a bogdanov–takens bifurcation in generic continuous...

A Bogdanov–Takens bifurcation in generic

continuous second order traffic flow models

Armando Carrillo∗ Joaqúın Delgado† Patricia Saavedra‡

Rosa Maria Velasco§ Fernando Verduzco¶

May 31, 2011

Keywords: Continuous traffic, bifurcation, traveling waves.

Abstract

We consider the continuous model of Kerner–Konäuser for traffic flow

given by a second order PDE for the velocity and density. Assuming

conservation of cars, traveling waves solution of the PDE are reduced to a

dynamical system in the plane. We describe the bifurcations set of critical

points and show that there is a curve in the set of parameters consisting

of Bogdanov–Takens bifurcation points. In particular there exists Hopf,

homoclinic and saddle node bifurcation curves. For each Hopf point a

one parameter family of limit cyles exists. Thus we prove the existence of

solitons solutions in the form of one bump traveling waves.

1 Introduction

Macroscopic traffic models are based in the analogy with a continuous 1-dimensionalflow. Conservation of number of cars leads to conservation of mass

∂ρ

∂t+

∂ρV

∂x= 0 (1)

where ρ(t, x) is density and V (t, x) the average velocity of cars. In the kineticmodels the continuity equation is supplied with the law of motion given by theNavier–Stokes equation

ρ

(

∂V

∂t+ V

∂V

∂x

)

=∂

∂x

(

η∂V

∂x

)

− ∂p∂x

+X (2)

∗Mathematics Department. Universidad de Sonora, MEX. Blvd. Luis Encinas y RosalesS/N, Col. Centro, Hermosillo, Sonora, México. [email protected].

†Mathematics Department. UAM–Iztapalapa. Av. San Rafael Atlixco 186, México, D.F.09340. Corresponding author. [email protected].

‡Mathematics Department. UAM–Iztapalapa, MEX. [email protected].§Physics Department. UAM–Iztapalapa, MEX. [email protected].¶Mathematics Department. Universidad de Sonora. [email protected].

1

To appear in Proceedings of the 9th conference on Traffic and Granular Flow TGF'11. Russia. September 28-October 11, 2011. V. V. Kozlov, A. Schadschneider, A. S. Bugaev and M. Schreckenberg (eds). Springer

where η is viscosity and p the local pressure, being proportional to the variance(“temperature”) of the traffic Θ(x, t), namely the average of the squared dif-ferences of the individual cars and the average velocity. The “external forces”in the Kerner-Konhauser are represented by driver’s tendency to acquire a safevelocity Ve(ρ) with a relaxation time τ ,

X = ρVe(ρ)− V

τ.

When Θ = Θ0 and η = η0 are supposed constants, the equation of motion (2)simplifies to

ρ

(

∂V

∂t+ V

∂V

∂x

)

= −Θ0∂ρ

∂x+

ρ(Ve(ρ)− V )τ

+ η0∂2V

∂x2(3)

We look for travelling wave solutions of (1), (3), then the change of variablesξ = x+ Vgt transform (1) into the quadrature

−Vgdρ

dξ+

dρV

dξ

which can be immediately solved

ρ(V + Vg) = Qg. (4)

Here the arbitrary constant Qq represents the local flux as measured by anobserver moving with the same velocity Vg as the travelling wave.

Following [4] introduce adimensional variables

z = ρmaxξ, v =V

Vmax, vg =

VgVmax

, qg =Qg

ρmaxVmax(5)

where ρmax and Vmax are some maximum reference values of the density de-pending on the empirical law Ve(ρ). Also let

ve(ρ) =Ve(ρ)

Vmax, θ0 =

Θ0V 2max

, λ =Vmaxη0

, µ =1

ρmaxη0τ. (6)

Here Ve(V ) is obtained through Ve(ρ) by means of (4). Substitution of (4) intothe equation of motion (3) yields dynamical system

dv

dz= y

dy

dz= λqg

[

1− θ0(v + vg)2

]

y − µqg(

ve(v) − vv + vg

)

(7)

In adimensional form, (4) becomes

ρ/ρmax =qg

v + vg(8)

Here and in what follows we will take the parameter values λ, µ as givenfor the model, and will analyze the dynamical behavior with respect to theparameters vg, qg.

