Analyses of driver’s anticipation effect in sensing relative fluxin a new lattice model for two-lane traffic systemArvind Kumar Gupta ∗, Poonam RedhuDepartment of Mathematics, Indian Institute of Technology Ropar, Rupnagar-140001, India
h i g h l i g h t s
• We presented a new two-lane lattice hydrodynamic model by considering the driver’s anticipation effect in sensing relative flux.• To describe the traffic behavior, linear and nonlinear stability analyses have been conducted.• The effect of anticipation coefficient is examined theoretically as well as numerically.
a r t i c l e i n f o
Article history:Received 24 April 2013Received in revised form 15 June 2013Available online 25 July 2013
Keywords:Traffic flowDriver’s anticipation effectTwo-lane system
a b s t r a c t
In this paper, a new lattice hydrodynamic traffic flowmodel is proposed by considering thedriver’s anticipation effect in sensing relative flux (DAESRF) for two-lane system. The effectof anticipation parameter on the stability of traffic flow is examined through linear stabilityanalysis and shown that the anticipation term can significantly enlarge the stability regionon the phase diagram. To describe the phase transition of traffic flow, mKdV equation nearthe critical point is derived through nonlinear analysis. The theoretical findings have beenverified using numerical simulation which confirms that traffic jam can be suppressedefficiently by considering the anticipation effect in the new lattice model for two-lanetraffic.
In recent years, the problem of traffic jam has attracted much attention of scientists and researchers. Therefore, a con-siderable variety of traffic models [1–17] have been discussed to study the complex phenomena in the past few decades.In 1998, Nagatani [18] proposed a simple lattice hydrodynamic model and derived mKdV equation to describe the trafficjam in terms of kink density wave near the critical point. The basic idea is that drivers adjust their velocity according tothe observed headway. Later, many extended [19–34] lattice models have been developed by considering different factorslike backward effect [19], lateral effect of the lane width [20] and anticipation effect of potential lane changing [21] etc. Re-cently, Peng [22] incorporated the effect of anticipation individual driving behavior and proposed a new lattice model. Kangand Sun [23] introduced a lattice hydrodynamic model by taking into account driver’s delay effect in sensing relative flux(DDSRF) and found that this effect has an important influence on the traffic jams. Most of the above cited models describesome traffic phenomena only on single lane.
Furthermore, Nagatani [35] extended his model to two-lane system and also to high-dimensional [36] traffic dynam-ics. Afterwards, some modifications have been made in two-lane lattice model by considering the optimal current differ-ence [37], and flow difference effect [38]. Very recently, Peng [39] analyzed the effect of driver’s anticipation in two-lanesystem.
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In real traffic, the driver often adjusts his velocity according to the observed traffic situation after sensing the relativeflux and always estimates his driving individual behavior. Most of highways comprise of more than one-lane, so it will bemore adequate to investigate this effect on a two-lane system. However, in the existing lattice models, driver’s anticipationeffect in sensing relative flux (DAESRF) in two-lane system has not been studied. This motivates us to develop a two-lanelattice model by incorporating the effect of driver’s anticipation individual behavior in sensing relative flux.
The paper is organized as follows. In the following section, a more realistic lattice model with DAESRF effect for two-lanetraffic system is presented. In Section 3, the linear stability analysis is studied for the proposed model. Section 4 is devotedto the nonlinear analysis in which mKdV equation is derived. Numerical simulations are carried out in Section 5 and finally,conclusions are given in Section 6.
2. A newmodel
A simplified version of continuummodel to describe the traffic phenomena on single lane, proposed by Nagatani [18] in1998 is
where ρ0 is the average density; a is the sensitivity of drivers; V (.) is the optimal velocity function; ρ(x+ δ) represents thedensity at position x + δ at time t; v and δ = 1/ρ0 are the velocity and the average headway, respectively.
The above simplified hydrodynamic model is further modified with dimensionless space x (let x = x∗/δ and x∗ indicatedas x hereafter) and expressed as
∂tρj + ρ0(ρjvj − ρj−1vj−1) = 0, (3)
∂t(ρjvj) = a[ρ0V (ρj+1) − ρjvj]. (4)
Eqs. (3) and (4) represent the lattice version of system of Eqs. (1) and (2), where j indicates site j on the one-dimensionallattice; ρj and vj, respectively, represent the local density and velocity at site j at time t .
