Mohamed Benyahia

Massimiliano D Rosini

Lack of BV bounds for approximate solutions to a two-phase transition model arising from vehicular traffic

Keywords: 2010 Mathematical Subject Classification. Primary: 35L65, 90B20, 35L45 Key words. Conservation laws, phase transitions, Lighthill-Whitham-Richards model, Aw-Rascle-

We consider wave-front tracking approximate solutions to a two-phase transition model for vehicular traffic. We construct an explicit example showing that the total variation in space of the solution blows up in finite time even for an initial datum with bounded total variation.

Introduction

In this paper we consider a two-phase transition model of hyperbolic conservation laws with point constraint on the density flux introduced in [START_REF] Dal Santo | General phase transition models for vehicular traffic with point constraints on the flow[END_REF]. The application of such model is, for instance, the modelling of vehicular traffic along a road with pointlike inhomogeneities characterized by limited capacity, such as speed bumps, construction sites, toll gates, etc.

We distinguish two phases: the congested phase Ω c and the free-flow phase Ω f . The model couples a 2 × 2 system of conservation laws for the congested phase, with a scalar conservation law for the free-flow phase. The coupling is achieved via phase transitions, namely discontinuities between two states belonging to different phases and satisfying the Rankine-Hugoniot conditions.

The first two-phase model has been proposed by Colombo [START_REF] Colombo | Hyperbolic phase transitions in traffic flow[END_REF]. Experimental data show that the density flux represented in the fundamental diagram is one-dimensional for high velocities, whereas it covers a twodimensional domain for low velocities, see [START_REF] Colombo | Hyperbolic phase transitions in traffic flow[END_REF]Figure 1.1]. For this reason it is reasonable to describe the traffic in the congested phase with a 2 × 2 system of conservation laws and the traffic in the free regime with a scalar conservation law.

Later on Goatin proposed in [START_REF] Goatin | The Aw-Rascle vehicular traffic flow model with phase transitions[END_REF] a two-phase model obtained by coupling the ARZ model by Aw, Rascle and Zhang [START_REF] Aw | Resurrection of "second order" models of traffic flow[END_REF][START_REF] Zhang | A non-equilibrium traffic model devoid of gas-like behavior[END_REF] for the congested phase Ω c , with the LWR model by Lighthill, Whitham and Richards [START_REF] Lighthill | On kinematic waves. II. A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF] for the free-flow phase Ω f . Recall that this model has been recently generalized in [START_REF] Benyahia | Entropy solutions for a traffic model with phase transitions[END_REF].

Both the models introduced in [START_REF] Colombo | Hyperbolic phase transitions in traffic flow[END_REF] and [START_REF] Goatin | The Aw-Rascle vehicular traffic flow model with phase transitions[END_REF] assume that Ω c ∩ Ω f = ∅. The first two-phase model considering a metastable phase Ω c ∩ Ω f = ∅ has been introduced in [START_REF] Blandin | A general phase transition model for vehicular traffic[END_REF]. Notice that, differently from [START_REF] Benyahia | Entropy solutions for a traffic model with phase transitions[END_REF][START_REF] Goatin | The Aw-Rascle vehicular traffic flow model with phase transitions[END_REF], for the models in [START_REF] Blandin | A general phase transition model for vehicular traffic[END_REF][START_REF] Colombo | Hyperbolic phase transitions in traffic flow[END_REF] the density flux vanishes at a unique maximal density, whose inverse corresponds to the average length of the vehicles. Here we do not consider a metastable phase and assume Ω c ∩ Ω f = ∅. We also assume that Ω f is characterized by a unique value V f for the velocity. At last we consider an heterogeneous traffic with vehicles having different lengths and allow the density flux to vanish at different densities.

These two-phase models have been recently generalized in [START_REF] Benyahia | A macroscopic traffic model with phase transitions and local point constraints on the flow[END_REF][START_REF] Dal Santo | General phase transition models for vehicular traffic with point constraints on the flow[END_REF] by considering Riemann problems, namely Cauchy problems for piecewise constant initial data with a single jump, coupled with a point constraint on the density flux, so that at the interface x = 0 the density flux of the solution must be lower than a given constant quantity F 0. This condition is referred to as unilateral point constraint on the density flux, see [START_REF] Rosini | Macroscopic models for vehicular flows and crowd dynamics: theory and applications[END_REF] and the references therein. In vehicular traffic, a point constraint accounts for pointlike inhomogeneities of the road and models, for instance, the presence of a toll gate across which the flow of the vehicles cannot exceed its capacity F .