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0.2 0.4 0.6 0.8 1.0Ρ

-0.2

0.2

0.4

0.6

0.8

1.0V

HaL

HbL

HcL

HdL

Figure 1: Fundamental diagrams of mean velocity V vs. density ρ. (a) Green-shields; (b) Greenberg; (c) Newell; (d) Kerner–Konḧauser.

2 Fundamental diagram

Dynamics of (7) depend on the explicit form of the constitutive relationshipdefined by the fundamental diagram Ve(ρ). A typical form of the curve Ve(ρ)must meet the following properties that we will state as hypotheses.

Hypothesis I. Ve(ρ) is a monotone decreasing function defined for ρ ≥ 0 andVe(ρjam) = 0 for some ρjam > 0.

Hypothesis II. Ve(ρ) is bounded from below.

Hypothesis III. limρ→0 Ve(ρ) = Vf > 0 and V′

e (ρ) = o(ρ2) as ρ → 0.

Hypothesis I is explicitly adopted in [1]; ρjam can be interpreted as jam densitywhere traffic is stuck. The condition V ′e (ρ) = o(ρ

2) in Hypothesis II allows fora plateau in the graph of the curve Ve(ρ) for small values of density; Vf is thenthe average velocity in free traffic and can be determined by speed limits.

Historically, diverse fundamental diagrams have been considered. Either fitto empirical data, or deduced from theoretical basis. Some examples are shownin Figure 1. Explicit formulas are:

1. Greenshields: V = Vmax(1 − ρ/ρjam).

2. Greenberg: V = Vmax ln(ρjam/ρ).

3. Newell: V = Vmax exp(−λρ).

4. Kerner–Könhauser: V = Vmax

(

1

1+exp( ρ/ρmax−0.250.06 )

− 3.72× 10−6)

.

It can be verified that of the four fundamental diagrams (1)–(4) all ex-cept Newell’s satisfy Hypothesis I, and only (3) and (4) satisfy hypothesis II.Kerner-Konäuser can be considered the most accurate fundamental diagram,since it was fitted from a large set of data using double induction detectors [1](Greenberg was fitted from 18 data circa 1959). Although Hypothesis III seems

3


reasonable, it is neither of the fundamental diagrams (1)–(4) satisfy this condi-tion. We argue that it can be mathematically sustained since experimental dataare available for strictly positive ρ; put it in another way, modern technologiessuch as GPS could be used to get more accurate data for small densities.

Thus one can take Kerner–Konäuser model as an empirical validated forρ > δ where δ is some positive small density, and modify it in such a way thatHypothesis III is true.

Hypothesis II implies that limρ→∞ Ve(ρ) exists. This limit is zero for Newell’sdiagram and equals −3.72×10−6Vmax for Kerner–Konäuser’s. Although ρ → ∞does not make sense physically, it will be useful from a mathematical point ofview. to allow this limit. This amounts to consider an arbitrary large valueof ρjam (as Newell’s diagram). In fact, in (8) v → −vg as ρ → ∞ and thiscorresponds to the limit of an observer moving with the wave seeing a standing.

We denote by ve(v) the expression ve(ρ) (see 6) when ρ is expressed through(8) as a function of v.

The following properties of ve(v) are readily obtained.

Lemma 1. Let Ve(ρ) satisfy Hypothesis I, II and II. Then ve(v) is a monotoneincreasing function of v and: (1) limv→−vg ve(v) exists; (2) limv→−vg v

′

e(v) = 0and, (3) limv→∞ ve(v) = vf .

Proof. Since (4) can be written as

v + vg =qg

ρ/ρmax(9)

it follows that ρ/ρmax is a decreasing function of v, and thus ve(v) is an in-creasing function of v, for v > −vg. If v → −vg then ρ → ∞ but then lim ve(ρ)exists; this proves (1). Using the chain rule

v′e(v) = v′

e(ρ(v))ρ′(v) = v′e(ρ(v))

(

− qg/ρmax(v + vg)2

)

= −ρmaxqg

v′e(ρ)

ρ2

the last term being o(ρ2) by Hypothesis III; this proves (2). Finally if v → ∞then ρ → 0 but then ve(ρ) → 0. This completes the proof.

