on other interesting discrete-time car following models and their synchronous dynamical behavior or synchronous traffic flows. Acknowledgments The project was supported by the Natural Science foundation of Shandong Province, China (CN, ZR2015AL004) (X. J. Wang). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

D.C. Gazis, R. Herman, R.W. Rothery, Non-linear follow the leader models of traffic flow, Oper. Res. 9 (1961) 545–567. Paul S. Addison, David J. Low, A novel nonlinear car-following model, Chaos 8 (1998) 791–799. D. Ling, P. Xiao, Analysis of stability and bifurcation of a c1ass of car following models, J. Transp. Eng. Inform. 7 (4) (2009) 6–11, (in Chinese). Xiaoyan Zhang, David F. Jarrett, Chaos in a dynamic model of traffic flows in an origin–destination network, Chaos 8 (1998) 503–513. Leonid A. Safonov, Elad TomerVadim, et al., Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic, Chaos 12 (2002) 1006–1014. Jie Sun, Zuduo Zheng, Jian Sun, Stability analysis methods and their applicability to car-following models in conventional and connected environments, Transp. Res. B 109 (2018) 212–237. Mark McCartney, Adiscrete time car following model and the bi-parameter logistic map, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 233–243. X.Y. Zhang, F.J. David, Stability analysis of the classical car following model, Transp. Res. B 31 (6) (1997) 441–462. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1995. J.P. Eckmann, D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys. 57 (1985) 617–656. J. Kaplan, J. Yorke, Chaotic behavior of multidimensional difference equations, in: H.O. Peitgen, H.O. Walther (Eds.), Functional Differential Equations and Approximation of Fixed Points, Springer, Berlin, 1979.