Existence results for unconstrained Cauchy problems for the above mentioned two-phase transition models have been proved in [START_REF] Benyahia | Entropy solutions for a traffic model with phase transitions[END_REF][START_REF] Blandin | A general phase transition model for vehicular traffic[END_REF][START_REF] Colombo | Global well posedness of traffic flow models with phase transitions[END_REF][START_REF] Goatin | The Aw-Rascle vehicular traffic flow model with phase transitions[END_REF]. The first existence result for the constrained Cauchy problem has been established in [START_REF] Benyahia | An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic[END_REF] for the case with a metastable phase. In the present paper, we focus on the constrained Cauchy problem without a metastable phase. More precisely, as in [START_REF] Benyahia | An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic[END_REF] we use the Riemann solvers established in [START_REF] Benyahia | Entropy solutions for a traffic model with phase transitions[END_REF] and [START_REF] Dal Santo | General phase transition models for vehicular traffic with point constraints on the flow[END_REF] in a wave-front tracking scheme to construct a sequence of approximate solutions {u n } n , which converges at the limit n → ∞ to a solution u. We show that the total variation (in space) of u n becomes arbitrarily large as n goes to infinity and that the total variation of solution u blows up in finite time, even if the initial datum u o has bounded total variation. The construction of our example relies on the fact that the assumption Ω c ∩ Ω f = ∅ allows to consider phase transitions from Ω c to Ω f and that the (density) flow of a non-classical shock can be lower than the maximal flow F allowed by the constraint. We stress that both these properties do not occur in the case Ω c ∩ Ω f = ∅ and this is exploited in [START_REF] Benyahia | An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic[END_REF]. Our example can be then useful to generalize the existence result proved in [START_REF] Benyahia | An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic[END_REF] to the case Ω c ∩ Ω f = ∅ and to understand which conditions on the initial data guarantee existence of global-in-time solutions.

For completeness in the following we briefly recall some results for conservation laws. We firstly focus on the scalar case. Kruzhkov [START_REF] Kruzhkov | First order quasilinear equations with several independent variables[END_REF] established global well-posedness results in the class of so-called entropy solutions to Cauchy problems for scalar conservation laws with initial data in L ∞ ∞ ∞ . Oleinik [START_REF] Oleinik | Discontinuous solutions of non-linear differential equations[END_REF] proved that if the flux is C 2 2 2 and uniformly convex, then the solution has bounded total variation, even if the initial datum is only in L ∞ ∞ ∞ . More recently, Ambrosio and De Lellis [START_REF] Ambrosio | A note on admissible solutions of 1D scalar conservation laws and 2D Hamilton-Jacobi equations[END_REF] improved Oleinik's result showing that the solution is a special function of bounded variation, at least except at most countably many times; we defer the reader to [3, §4] for the definition of special functions of bounded variation. Moreover by Schaeffer's regularity theorem [START_REF] Schaeffer | A regularity theorem for conservation laws[END_REF] the solution has locally at most a finite number of discontinuity curves. This result is sharp in the sense that the regularity does not hold for every smooth initial datum, see the examples constructed in [36, §5] and [START_REF] Adimurthi | Structure of entropy solutions to scalar conservation laws with strictly convex flux[END_REF].

About systems of conservation laws in one space dimension, the pioneering work by Glimm [START_REF] Glimm | Solutions in the large for nonlinear hyperbolic systems of equations[END_REF] established global in time existence of distributional solutions to Cauchy problems for strictly hyperbolic systems, with each characteristic field being either genuinely nonlinear or linearly degenerate and initial data having sufficiently small total variation. Uniqueness results were obtained by Bressan and several collaborators in a series of papers, we defer the reader to book [START_REF] Bressan | Hyperbolic systems of conservation laws[END_REF]. Several regularity results that apply to scalar conservation laws with convex fluxes have been extended to systems of conservation laws with genuinely nonlinear vector fields, see for instance Glimm and Lax [START_REF] Glimm | Decay of solutions of systems of nonlinear hyperbolic conservation laws[END_REF], Liu [START_REF] Liu | Decay to N -waves of solutions of general systems of nonlinear hyperbolic conservation laws[END_REF], Bressan and Colombo [START_REF] Bressan | Decay of positive waves in nonlinear systems of conservation laws[END_REF], and Bianchini and Caravenna [START_REF] Bianchini | SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws in one space dimension[END_REF].

As a matter of fact, a satisfactory well-posedness theory for hyperbolic systems of conservation laws is available only for data with small total variation. In the case of large data the well-posedness holds also as long as the total variation remains bounded [START_REF] Bressan | Unique solutions of 2 × 2 conservation laws with large data[END_REF][START_REF] Lewicka | Well-posedness for hyperbolic systems of conservation laws with large BV data[END_REF]. The main remaining open problem is whether, for initial data with large total variation, the total variation remains uniformly bounded or can blow up in finite time. Examples of finite time blow up have been constructed in [START_REF] Baiti | Blowup in L ∞ for a class of genuinely nonlinear hyperbolic systems of conservation laws[END_REF][START_REF] Bressan | Lack of BV bounds for approximate solutions to the p-system with large data[END_REF][START_REF] Caravenna | s regularity theorem for scalar conservation laws does not extend to systems[END_REF][START_REF] Jenssen | Blowup for systems of conservation laws[END_REF].

The paper is organized as follows. In the next section we recall the two-phase transition model under consideration. In Section 3 we recall the Riemann solvers that are used then in the last section to construct our example.

The model

Before introducing model (2), we collect the parameters of the model together with their properties, see Figure 1.