In the literature of traffic flow modeling, multi-lane models are found to be more accurately describing the flow as mostof the road networks are made up of two or more lanes. In this regard, Nagatani [18] further extended single lane latticemodel to describe two-lane traffic by incorporating lane change effect in the continuity equation. On a two-lane highway,the lane change occurs only in the following possibilities:
1. if the density at site j − 1 on the second lane is higher than that at site j on the first lane, the lane changing occurs fromthe second lane to the first lane and will be proportional to their density difference as follows:
γ |ρ20V
′(ρ0)|ρ2,j−1(t) − ρ1,j(t)
.
2. if the density at site j on the first lane is higher than that at site j + 1 on the second lane, the lane changing occurs fromthe first lane to the second lane and will be proportional to their density difference as follows:
γ |ρ20V
′(ρ0)|ρ1,j(t) − ρ2,j+1(t)
.
Here, ρ1,j(t) and ρ2,j(t) are the densities on the first and second lane, respectively. The proportionality constantγ |ρ2
0V′(ρ0)|
is chosen in such away that it becomes dimensionless. Based on the above lane changing rules, the continuity
equation for two-lane traffic can be obtained in the same fashion as in Ref. [40] and is given by
∂tρj + ρ0(ρjvj − ρj−1vj−1) = γ |ρ20V
′(ρ0)|(ρj+1 − 2ρj + ρj−1) (5)
where ρj =ρ1,j+ρ2,j
2 and ρjvj =ρ1,jv1,j+ρ2,jv2,j
2 .Under the assumption that evolution equation of traffic current on each lane will not be affected by lane changing, the
evolution equation for two-lane traffic [40] was incorporated as
∂t(ρjvj) = a[ρ0V (ρj+1) − ρjvj]. (6)
Recently, Peng [39] proposed a new evolution equation with considering the driver anticipation effect (DAE) for two-lanesystem. The idea is that driver adjusts his running behavior by observing varying condition of flux at site j + 1. The newlattice version of evolution equation with DAE can be expressed as
where Qj+1 = ρj+1vj+1, represents the traffic flux at site j + 1 at time t .But, in real traffic flow, driver always senses the traffic relative information at time t and makes a decision to adjusts
velocity of his vehicle at later time t + τ1, where τ1 is the delay of driver response in sensing headway. Then, due to thedelay of car motion, vehicle moves at time t + τ1 + τ2, where τ2 is the delay time of car motion. So, the total delay time canbe divided into two parts τ1 and τ2. For simplicity, we choose the linear relationship between driver’s response delay τ1 andthe total delay time τ as τ1 = ατ , where α is the anticipation coefficient corresponds to individual behavior and τ = 1/a
5624 A.K. Gupta, P. Redhu / Physica A 392 (2013) 5622–5632
denotes the delay time which allows for the time lag, that it takes the traffic current to reach the optimal current whenthe traffic is varying. However, individual difference anticipation driver behavior was not considered in two-lane Peng’smodel [22]. In view of above reason, we propose a new evolution equation with consideration of anticipation driving effectin sensing relative flux (DAESRF) on a two-lane system as follows:
Based on the sign of anticipation coefficient α, our model can explore different characteristics of driver’s behavior ontwo-lane highway. Here, α > 0 represents the anticipation driving behavior or the drivers forecast effect in a traffic systemwith ITS. The idea is that driver adjusts driving individual speed to the anticipation optimal speed at time t +ατ after delaytime τ in advance. So, the bigger value of α corresponds to the skillful driver in the model.
For α < 0, i.e. negative anticipation coefficient corresponds to the explicit driver’s physical delay in sensing relative fluxeffect, recently proposed by Kang and Sun [23] in one-lane latticemodel.Whenα = 0, the newmodel reduces to Peng’s [39].For simplicity, using the Taylor series expansion and neglecting the nonlinear terms, the new evolution equation can beobtained as follows:
where △Qj(t) = ρj+1(t)vj+1(t) − ρj(t)vj(t). By taking the difference form of Eqs. (5) and (9) and eliminating speed vj, thedensity equation is obtained as
where △ρj(t) = ρj+1(t) − ρj(t) and ∆ρj(t) = ρj(t + τ) − ρj(t).