The above lemma shows that given “reasonable” fundamental diagram (namelysatisfying Hypothesis I-III) function ve(v) is sigmoidal: within the interval[−vg,∞] ve(v) approaches its limits as v → −vg,∞ asymptotically flat.

2.1 Critical points

Critical points of system (7) are given by the equations y = 0 and ve(v) = v.The following results describes a generical situation.

Proposition 1. Let Ve(ρ) satisfy Hypothesis I–III. Then there exists either 1,2 or 3 critical points of system (7).

4


-0.5 -0.25 0.25 0.5 0.75 1 1.25 1.5

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.5 -0.25 0.25 0.5 0.75 1 1.25 1.5

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.5 -0.25 0.25 0.5 0.75 1 1.25 1.5

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.5 -0.25 0.25 0.5 0.75 1 1.25 1.5

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 2: Possible intersections ve(v) = v leading to critical points of system (7).Only the graph to the right of the vertical curve v = −vg must be considered.From left to right from top to bottom, the number of critical points are: zero;one; three and two. A non–generic situation of tangency at the inflection pointis not shown.

Proof. Since the graph of ve(v) is sigmoidal, and the critical points are givenby the intersection of the graphs of ve(v) and the straight line v = v the resultfollows.

Figure 2 shows several possibilities for the intersection of the curve ve(v) = v,that is there exists either 0, 1, 2 or 3 critical points.

3 The Bogdanov–Takens bifurcation

In this section we will show that the Kerner-Konäuser ode (7) undergoes aBogdanov-Takens (BT) bifurcation. The BT bifurcation occurs in a two param-eter dynamical system in the plane (see also it generalization to n-dimensions[?]) when for some value of the parameter vector the dynamical system has crit-ical point with a double non–semisimple eigenvalue and some non-degeneracyconditions are satisfied. We state the conditions and its normal form accordingto Kuznetzov [3].

Theorem 1 (Bogdanov–Takens). Suppose that a planar system

ẋ = f(x, α), x ∈ R2, α ∈ R2, (10)

with smooth f , has at α = 0 the equilibrium x = 0 with a double zero eigenvalue

λ1,2(0) = 0.

Assume that the following genericity conditions are satisfied

5


(BT.0) The Jacobian matrix A(0) = fx(0, 0) 6= 0;

(BT.1) a20(0) + b11(0) 6= 0;

(BT.2) b20(0) 6= 0;

(BT.3) The map

(x, α) 7→(

f(x, α), tr

(

∂f(x, α)

∂x

)

, det

(

∂f(x, α)

∂x

))

is regular at the point (x, α) = (0, 0).

Then there exists smooth invertible transformations smoothly depending onthe parameters, a direction–preserving time reparametrization, and smoothinvertible parameters changes, which together reduce the system to

η̇1 = η2 (11)

η̇2 = β1 + β2η1 + η21 + sη1η2 +O(||η||3) (12)

where s = sign[b20(0)(a20(0) + b11(0))] = ±1.

Remark 1. The coefficients appearing in the hypotheses (BT.1), (BT.2) aredefined as follows: Let v0, v1 be generalized right eigenvectors of A0 = A(0)and w0, w1 generalized left eigenvectors, such that A0v0 = 0, A0v1 = v0 andATw1 = 0, A

Tw0 = w1. Then by a linear change of coordinates x = y1v0+ y2v1system (10) can be expanded in a Taylor series as

ẏ1 = y2 + a00(α) + a10(α)y1 + a01(α)y2

+1

2a20(α)y

21 + a11(α)y1y2 +

1

2a02(α)y

22 + P1(y, α)

ẏ2 = b00(α) + b10(α)y1 + b01(α)y2

+1

2b20(α)y

21 + b11(α)y1y2 +

1

2b02(α)y

22 + P2(y, α)

where akl(α), P1,2(y, α) are smooth functions of their arguments and

a00(0) = a10(0) = a01(0) = b00(0) = b10(0) = b01(0) = 0.