ρ R ρ - f ρ + f ρ - c ρ + c f - f f + f f - c f + c f Ω - f Ω + f Ω c V f V c ρ R f u * (u , u r ) u r v -(u r ) v + (u r ) u ω f (u ) ω c (u ) v V f V c w - w --1 w + w Ω - f Ω + f Ω c v -(u r ) v + (u r ) u * (W(u ), v r ) ω f (u ) ω c (u ) u r u ρ
in Ω f , then Ω f . = u ∈ R 2 + : ρ ρ + f , v = V f , where R + . = [0, ∞). If V c ∈ (0, V f ) is the maximal velocity in the congested phase Ω c and it is reached for densities ranging in [ρ - c , ρ + c ] with 0 < ρ - c < ρ + c , then Ω c . = u ∈ R 2 + : v V c , w -v + p(ρ) w + , where w ± . = p(ρ ± c ) + V c . Above p ∈ C 2 ((0, ∞); R
) is an anticipation factor, which takes into account drivers' reactions to the state of traffic in front of them, see [START_REF] Aw | Resurrection of "second order" models of traffic flow[END_REF]. We assume that p (ρ) > 0, 2p (ρ) + p (ρ) ρ > 0 for every ρ > 0.

(1)

Typical choices for p are p(ρ) . [START_REF] Aw | Resurrection of "second order" models of traffic flow[END_REF][START_REF] Goatin | The Aw-Rascle vehicular traffic flow model with phase transitions[END_REF].

= ρ γ with γ > V f /(w --V f ) > 0, or p(ρ) . = V ref ln(ρ/ρ max ) with V ref > V f and ρ max > 0, see
Assume that w + = p(ρ + f ) + V f and that ρ -

f . = p -1 (w --V f ) is well defined. Let R .
= p -1 (w + ) > 0 be the maximal density (in the congested phase). Introduce

f ± f . = V f ρ ± f , f ± c . = V c ρ ± c ,
and assume that

f + c ∈ (f - f , f + f ), so that 0 < f - c < f - f < f + c < f + f . Define Ω . = Ω f ∪ Ω c , Ω - f . = u ∈ Ω f : ρ ∈ [0, ρ - f ) , Ω ex c . = u ∈ R 2 + : v V f , w -v + p(ρ) w + , Ω + f . = u ∈ Ω f : ρ ∈ [ρ - f , ρ + f ] , Ω ex . = Ω - f ∪ Ω ex c . Notice that Ω ex c ∩ Ω f = Ω + f . We assume that v < p (ρ) ρ for every (ρ, v) ∈ Ω ex c .
Let the (extended) Lagrangian marker w : Ω ex → [w --1, w + ] and W : Ω ex → [w -, w + ] be defined by

w(u) . =    v + p(ρ) if u ∈ Ω ex c , w --1 + ρ ρ - f if u ∈ Ω - f , W(u) . = max{w -, w(u)}.
The constrained PT model is then represented by the constrained Cauchy problem for

Free-flow (LWR)      u = (ρ, v) ∈ Ω f , ρ t + (ρ V f ) x = 0, v = V f , Congested flow (ARZ)      u = (ρ, v) ∈ Ω c , ρ t + (ρ v) x = 0, ρ W(u) t + ρ W(u) v x = 0, (2a)
with initial condition

u(0, x) = u o (x), (2b)
and point constraint on the density flux at x = 0

f u(t, 0 ± ) F, (2c)
where F 0 is a given constant quantity.

In Section 4 we consider PT model ( 2) with F ∈ (f + c , f + f ) and choose an initial datum u o with bounded total variation such that the total variation of the corresponding solution u blows up in finite time. The solution u is obtained as the limit of approximate solutions u n constructed by applying a wave-front tracking algorithm analogous to that used in [START_REF] Benyahia | An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic[END_REF], see [START_REF] Bressan | Hyperbolic systems of conservation laws[END_REF][START_REF] Holden | Front Tracking for Hyperbolic Conservation Laws[END_REF] and the references therein. Our wave-front tracking algorithm relies on the Riemann solvers R, R F : Ω 2 → BV(R; Ω) for the unconstrained and constrained Riemann problems for [START_REF] Ambrosio | A note on admissible solutions of 1D scalar conservation laws and 2D Hamilton-Jacobi equations[END_REF]. For completeness we recall their definitions in the next section.

The Riemann solvers

In this section we give the definitions of the Riemann solvers R ARZ : Ω 2 c → BV(R; Ω c ), R, R F : Ω 2 → BV(R; Ω) for the ARZ model and for the PT models without or with point constraint, respectively defined in [START_REF] Aw | Resurrection of "second order" models of traffic flow[END_REF][START_REF] Benyahia | Entropy solutions for a traffic model with phase transitions[END_REF][START_REF] Dal Santo | General phase transition models for vehicular traffic with point constraints on the flow[END_REF]. We recall that in general a Riemann solver RS : D 2 → BV(R; D) gives the solution corresponding to any Riemann problem, namely to any Cauchy problem with initial datum

u o (x) = u if x < 0, u r if x 0, (3)
with u , u r ∈ D, by letting u(t, x) .

= RS[u , u r ](x/t).