3. Linear stability analysis
To investigate the driver’s anticipation effect on jamming transition of traffic flow, we conducted linear stability analysisin this section. For this, the state of uniform traffic flow is taken as ρ0 and optimal velocity V (ρ0), where ρ0 is a constant.Hence, the steady-state solution of the homogeneous traffic flow is given by
ρj(t) = ρ0, vj(t) = V (ρ0). (11)
Let yj(t) be a small perturbation to the steady-state density on site j. Then,
ρj(t) = ρ0 + yj(t). (12)
Putting this perturbed density profile into Eq. (10) and linearizing it, we get
yj(t + 2τ) − yj(t + τ) + τρ20V
′(ρ0) △ yj(t) + ατρ20V
′(ρ0) △ (∆yj(t))
+ κρ0τγ |ρ20V
′(ρ0)|[α △3 yj−1(t − τ) + (1 + α) △
3 yj−1(t)]
+ κρ0[α △ (∆yj(t − τ)) − (1 + α) △ (∆yj(t))]
− τγ |ρ20V
′(ρ0)| △2 yj(t + τ) = 0. (13)
Substituting yj(t) = exp(ikj + zt) in Eq. (13), we obtain
e2τ z − eτ z+ τρ2
0V′(ρ0)[eik − 1] + ατρ2
0V′[eik+τ z
− eik − eτ z+ 1]
+ κρ0ατγ |ρ20V
′(ρ0)|[e2ik−τ z− 3eik−τ z
+ e−τ z− e−ik−τ z
]
+ κρ0α[eik − eik−τ z− 1 + e−τ z
] − κρ0(1 + α)[eik − eik−τ z+ e−τ z
+ 1]− κρ0(1 + α)τγ |ρ2
0V′(ρ0)|[e2ik − 3eik + 2 − e−ik
]
− τγ |ρ20V
′(ρ0)|[eik+τ z− 2eτ z
+ e−ik+τ z] = 0. (14)
Inserting z = z1(ik)+ z2(ik)2 . . . into Eq. (14), we will obtain the first-order and second-order terms of the coefficient ik and(ik)2, respectively, we get
z1 = −ρ20V
′(ρ0), (15)
z2 = −3τ z212
−ρ20V
′(ρ0)
2− ταρ2
0V′(ρ0)z1 + κρ0z1 + γ |ρ2
0V′(ρ0)|. (16)
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When z2 < 0, the uniform steady-state flow becomes unstable for long-wavelength waves. For z2 > 0 the uniform flowbecomes stable. Thus, the stability condition for the steady-state is
τ = −1 + 2γ + 2κρ0
ρ20V ′(ρ0)(3 − 2α)
. (17)
The instability condition for the homogeneous traffic flow can be described as
τ > −1 + 2γ + 2κρ0
ρ20V ′(ρ0)(3 − 2α)
. (18)
As α = 0, the above instability criteria (Eq. (18)) will become the same as that of Peng’s model [39].Eq. (18) clearly shows that lane changing parameter γ and anticipation coefficient α play an important role in stabilizing
the traffic flow. Solid curves in Fig. 1(a) and (b) are the neutral stability curves in the phase space correspond to γ = 0and γ = 0.1, respectively, for different values of α. It can be easily depicted from the figures that the amplitude of thesecurves decreases with an increase in α which means that larger value of α leads to the enlargement of stability region andhence, the traffic jam is suppressed efficiently. For negative values of α, when explicit driver’s physical delay is present,the instability region increases, which is in accordance with the result of Kang and Sun [23] for one-lane traffic model. Oncomparing Fig. 1(a) and (b), it can also be concluded that the increase in the value of γ results in a further increase in thestable region which means that lane changing also reduces traffic jams significantly. This phenomenon is much closer tothe real traffic as under the congested flow, vehicles try to accommodate themselves in a less denser lane by changing theirlane quite frequently to overcome the jam situation.