Remark 2. The normal form (12) is a universal unfolding, that is all nearbydynamical systems satisfying the hypotheses of the theorem is topologicallyequivalent to the normal form with the O(||η||3) terms deleted. Figure showsthe different scenarios depending in the parameter space β1–β2.

Lemma 2. Let Ve(ρ) satisfy Hypothesis I–III. Suppose that for some value ofthe parameters vg, qg the curves ve(v) and v = v have a tangency at v = vc.Then there exists θ0 such that system (7) has a critical point with a double zeroeigenvalue and its linear part is not semisimple.

6


Figure 3: Bogdanov–Takens bifurcation

Proof. The linear part at vc is

A0 =

(

0 1

− qgµ(v′

e(vc)−1)v+vg

qg

(

1− θ0(vc+vg)2)

λ

)

since the curves v = ve(v), v = v have a tangency at v = vc then v′

e(vc) = 1 andA0 reduces to

A0 =

(

0 1

0 qg

(

1− θ0(vc+vg)2)

λ

)

choosing θ0 = (vc + vg)2 reduces to

A0 =

(

0 10 0

)

(13)

Theorem 2. Let Ve(ρ) satisfy Hypothesis I–III. Suppose that for some value ofthe parameters vg, qg the curves ve(v), v = v have a tangency at v = vc withv′′e (vc) 6= 0. Then there exists θ0 such that system (7) has the linear part (13).Moreover, if the following generic condition

(

∂2ve∂v2

− ∂2ve

∂vg∂v

)

∂ve∂qg

+∂2ve∂qg∂v

∂ve∂vg

6= 0 (14)

is satisfied at the critical value of the point and parameters, then the systemundergoes a Bogdanov-Takens bifurcation.

7


Proof. According to the remark after the Bogdanov–Takens theorem, the gen-eralized right eigenvectors v0, v1 and the generalized left eigenvectors w1, w0as the canonical vectors (1, 0), (0, 1) in R2, respectively, and the coordinates(y1, y2) as (v, y) coordinates of the dynamical system (7). Then, denotig asf(y1, y2) the vector field defining (7), the coefficients appearing in conditions(BT.1), (BT.2) are

a20(0) =∂2

∂y21w0 · f(y1, y2) =

∂2

∂v2y = 0

b20(0) =∂2

∂y21w1 · f(y1, y2)

=∂2

∂v2

(

λqg

[

1− θ0(v + vg)2

]

y − µqg(

ve(v)− vv + vg

))

= −µqgv′′

e (vc)

vc + vg

b11(0) =∂2

∂y1∂y2w1 · f(y1, y2)

=∂2

∂y1∂y2

(

λqg

[

1− θ0(v + vg)2

]

y − µqg(

ve(v) − vv + vg

))

=2µqgθ0

(vc + vg)3

we immediately verify that

a20(0) + b11(0) =2µqgθ0

(vc + vg)36= 0

b20 = −µqgv

′′

e (vc)

vc + vg6= 0

the last condition being assumed by hypothesis.A straightforward computation shows that the matrix of the map in condi-

tion (BT.3) at v = vc, y = 0, vg, qg is

0 1 0 0

0 0 −µqgve

′

vg

vc+vg−

µqgve′

qg

vc+vg2λqgvc+vq

02λqgvc+vg

0

µqgve′′

vvc+vg

−

2λqgvc+vq

−

µqgve′

vg

(vc+vg)2+

µqgve′′

vg,v

vc+vg−

µqgve′

qg

(vc+vg)2+

µqgve′′

qg,v

vc+vg

where ve′, ve

′′ denote first and second order partial derivatives with respectto the subindex variables and are evaluated at (qg, vg, vc). The nonvanishing ofthe determinant is equivalent to the condition

(ve′′

v − ve′′vg ,v)ve′

qg + ve′

vgve′′

qg ,v 6= 0 (15)

which is the same as (14).