We introduce the following functions, see Figure 1:

r f : [w -, w + ] → [ρ - f , ρ + f ], r f (w) . = p -1 w -V f , r c : [w -, w + ] → [ρ - c , ρ + c ], r c (w) . = p -1 w -V c , ω f : [w -, w + ] → Ω + f , ω f (w) . = (r f (w), V f ), ω c : [w -, w + ] → Ω c , ω c (w) . = (r c (w), V c ), v ± : [0, V c ] → Ω c , v ± (v) . = (p -1 (w ± -v), v), u * ≡ (r * , v * ) : [w -, w + ] × [0, V c ] → Ω c , u * (w, v) . = (p -1 (w -v), v), Λ : (u , u r ) ∈ Ω 2 : ρ = ρ r → R, Λ(u , u r ) . = f (u r ) -f (u ) ρ r -ρ .
For any w ∈ [w -, w + ], let

λ w : [r f (w), p -1 (w)] → [-p -1 (w) p (p -1 (w)), V f -r f (w) p (r f (w))]
be defined by λ w (ρ) . = w -p(ρ) -ρ p (ρ) and let

R w : [-p -1 (w) p (p -1 (w)), V f -r f (w) p (r f (w))] → [r f (w), p -1 (w)]
be its inverse function. Define [a] + . = max{a, 0}.

Some comments on the above definition are in order. In case (ARZ.1) we have r f (w(u ,r ))

ρ < ρ r p -1 (w(u ,r )) and R ARZ [u , u r ] has only the 1-shock S 1 (u , u r ). In case (ARZ.2) we have r f (w(u ,r )) ρ r < ρ p -1 (w(u ,r )) and R ARZ [u , u r ] has only the 1-rarefaction R 1 (u , u r ). In case (ARZ.3) we have v = v r and R ARZ [u , u r ] has only the 2-contact discontinuity C 2 (u , u r ). In case (ARZ.4) we have r f (w(u ,r )) ρ < r * (u , u r ) p -1 (w(u )) and R ARZ [u , u r ] is the juxtaposition of S 1 (u , u * (u , u r )) and C 2 (u * (u , u r ), u r ). In case (ARZ.5) we have r f (w(u ,r )) r * (u , u r ) < ρ p -1 (w(u )) and R ARZ [u , u r ] is the juxtaposition of R 1 (u , u * (u , u r )) and C 2 (u * (u , u r ), u r ). Definition 3.2. The Riemann solver R PT : Ω 2 → BV(R; Ω) for PT model (2a), (2b), ( 3) is defined as follows:

(PT.1) If u , u r ∈ Ω f , then R PT [u , u r ](ν) . = u if ν < Λ(u , u r ), u r if ν Λ(u , u r ). (PT.2) If u , u r ∈ Ω c , then R PT [u , u r ] ≡ R ARZ [u , u r ]. (PT.3) If u ∈ Ω c and u r ∈ Ω f , then R PT [u , u r ](ν) . =      R ARZ [u , ω c (w(u ))](ν) if ν < Λ(ω c (w(u )), ω f (w(u ))), ω f (u ) if Λ(ω c (w(u )), ω f (w(u ))) ν < V f , u r if ν V f . (PT.4) If u ∈ Ω - f and u r ∈ Ω c , then R PT [u , u r ](ν) . = u if ν < Λ(u , v -(v r )), R ARZ [v -(v r ), u r ](ν) if ν Λ(u , v -(v r )). (PT.5) If u ∈ Ω + f and u r ∈ Ω c , then R PT [u , u r ] ≡ R ARZ [u , u r ].
We underline that R PT [u , u r ] has a phase transition in case (PT.3) between ω c (w(u )) and ω f (w(u )), in case (PT.4) between u and v -(v r ), and in case (PT.5) between u and u * (w(u ), v r ).

Fix

F ∈ (f + c , f + f ). Since (t, x) → R PT [u , u r ](x/t)
does not in general satisfy constraint condition (2c), we introduce

D 1 . = (u , u r ) ∈ Ω 2 : f R PT [u , u r ](t, 0 ± ) F , D 2 . = Ω 2 \ D 1 = (u , u r ) ∈ Ω 2 : w(u ) > w F , u r ∈ Ω f , where w F . = w(u F ) ∈ (w -, w + ) with u F . = (F/V f , V f ) ∈ Ω + f . ρ f F ǔ(w) û(w) u F ω f p -1 (w), 0 p -1 (w)
Figure 2: Geometrical meaning of û and ǔ defined in (4).

Define û : (w

F , w + ] → Ω c , w → û(w) . = ω c (w), ǔ : (w F , w + ] → Ω f , w → ǔ(w) . = f û(w) V f , V f , (4)
see Figure 2. Notice that for any w ∈ (w

F , w + ] f ǔ(w) = f û(w) < F. (5)
Definition 3.3. The constrained Riemann solver R F : Ω 2 → BV(R; Ω) for constrained PT model (2) is defined as follows:

R F [u , u r ](x) . =        R PT [u , u r ](ν) if (u , u r ) ∈ D 1 , R PT [u , û(w(u ))](ν) if ν < 0 R PT [ǔ(w(u )), u r ](ν) if ν 0 if (u , u r ) ∈ D 2 .
Remark 3.1. For any u , u r ∈ Ω we have that u(t, x) ≡ R F [u , u r ](x/t) has the following properties:

• Any discontinuity of u away from x = 0 is classical, i.e., it satisfies the Lax entropy inequalities.