4. Nonlinear stability analysis
Using reduction perturbation method, we investigate the evolution characteristic of traffic jam around the critical point(ρc, ac) on coarse-grained scales. Long-wavelength expansion method is used to understand the slowly varying behaviornear the critical point. The slow variables X and T for a small positive scaling parameter ϵ (0 < ϵ ≪ 1) are defined as
X = ϵ(j + bt), T = ϵ2t (19)
where b is a constant to be determined. Let ρj satisfy the following equation:
ρj(t) = ρc + ϵR(X, T ). (20)
By expanding Eq. (10) to fifth order (see Appendix) of ϵ with the help of Eqs. (19) and (20), we obtain the followingnonlinear equation:
ϵ2k1∂XR + ϵ3k2∂2XR + ϵ4(∂TR + k3∂3
XR + k4∂XR3) + ϵ5(k5∂T∂XR + k6∂4XR + k7∂2
XR3) = 0. (21)
The coefficients ki (i = 1, 2, . . . , 7) are given in the Table 1, where V ′=
dV (ρ)
dρ |ρ=ρc , V′′′
=dV3(ρ)
dρ3 |ρ=ρc . Near the criticalpoint (ρc, ac), the value of τ is set as
τ = τc(ϵ2+ 1). (22)
By taking b = −ρ2c V
′ and eliminating the second-order and third-order terms of ϵ, we obtain
ϵ4(∂TR − g1∂3XR + g2∂XR3) + ϵ5(g3∂2
XR + g4∂4XR + g5∂2
XR3) = 0 (23)
where the coefficients gi (i = 1, 2, . . . , 5) are shown in Table 2.In order to determine the value of propagation velocity for the kink–antikink solution, it is necessary to satisfy the
following condition:
(R′
0,M[R′
0]) ≡
∞
−∞
dXR′
0M[R′
0] = 0 (24)
with M[R′
0] = M[R′]. By solving Eq. (24), the selected value of c is
c =5g2g3
2g2g4 − 3g1g5. (25)
Hence, the kink–antikink solution is given by
ρj = ρc + ϵ
g1cg2
tanh
c2(X − cg1T )
(26)
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a b
Fig. 1. Phase diagram in parameter space (ρ, a), κ = 0.1 for (a) γ = 0, and (b) γ = 0.1, respectively.
The kink–antikink solution represents the coexisting phase including both freely moving phase and congested phase whichcan be described by ρj = ρc ± A, respectively in the phase space (ρ, a). For a particular case, when α = 0, the resultsbecome similar to those found by Peng [39]. The dashed lines in Fig. 1 represent the coexisting curves which divide thephase plane into three regions: stable, metastable and unstable. In the stable region, the traffic flowwill remain stable undera disturbance while in metastable and unstable region; a small disturbance will lead to the congested traffic. In addition,with an increase of γ and anticipation coefficient α, the corresponding neutral and coexisting curves both lower down,which means that the effect of both the parameters can further stabilize the traffic flow. On the other side, for negativevalue of α, the amplitude of these curves increases which reduces the stability region of traffic flow.
5. Numerical simulation
To checkwhether themodel is capable of describing traffic flow dynamic and validate linear aswell as nonlinear analysis,numerical simulation is carried out for the newmodel with periodic boundary conditions. The initial conditions are adopted
A.K. Gupta, P. Redhu / Physica A 392 (2013) 5622–5632 5627
a b
c d
e
Fig. 2. Spatiotemporal evolutions of density when γ = 0, κ = 0.1 for (a) α = 0, (b) α = 0.1, (c) α = 0.2, (d) α = 0.3, and (e) α = 0.4, respectively.
as follows:
ρj(1) = ρj(0) =
ρ0; j =
M2
,M2
+ 1
ρ0 − σ ; j =M2
ρ0 + σ ; j =M2
+ 1
where, σ is the initial disturbance, M is the total number of sites taken as 100 and other parameters are set as follows:σ = 0.1, a = 2.3, τ =
1a .
The optimal velocity function given by Nagatani [40] is adopted.
V (ρ) =Vmax
2
tanh
1ρ
−1ρc
+ tanh
1ρc
(28)
where vmax and ρc denote the maximal velocity and the safety critical density, respectively. The optimal velocity function ismonotonically decreasing, has an upper bound and a turning point at ρ = ρc = ρ0. For computation, maximal velocity andcritical density are set at 2 and 0.25, respectively.