8


0

0.1

0.20.3

qg

-1-0.500.5

vg

0

0.5

1

x

0

0.1

0.20.3

0

0.5

0.05 0.1 0.15 0.2 0.25 0.3gq

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

vg

Figure 4: Surface of critical points and singular locus of the projection. Thesolid curve is γ.

4 The Kerner-konäuser model

For the Kerner-konäuser model one takes

ve(qg, vg, v) =1

1 + exp

[ qgv+vg

−d2

d3

] + d1 (16)

with the empirical values [1]

d1 = −3.72× 10−6, d2 = 0.25, d3 = 0.06

The set of critical points and parameters

{(qg, vg, vc) | ve(vc)− vc = 0}

form a three-dimensional surface, given as the zero level set of a functionF (qg, vg, vc) = ve(qg, vg, vc) − vc = 0. It is shown in Figure 1. There thevertical axis is x = vg + vc and we restrict to x > 0. The solid curves whichwill referred as γ are the loci where the projection (vg, qg, vc) 7→ (vg, qg) is notsurjective, namely the curve defined by the set of equations

F (qg, vg, vc) = ve(vc)− vc = 0,Fv(qg, vg, vc) = v

′

e(vc)− 1 = 0.

These were computed numerically by solving the above equations. Their pro-jections on the qg–vg plane is a cusp curve (see Figure 4) and characterize thevalues of parameters (qg, vg) such that the curve ve(v) is tangent to the graphof the identity. On the other hand the solid curve in the three dimensionalspace qg–vg–vc (or qg–vg–x) can be lifted to a curve γ̃ in four dimensional spaceθ0–qg–vg–vc where θ0 =

√vc + vg and thus γ can be viewed as the projection of

γ̃ and represent values (θ0, qg, vg) such that the linear part at the critical pointhas the form (13).

9


Proposition 2. The conditions of theorem 2 are satisfied for the Kerner–Konäuser model (16) whenever d1 is small an negative, and d2, d3 > 0.

Proof. the condition v′′e (vc) = 0 reduces to

eqgd3x (qg − 2d3x)− ed2/d3(qg + 2d3x) = 0, (17)

and the nondegenericity condition (14) to

eqgd3x (qg − d3x) + ed2/d3(qg + d3x) = 0. (18)

The change of variable qg = zx (recall x = v+vg, thus z = ρ/ρmax) and reduceseach equation to the trivial one x = 0 which is discarded since v 6= −vg, or

2d3ed2/d3 + (2d3 − z)ez/d3 + ed2/d3z = 0 (19)

d3ed2/d3 + (d3 − z)ez/d3 + ed2/d3z = 0 (20)

Taking the difference of these yields d3ez/d3 = −d3ed2/d3 . Since all constants

d1, d2, d3 are positive, this last equation has no solution for z > 0.

Corolary 1. The Kerner–Konäuser model (3) with the fundamental diagram(16) under the assumption (1) has traveling wave solutions in the unboundeddomain x ∈ (−∞,∞) and in the bounded domain [0, L] with periodic boundaryconditions.

Proof. From the BT Theorem 2, it follows that there exists Hopf limit cycles(v(z), y(z)), where z = ρmaxξ and ξ = x+ Vgt (see (5)). This yields a travelingwave solution of (3) in the form V (x, t) = Vmaxv(x+Vgt). If T is commensurablewith L then the traveling wave satisfies also periodic boundary conditions.

References

[1] Kerner, B.S. and Konhäuser, P. Cluster effect in initially homogeneous trafficflow. Phys. Rev. E, 48, No.4, R2335–R2338, 1993.

[2] Kerner, B.S., Konhäuser, P. and Schilke, P. Deterministic spontaneous ap-pearence of traffic jams in slighty inhomogeneous traffic flow. Physical Re-view E, 51, No. 6, 6243–6248, 1995.

[3] Kuznetzov, Y. “Elements of Applied Bifurcation Theory”, Appl. Math. Ser.112, 3d. ed., Springer

[4] P. Saavedra & Velasco R.M. Phase space analysis for hydrodynamic trafficmodels. Physical Review E 79, 0066103, 2009.

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