• Non-classical discontinuities of u may occur only at x = 0, and in this case the density flux at x = 0 does not exceed the maximal flux F allowed by the constraint. We stress that differently from the non-classical discontinuities performed by solutions of the constrained LWR model or of the constrained ARZ model, see [START_REF] Andreianov | Solutions of the Aw-Rascle-Zhang system with point constraints[END_REF][START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF], the non-classical shocks performed by solutions of constrained PT model (2) do not have density flux equal to F , see [START_REF] Aw | Resurrection of "second order" models of traffic flow[END_REF].

• Any discontinuity δ(t) of x → u(t, x) satisfies the first Rankine-Hugoniot jump condition

ρ t, δ(t) + -ρ t, δ(t) -δ(t) = f u(t, δ(t) + ) -f u(t, δ(t) -) ,
and if δ(t) = 0, then it satisfies also the second Rankine-Hugoniot jump condition

ρ t, δ(t) + W u t, δ(t) + -ρ t, δ(t) -W u t, δ(t) - δ(t) = f u t, δ(t) + W u t, δ(t) + -f u t, δ(t) -W u t, δ(t) -.
The (extended) linearized momentum ρ W(u) is in general not conserved across non-classical shocks, even if they are between states in Ω ex c . As a consequence, a solution to (2) taking values in Ω ex c is not necessarily a weak solution to ARZ model. This is in the same spirit of the solutions considered in [START_REF] Benyahia | A macroscopic traffic model with phase transitions and local point constraints on the flow[END_REF][START_REF] Dal Santo | General phase transition models for vehicular traffic with point constraints on the flow[END_REF][START_REF] Delle Monache | Stability estimates for scalar conservation laws with moving flux constraints[END_REF][START_REF] Dymski | Existence of BV solutions for a non-conservative constrained Aw-Rascle-Zhang model for vehicular traffic[END_REF][START_REF] Garavello | The Aw-Rascle traffic model with locally constrained flow[END_REF] for traffic through locations with reduced capacity.

Blow-up example

The following example shows that taking initial data in BV does not ensure the global existence of a solution to constrained Cauchy problem [START_REF] Ambrosio | A note on admissible solutions of 1D scalar conservation laws and 2D Hamilton-Jacobi equations[END_REF].

ρ f R F u 2 f u 4 f u 0 f u 3 f u 1 f u 2 c u 4 c u 0 c u 3 c u 1 c x t 0 x 1 x 2 x 3 u 0 c u 1 c u 2 c u 0 c u 0 f u 0 f u 1 f u 2 f u 0 f u 1 p u 1 c u 2 c u 0 c u 2 f u 2 f u 0 f u 0 f x t 0 x 1 x 2 x 3
Fix F ∈ (f + c , f + f ). Introduce u 0 f . = u F ∈ Ω + f , u 0 c . = ω c (u F ) ∈ Ω c , u 0 p . = f (u 0 c ) V f , V f ∈ Ω f ,
and let

u i f ∈ Ω + f , u i c ∈ Ω c , u i p ∈ Ω f , i ∈ N, be defined by w(u i c ) = w(u i f ) = w(u 0 f ) + (-2) 1-i w, v i c = V c , f (u i p ) = f (u i c ),
where w . = min{w +w(u 0 f ), 2(w(u 0 f ) -w -)} > 0. Clearly for any i ∈ N we have

f (u 2i f ) < F = f (u 0 f ) < f (u 2i-1 f
).

Let x i . = -1 + 2 -i , i ∈ N, and consider the initial datum

u o (x) . =          u 0 f if x > 0, u 0 c if x 1 < x 0, u i c if x i+1 < x x i , i ∈ N, u 0 c if x -1.
We approximate the initial datum with

u o n (x) . =          u 0 f if x > 0, u 0 c if x 1 < x 0, u i c if x i+1 < x x i , i ∈ N ∩ [1, n], u 0 c if x x n+1 .
Notice that u o ∈ BV(R; Ω) because p -1 : (0, R) → R is Lipschitz by [START_REF] Adimurthi | Structure of entropy solutions to scalar conservation laws with strictly convex flux[END_REF] and

TV(v o ) = V f -V c , TV w(u o ) = w + ∞ i=1 3 2 i w = 4 w.
Furthermore we have

u o n L ∞ ∞ ∞ u o L ∞ ∞ ∞ , TV(u o n ) TV(u o ), lim n→∞ u o n -u o L 1 1 1 (R;Ω) = 0.
We approximate the solution u corresponding to the initial datum u o with the solution u n corresponding to the approximate initial datum u o n . Clearly u n is the juxtaposition of solutions to Riemann problems at the discontinuities of u o n and at the interactions (between two waves or a wave and the interface x = 0). We describe in detail only the case n = 2, see Figure 3, the general case is analogous. Notice that any contact discontinuity (u h f , u k f ) has speed (of propagation) V f and any contact discontinuity (u h c , u k c ) has speed V c .

• For times sufficiently small u 2 has the phase transition pt 0 0 . = (u 0 c , u 0 f ) with negative speed started at x = 0, and three contact discontinuities cd 0

1 . = (u 1 c , u 0 c ), cd 0 2 . = (u 2 c , u 1 c ) and cd 0 3 . = (u 0 c , u 2
c ) started at x 1 , x 2 and x 3 , respectively.