Fig. 2 shows the simulation results of density evolution after 104 time steps for different values of α when lane changingis not allowed (γ = 0). It is clear from the Fig. 2(a)–(d) that initial disturbance leads to the kink–antikink soliton which
5628 A.K. Gupta, P. Redhu / Physica A 392 (2013) 5622–5632
a b
c d
e
Fig. 3. Density profile at time t = 10 300 s when γ = 0, κ = 0.1 for (a) α = 0, (b) α = 0.1, (c) α = 0.2, (d) α = 0.3, and (e) α = 0.4, respectively.
propagates in the backward direction. Due to this, initial small amplitude disturbance evolves into congested flow as thestability condition is not satisfied. In the stable region, a small amplitude perturbation to the homogeneous density diesout and stop-and-go wave disappears for α = 0.4. It is also clear from Fig. 2(a)–(e) that anticipation driving behavior hasefficiently suppressed the traffic jam and also validate the theoretical findings.
Fig. 3 describes the density profile at time t = 10 300 s corresponding to panel of Fig. 2. From Fig. 3(a) to (d), the numberof stop-and-go waves decreases with an increase of anticipation coefficient. The region of free flow turns wide and theamplitude of density waves is weakened with the increase in anticipation coefficient which means that the anticipationeffect enhances the stability of the traffic flow. For α = 0.4, the traffic jam disappears and flow becomes uniform. Therefore,driver’s anticipation effect enhances the stability of system.
Figs. 4 and 5 show the simulation results under the different values of α when lane changing is permitted (γ = 0.1). Theresults corresponding to γ = 0.1 are qualitatively similar to those obtained for γ = 0. Parallel to no lane changing situation,initially kink–antikink soliton occurs and propagates in the backward direction. Since, the stability condition is not satisfied,the initial small perturbation on the uniform flow evolves into congested flow. It is clear from the density profile (Fig. 5(c))that the traffic flow becomes stable for small value of anticipation coefficient for lane changing in comparison to withoutlane changing situation.
A.K. Gupta, P. Redhu / Physica A 392 (2013) 5622–5632 5629
a b
c
Fig. 4. Spatiotemporal evolutions of density when γ = 0.1, κ = 0.1 for (a) α = 0, (b) α = 0.1, and (c) α = 0.2, respectively.
a b
c
Fig. 5. Density profile at time t = 10 300 s when γ = 0.1, κ = 0.1 for (a) α = 0, (b) α = 0.1, and (c) α = 0.2, respectively.
5630 A.K. Gupta, P. Redhu / Physica A 392 (2013) 5622–5632
a b
Fig. 6. Spatiotemporal evolutions of density when γ = 0, κ = 0.1 for (a) α = −0.1, and (b) α = −0.2, respectively.
a b
Fig. 7. Density profile at time t = 10 300 s when γ = 0, κ = 0.1 for (a) α = −0.1, and (b) α = −0.2, respectively.
a b
Fig. 8. Spatiotemporal evolutions of density when γ = 0.1, κ = 0.1 for (a) α = −0.1, and (b) α = −0.2, respectively.
Figs. 6 and 7 show the simulation results when the anticipation coefficient takes the negative value and lane changingis not permitted (γ = 0). This situation corresponds to the explicit driver’s delay in sensing relative flow effect. It is clearfrom the Fig. 6(a) and (b), DDSRF effect plays an important role in traffic congestion. The number of stop-and-go waves aswell as their amplitude increase with an increase in the DDSRF effect, which means that jam, can easily appear as the driverhas long delay response.
Figs. 8 and 9 show the simulation results when the anticipation coefficient takes the negative value and lane changing ispermitted (γ = 0.1). The results corresponding to γ = 0.1 are qualitatively similar to those obtained for γ = 0. It is alsoclear from the density profile that the traffic becomes more congested in the case of no lane changing as comparison to thelane changing. Therefore, from theoretical and simulation results, it is reasonable to conclude that traffic jam can efficientlysuppressed by considering the driver’s behavior on a two-lane system.
A.K. Gupta, P. Redhu / Physica A 392 (2013) 5622–5632 5631
a b
Fig. 9. Density profile at time t = 10 300 s when γ = 0.1, κ = 0.1 for (a) α = −0.1, and (b) α = −0.2, respectively.
6. Conclusion
We presented a new lattice hydrodynamic model of traffic flow by considering anticipation driving individual behaviorin sensing relative flux. The traffic behavior has been analyzed through linear and nonlinear analysis. We derived the mKdVequation to describe the traffic jam near the critical point and obtained kink–antikink soliton solution related to the trafficflow density. Phase diagrams in the density–sensitivity space with the neutral stability curves and the coexisting curvesare given. It is concluded that the anticipation coefficient corresponds to driver’s behavior in sensing relative flux increasessignificantly the stability of traffic flow in two-lane system. On the other hand, negative value of anticipation coefficient,which corresponds to driver’s delay response in sensing relative flux effect, reduces the stable region and increases conges-tion. The simulation results are in good accordance with the theoretical findings. Therefore, it is reasonable to conclude thatanticipation effect plays an important role in stabilizing/destabilizing the traffic flow on two-lane highway and this effectshould be considered in traffic flow modeling on two-lane highway.
Acknowledgment
The second author acknowledges Council of Scientific and Industrial Research (CSIR), India for providing financialassistance.
Appendix
In this appendix, we give the expansion of each terms in Eq. (10) using Eqs. (19) and (20) to the fifth-order of ϵ.
ρj(t + τ) = ρc + ϵRϵ2b∂XR +ϵ3
2(2bτ)2∂2
XR +ϵ4
6(2bτ)3∂3
XR + ϵ4τ∂TR
+ϵ5
24(2bτ)5∂5
XR + ϵ5bτ 2∂T∂XR. (A.1)
ρj(t + 2τ) = ρc + ϵRϵ22b∂XR +ϵ3
2(2bτ)2∂2
XR +ϵ4
6(2bτ)3∂3
XR + ϵ4τ∂TR
+ϵ5
6(2bτ)5∂5
XR + ϵ54bτ 2∂T∂XR. (A.2)
ρj+1(t) = ρc + ϵR + ϵ2∂XR +ϵ3
2∂2XR +
ϵ4
6∂3XR +
ϵ5
24∂4XR. (A.3)
ρj−1(t) = ρc + ϵR − ϵ2∂XR +ϵ3
2∂2XR −
ϵ4
6∂3XR +
ϵ5
24∂4XR. (A.4)
ρj+1(t) − 2ρj(t) + ρj−1(t) = ϵ3∂2XR +
ϵ5
12∂4XR. (A.5)
The expansion of optimal velocity function at the turning point is
V (ρj) = V (ρc) + V ′(ρc)(ρj − ρc) +V ′′′(ρc)
6(ρj − ρc)
3. (A.6)
V (ρj+1) = V (ρc) + V ′(ρc)(ρj+1 − ρc) +V ′′′(ρc)
6(ρj+1 − ρc)
3. (A.7)
5632 A.K. Gupta, P. Redhu / Physica A 392 (2013) 5622–5632
Using Eqs. (A.6) and (A.7), we get
V (ρj+1) − V (ρj) = V ′(ρc)
ϵ2∂XR +
ϵ3
2∂2XR +
ϵ4
6∂3XR +
ϵ5
24∂4XR
+V ′′′(ρc)
6
ϵ4∂XR3
+ϵ5
2∂2XR
3
. (A.8)
By inserting (A.1), (A.2), (A.5) and (A.8) into Eq. (10), we obtain Eq. (21).
References
[1] A.K. Gupta, S. Sharma, Chin. Phys. B 21 (2010) 015201.[2] A.K. Gupta, V.K. Katiyar, Physica A 368 (2006) 551.[3] A.K. Gupta, V.K. Katiyar, J. Phys. A: Math. Gen. 38 (2005) 4069.[4] A.K. Gupta, V.K. Katiyar, Physica A 371 (2006) 674.[5] H.J. Huang, T.Q. Tang, Z.Y. Gao, Acta Mech. Sin. 22 (2006) 132.[6] R. Jiang, Q.S. Wu, Z.J. Zhu, Trans. Res. B 36 (2002) 405.[7] T.Q. Tang, C.Y. Li, H.J. Huang, Phys. Lett. A 374 (2010) 3951.[8] T.Q. Tang, C.Y. Li, Y.H. Wu, H.J. Huang, Physica A 390 (2011) 3362.[9] T.Q. Tang, H.J. Huang, Chin. Sci. Bull. 49 (2004) 2097.