• The first interaction occurs between cd 0 1 and pt 0 0 ; the result is the phase transition pt 0 1 . = (u 1 c , u 1 f ) with negative speed and the contact discontinuity cd

1 1 . = (u 1 f , u 0 f ). • When cd 1
1 interacts with the interface x = 0, the result is the phase transition pt 1 0 . = (u 1 f , u 1 c ) with negative speed, the stationary non-classical shock ns 1 0 . = (u 1 c , u 1 p ) and the contact discontinuity

cd 2 1 . = (u 1 p , u 0 f ) with speed V f .
• When cd 0 2 interacts with pt 0 1 , the result is the phase transition pt 0 2 . = (u 2 c , u 2 f ) with negative speed and the contact discontinuity cd

. = (u 2 f , u 1 f ). • When cd 1
2 interacts with pt 1 0 , the result is the phase transition pt • When cd 2 2 interacts with ns 1 0 , the result is the phase transition pt 2 0 . = (u 2 c , u 2 f ) with negative speed and the contact discontinuity cd 3 2 . = (u 2 f , u 1 p ).

• When cd 0 3 interacts with pt 0 2 , the result is the phase transition pt 0 3 . = (u 0 c , u 0 f ) with negative speed and the contact discontinuity cd • When cd 2 3 interacts with pt 2 0 , the result is the phase transition pt 2 1 . = (u 0 c , u 0 f ) with negative speed and the contact discontinuity cd 3 3 . = (u 0 f , u 2 f ). • Finally cd 3 3 crosses the interface x = 0.

It is now clear how to construct u n for n > 2. We point out that as n grows, the number of contact discontinuities cd 0 i . = (u i c , u i-1 c ) appearing at t = 0 increases. After some interactions, cd 0 i eventually evolves into a contact discontinuity cd i i that reaches the interface x = 0 at some time t = t i , creating in this way a phase transition pt i 0 with negative speed. Clearly t i < t i+1 < t n < T c . = 1/V c because any wave in x < 0 moving with positive speed has speed (of propagation) bounded from below by V c and starts at time t = 0 at some x ∈ (-1, 0). Since the jump across each phase transition increases the total variation by at least V f -V c and for any t > T c the approximate solution u n (t, •) contains at least n + 1 phase transitions, we have that TV u n (T c , •)

(n + 1) V f -V c .
As a consequence lim n→∞ TV u n (T c , •)

lim n→∞ (n + 1) V f -V c = ∞,
thus the total variation of the solution u explodes in a finite time.

Figure 1 :

 1 Figure 1: Parameters and notations. Let ρ 0 and v 0 be the density and the velocity of the vehicles, respectively. Denote u . = (ρ, v) and let f (u) . = v ρ be the density flux. If V f > 0 is the unique velocity in the free-flow phase Ω f and ρ + f is the maximal densityin Ω f , then Ω f . = u ∈ R 2 + : ρ ρ + f , v = V f , where R + . = [0, ∞). If V c ∈ (0, V f )is the maximal velocity in the congested phase Ω c and it is reached for densities ranging in [ρ - c , ρ + c ] with 0 < ρ - c < ρ + c , then

Figure 3 :

 3 Figure 3: The approximate solution u 2 . The dashed lines correspond to phase transitions.

1

 1

2

 2

1

 1

3 . 2 . 3 .

 323 = (u 0 f , u 2 f ). • When cd1 3 interacts with pt 1 1 , the result is the phase transition pt 1 = (u 0 f , u 0 c ) with negative speed and the contact discontinuity cd 2 = (u 0 c , u 2 c ).

Acknowledgements

The second author is member of GNAMPA. He acknowledges the support of the National Science Centre, Poland, Project "Mathematics of multi-scale approaches in life and social sciences" No. 2017/25/B/ST1/00051, of the INdAM-GNAMPA Project 2019 "Equazioni alle derivate parziali di tipo iperbolico o non locale ed applicazioni" and of University of Ferrara, FIR Project 2019 "Leggi di conservazione di tipo iperbolico: teoria ed applicazioni".

 3) is defined as follows:

(ARZ.6) If u = u r ∈ Ω ex c , then R ARZ [u , u r ] ≡ u ,r .


Safieddine Bouali
email: safieddine.bouali@isg.rnu.tn

A Versatile Six-wing 3D Strange Attractor

To create additional wings to a given strange attractor, several methods based on the heteroclinic loop or switching controls for example are applied, but complicate the approach and require the extension of the system to one or more other dimensions of the phase space. This deflects us from the objectives of research on low-dimensional chaotic systems. Remaining in this narrow area of 3D phase spaces to invent multi-wing attractors constitutes the main scope of the present paper. Indeed, we present a rapid investigation of a very simple autonomous 3D system of firts-order differential equations with a rich variety of phase portraits. This new intentionally constructed model exhibits double, four-or even six-wing strange attractors. We point out that under the influence of the scalar parameters, such versatile chaotic attractors are obtained. A similar sequence was likewise observed for the periodic behaviors. Besides, both chaotic or regular featured trajectories are found to be in bilateral agreement even when the morphology of the portrait changes. Obviously, we present the basic attributes of the system and its bifurcation diagram. Eventually, we emphasize that the study of the relationship between the written differential equations and the observed characteristics of attractors remains undervalued.