[10] T.Q. Tang, H.J. Huang, G. Xu, Physica A 387 (2008) 6845.[11] T.Q. Tang, H.J. Huang, H.Y. Shang, Phys. Lett. A 374 (2010) 1668.[12] T.Q. Tang, H.J. Huang, S.C. Wong, R. Jiang, Acta Mech. Sin. 23 (2007) 49.[13] T.Q. Tang, H.J. Huang, S.C. Wong, R. Jiang, Acta Mech. Sin. 24 (2008) 399.[14] T.Q. Tang, H.J. Huang, S.C. Wong, X.Y. Xu, Physica A 376 (2007) 649.[15] T.Q. Tang, H.J. Huang, X.Y. Xu, Y. Xue, Chin. Phys. Lett. 24 (2007) 1410.[16] T.Q. Tang, H.J. Huang, Z.Y. Gao, Phys. Rev. E 72 (2005) 066124.[17] T.Q. Tang, S.C. Wong, H.J. Huang, P. Zhang, J. Adv. Transport. 43 (2009) 245.[18] T. Nagatani, Physica A 261 (1998) 599.[19] H.X. Ge, R.J. Cheng, Physica A 387 (2008) 6952.[20] G.H. Peng, X.H. Cai, B.F. Cao, C.Q. Liu, Phys. Lett. A 375 (2011) 2823.[21] G.H. Peng, X.H. Cai, C.Q. Liu, M.X. Tuo, Phys. Lett. A 376 (2012) 447.[22] G.H. Peng, Commu. Nonlinear Sci. Numer. Simul. (2013) http://dx.doi.org/10.1016/j.cnsns.2013.03.007.[23] Y.R. Kang, D.H. Sun, Nonlinear Dynam. 71 (2013) 531.[24] J.F. Tian, Z.Z. Yuan, B. Jia, M.H. Li, G.J. Jiang, Physica A 391 (2012) 4476.[25] H.X. Ge, S.Q. Dai, Y. Xue, L.Y. Dong, Phys. Rev. E 71 (2005) 066119.[26] H.X. Ge, Physica A 388 (2009) 1682.[27] H.X. Ge, R.J. Cheng, L. Lei, Physica A 389 (2010) 2825.[28] X.L. Li, Z.P. Li, X.L. Han, S.Q. Dai, Commu. Nonlinear Sci. Numer. Simul. 14 (2009) 2171.[29] Z.P. Li, X.L. Li, F.Q. Liu, Int. J. Mod. Phys. C 19 (2008) 1163.[30] G.H. Peng, X.H. Cai, C.Q. Liu, B.F. Cao, Phys. Lett. A 375 (2011) 2153.[31] G.H. Peng, X.H. Cai, B.F. Cao, C.Q. Liu, Physica A 391 (2012) 656.[32] G.H. Peng, F.Y. Nie, B.F. Cao, C.Q. Liu, Nonlinear Dynam. 67 (2012) 1811.[33] T. Nagatani, Physica A 264 (1999) 581.[34] Y. Xue, Acta Phys. Sin. 53 (2004) 25.[35] T. Nagatani, Phys. Rev. E 59 (1999) 4857.[36] T. Nagatani, Physica A 272 (1999) 592.[37] G.H. Peng, Commu. Nonlinear Sci. Numer. Simul. 18 (2013) 559.[38] T. Wang, Z.Y. Gao, X.M. Zhao, J.F. Tian, W.Y. Zhang, Chin. Phys. B 21 (2012) 070507.[39] G.H. Peng, Nonlinear Dynam. (2013) http://dx.doi.org/10.1007/s11071-013-0850-7.[40] C. Tian, D.H. Sun, S.H. Yang, Chin. Phys. B 20 (2011) 88902.