I. Introduction

Techniques of mutating known doublewing attractors into multi-scroll shapes are still experienced at least from 20 years [START_REF] Aziz-Alaoui | Differential Equations with Multispiral Attractors[END_REF], and recent papers reported that such theoretical (and experiemental) field remains a challenging task [START_REF] Tahir | A Novel No-Equilibrium Chaotic System with Multiwing Butterfly Attractors[END_REF][START_REF] Chen | A New Multi-wing Chaotic System and Its Fractional-Order Form[END_REF][START_REF] Wang | Four-wing attractors in a novel chaotic system with hyperbolic sine nonlinearity[END_REF] [START_REF] Zhang | Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability[END_REF]. Having in mind that it is possible to achieve multi-scroll chaotic attractors by establishing unstable points in a chaotic system [START_REF] Elwakil | A four-wing butterfly attractor from a fully autonomous system[END_REF], several other methods to increase the number of wings were accurately analyzed [START_REF] Lü | Generating multiscroll chaotic attractors: Theories, methods and applications[END_REF]]. Among them, the piecewise-linear switching system yielding to multi-scroll attractors appears as the most applied framework [START_REF] Han | Generation Method of Multipiecewise Linear Chaotic Systems Based on the Heteroclinic Shil'nikov Theorem and Switching Control[END_REF]. This paper will also introduce a chaotic model with an increasing number of wings, however without these sophisticated techniques. Indeed, our main concern is to build a minimalist model related to the number of terms in the right-hand side of the equations with the lowest nonlinearity. Known as the parcimony principle, such goal can lead to a non-wasteful 3D system in the sense of economic use of mathematical relations. The ultimate purpose of this theoretical exploration is to determine a simple 3D strange attractor with different number of rolls in relation to scalar parameters rejecting obviously the bias of additional differential equations or artifacts (wave functions, homoclinic loops, ...). Versatile number of wings and more complexity could be expected from the designing our mathematical application.

II. The 3D Chaotic System

Our new system written in three first-order differential equations modifies the model of [START_REF] Bouali | A 3D Strange Attractor with a Distinctive Silhouette[END_REF] by adding a nonlinear term in the right-hand side of the third equation:

{ 𝒅𝒙 𝒅𝒕 = 𝛂 𝐳 -𝐱(𝟏 -𝐲) 𝒅𝒚 𝒅𝒕 = -𝐲 (𝟏 -𝐱 𝟐 ) 𝒅𝒛 𝒅𝒕 = -𝛃 𝐱 -𝛍 𝐳(𝟏 -𝐲)
where x, y, and z are the state variables of the system, and P (α, β, µ), the set of parameters. The model embeds two quadratic nonlinearities and only one cubic term, respectively, xy, yz, and yx 2 .

II.1. Basic mathematical attributes

The equilibria coordinates could be found by setting:

𝒅𝒙 𝒅𝒕 = 𝒅𝒚 𝒅𝒕 = 𝒅𝒛 𝒅𝒕 = 0.
Thus, the following system leads to the solutions:

{ α z -x(1 -y) = 0 -y (1 -x 2 ) = 0 -β x -µ z(1 -y) = 0
For the set of the parameters P (α, β, µ) = (0.1, 0.1, 1), the coordinates of the three equilibria are:

S0 (x0, y0, z0) = (0, 0, 0), S1 (x1, y1, z1) = (1, 1-(0.1) i, i), and S2 (x2, y2, z2) = (-1, 1+ (0.1) i, i).
Besides, in order to investigate the stability of S0 (x0, y0, z0), J the Jacobian is:

J= [ (𝟏 -𝐲) 𝒙 𝛂 𝟐𝒙𝒚 (𝟏 -𝐱 𝟐 ) 𝟎 -𝛃 µ 𝐳 -µ (𝟏 -𝐲) ]
It is easy to found that the corresponding characteristic equation, |J -λI| = 0 at S0 is:

(1-λ) (λ 2 + 2λ +1.01) = 0

Thus, the eigenvalues are : λ1 = 1, λ2 = -1 + 0.1 i, and λ3 = -1 -0.1 i

We notice that λ1, the real eigenvalue, is positive reporting that S0 is unstable with Index-1. Therefore, the emergence of chaos becomes confirmed knowing that the Shilnikov criteria [START_REF] Shilnikov | Methods of Qualitative Theory in Nonlinear Dynamics[END_REF]] requiring at least one unstable equilibrium are fulfilled.

II.2. The dissipativity

The dissipative nature of the 3D system could be derived from the divergence nature of the whole vector field:

𝒅𝒊𝒗. (𝑽𝒐𝒍𝒖𝒎𝒆) = 𝝏𝑽𝒐𝒍𝒖𝒎𝒆 𝝏𝒕 ⁄ 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝑻𝒓(𝑱)
To this end, Tr (J), the sum of the diagonal terms of the Jacobian should be negative to attest the dissipativity of the flow:

𝑻𝒓( 𝑱 ) = -(𝟏 -𝐲)(𝟏 + µ) + (𝟏 -𝐱 𝟐 ) < 0
where µ real value. Dissipativity and volume contraction of the flow are accurately identified when these state variables of the flow, x, and y (and not including z) satisfy the required condition:

(𝟏 -𝐱 𝟐 ) < (𝟏 -𝐲)(𝟏 + µ)

In the specific case where the condition is actually fulfilled, orbits are ultimately limited in a specific fractal-dimensional subspace of zero volume. That is the case at the neighborhood of S0.

On the other hand, the Lyapunov exponents spectrum shows the chaotic nature of the system since for µ, varying from 0.1 to 1.4, the dominant Lyapunov exponent reaches a positive value (Fig. 1).

III. Strange Attractors Evolving from Two-to Four or Six-wing

We simulate the system with the parameter set P (α, β, µ)= (1, -0.1, 1) that lead to an unconventional double-wing attractor (fig. 2a). Surprisingly, the modification of the parameter set to P (α, β, µ)= (0.1, 0.2, 1) displays a portrait of four-wing attractor (fig. 2b). Furthermore, for P (α, β, µ)= (0.1, 0.1, 1), the third representation depicts an identifiable six-wing strange attractor having a very complex dynamics (fig. 2c). We can notice that the rolls could be gathered in non identical pairs exhibiting a certain degre of symmetry. We can also indicate that almost all dynamics are highly volatile in relation the initial conditions. A small gap can project any simulation to a vanishing path. The volatility of the number of wings in the narrow chaos bubbles is also observed for the regular movements in the wide periodicity windows, as depicted in the bifurcation diagrams of x and z (Fig. 3). The variation of the parameter set P leads to a sequence of periodic cycles strongly similar to that shown in chaotic patterns.

The sample of periodic behaviors in Figure 4 displays even and likewise odd rolls. For instance, the featured regular dynamics are obtained by the modification of β and µ, when α keeps the value 0.1. It is worth mentioning that further analysis should be done to encompass the whole characteristics of this new 3D dynamical system. Despite its apparently simple formulation, the model leads also toward typical solenoid-like dynamics for particular parameter sets (figure 5). Whether chaotic or regular, the dynamics reported illustrate the ability of 3D systems to adopt a wide range of patterns and shapes. 1,0.1,and IC (x0,y0,z0)= (-0.446, 2.826, -0.391,and (b) a periodic behavior for P(α, β, µ)= (0.1, 0.2, 0.2).

Our model could be described as versatile not only for its sensitive dependence of the number of wings on the variation of the parameters but also for the extreme volatility of the dynamics in relation with the asymptotic stability of the attractors. Indeed, small gaps of the initial conditions are omitted and the attractors disappear. Thus, for the attractor with six wings, starting a simulation with IC (-0.794, 0.001, -0.941) and not IC (-0.794, 0.00132, -0.941) does not replicate the related dynamics.

This tiny difference Δy = 0.00032 is sufficient to deflect the trajectory out from the basin of attraction.

IV. Concluding Remarks

Generating grid multi-wing attractors throught heteroclinic loops, by adding a piecewise linear function, and other advanced methods as coupling 3D systems with additional first-order differential equations were widely investigated.

However, our research does not lie into such mainstream, but explores directly what system allows the variation of the number of wings without heavy techniques. To this end, generating multiwing chaotic attractors without complicatedness can be achieved. We have shown that the multiplication of wings to attractors, relevant for research purposes, could be controled by scalar parameters. The carefully selection of the equations governing the dynamics deserves to be practiced in this direction. Our results are expected to develop further experimentations. Indeed, the discovery of such new system contributing towards knowledge of the relation between the choice of right-hand terms of the system equations and global complex dynamical behaviors are quite subtle and deserve to be deeply studied.

Fig. 1 .

 1 Fig.1. Spectrum of the largest Lyapunov exponent. Its value varies in the vicinity of the unit, which testifies to the chaoticity of the 3D model.

Fig. 2 .

 2 Fig. 2. Multiwing strange attractors. (a) 2-wing attractor of the 3D system with P(α, β, µ)= (1, -0.1, 1) and IC (x0, y0, z0)= (1.9053, 0.0038, -0.2332), (b) 4-wing attractor of the 3D system with P(α, β, µ)= (0.1, 0.2, 1) and ICb (x0, y0, z0)= (0.01, 2.734, 0.0456), and (c) 6-wing attractor of the 3D system with P(α, β, µ)= (0.1, 0.1, 1) and ICc (x0, y0, z0)= (-0.794, 0.00132, -0.941).

Fig. 3 .

 3 Fig. 3. The bifurcation diagrams of x and z show wide windows of periodicity. The system parametrized with P (α, β, µ)= (0.1, 0.1, s), and s the control parameter є ]0, 1.1 [ leads to large bubbles of stability when the trajectories of (a) the variable x, and (b) the variable z cross the plane 𝑦 ̅ = 3.

Fig. 4 .

 4 Fig. 4. A Sample of the Periodic Motions. (a) A basic periodicity observed for P(α, β, µ)= (0.1, -0.1, 1), (b) period-4 dynamics for P(α, β, µ)= (0.1, 0.2, 0.7), (c) period-6 for P(α, β, µ)= (0.1, 0.1, 0.5), and (d) period-8 for P(α, β, µ)= (0.1, 0.1, 0.7).

Fig. 5 .

 5 Fig.5. The 3D system exhibits solenoid-like phase portraits. (a) A chaotic attractor is displayed for P(α, β, µ)= (0.1, 0.1, and IC (x0, y0, z0)=(-0.446, 2.826, -0.391,and (b) a periodic behavior for P(α, β, µ)= (0.1, 0.2, 0.2).