W Oukil

Ph Thieullen

A Kessi

Invariant cone and synchronization state stability of the mean field models

Keywords: Stability, coupled oscillators, mean field, interconnected systems, synchronization, Winfree model

In this article we prove the stability of some mean field systems similar to the Winfree model in the synchronized state. The model is governed by the coupling strength parameter κ and the natural frequency of each oscillator. The stability is proved independently of the number of oscillators and the distribution of the natural frequencies. The main result is proved using the positive invariant cone method for the linearized system. This method can be applied to other mean field models as in the Kuramoto model.

Introduction and Main result

In 1967 Winfree [START_REF] Winfree | Biological rhythms and the behavior of populations of coupled oscillators[END_REF] proposed a mean field model describing the synchronization of a population of organisms or oscillators that interact simultaneously [START_REF] Ariaratnam | Phase diagram for the Winfree model of coupled nonlinear oscillators[END_REF][START_REF] Quinn | Singular unlocking transition in the Winfree model of coupled oscillators[END_REF]. The mean field models are used, for example, in the Neurosciences to study of neuronal synchronization in the brain [START_REF] Cumin | Generalising the Kuramoto model for the study of neuronal synchronization in the brain[END_REF][START_REF] Ermentrout | The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators[END_REF]. The Winfree model is given by the following differential equation

      
ẋi = ω i -κσ(x)R(x i ), t ≥ 0, x = (x 1 , . . . , x n ), σ(x) := 1 n n j=1 P (x j ), ∀x = (x 1 , . . . , x n ) ∈ R n , sup x∈R P (x)R(x) > 0, P, R ∈ C 2 (R) are 2π-periodic, [START_REF] Ariaratnam | Phase diagram for the Winfree model of coupled nonlinear oscillators[END_REF] where n ≥ 1 is the number of oscillators, σ(x) is the mean field interaction, x i (t) is the phase of the i-th oscillator, and x(t) = (x 1 (t), . . . , x n (t)) is the global state of the system. We assume that the natural frequencies are chosen indifferently in some interval about ω = 1, ω i ∈ (1 -γ, 1 + γ), where γ ∈ (0, 1).

(

) 2
The coupling strength κ is taken in the interval (0, 1). Let

M := 16 max{ P (i) ∞ R (j) ∞ : 0 ≤ i, j ≤ 2}, (3)
be a constant used explicitly in some estimates measuring the size of the mean field. We first define the notions of invariance and stability. Let n ∈ N * and F : R n → R n be a C 1 vector field. Denote DF its Jacobian and assume max{ sup

z∈R n F (z) , sup z∈R n DF (z) } < ∞.
where . is the usual matrix norm. Let φ t : R n → R n be the flow of the autonomous system ẋ = F (x), t ≥ 0.

Definition 1 (Invariance). Let C ⊂ R n be an open set. We say that C is φ t -positively invariant for the system (4), if φ t (C) ⊂ C for all t ≥ 0.

Definition 2 (Stability). Let C ⊂ R n be an open set. We say that the system(4) is φ t -positively stable on C, if C is φ t -positively invariant and

∃λ > 0, ∀x ∈ C, ∃δ > 0, ∀y ∈ C : x -y < δ =⇒ φ t (x) -φ t (y) ≤ λ x -y , ∀t ≥ 0.
Let Φ t be the flow of the Winfree model [START_REF] Ariaratnam | Phase diagram for the Winfree model of coupled nonlinear oscillators[END_REF]. The existence of a synchronization state in the Winfree model is proved in [START_REF] Oukil | Synchronization hypothesis in the Winfree model[END_REF] for every number n of oscillators and every choice of natural frequencies. We recall the main synchronization hypothesis used in [START_REF] Oukil | Synchronization hypothesis in the Winfree model[END_REF],

2π 0 P (s)R ′ (s) 1 -κP (s)R(s) ds > 0, ∀κ ∈ (0, κ * ), (H)
where κ * is the locking bifurcation critical parameter κ * defined by

κ * := sup x∈R P (x)R(x) -1 . (5)
We proved in [START_REF] Oukil | Synchronization hypothesis in the Winfree model[END_REF] there exists an open set

U ⊂ (γ, κ) ∈ (0, 1) × (0, κ * ) : 1 -γ - κ κ * > 0 containing in its closure {0} × [0, κ * ],
such that for every n ≥ 1 and every parameter (γ, κ) ∈ U there exist two constants D ∈ (0, 1) and α(γ, κ, D),

     1 -γ -M κD -κ/κ * > 0, α(γ, κ, D) := 2γ + M κD 2 1 -κ/κ * + (2γ + M κD 2 + M κD)(γ + M κD) (1 -γ -M κD -κ/κ * )(1 -κ/κ * ) , (6)
and a C 2 2π-periodic function ∆ γ,κ : R → (0, D) solution of

d ds ∆ γ,κ (s) = - κP (s)R ′ (s) 1 -κP (s)R(s) ∆ γ,κ (s) + α(γ, κ, D), (7)
and a Φ t -positively invariant open set C n γ,κ independent of choice of the natural frequencies (ω i ) n i=1 ,

C n γ,κ := x = (x i ) n i=1 ∈ R n : max i,j |x j -x i | < ∆ γ,κ 1 n n i=1 x i . (8)
The following main result asserts that C n γ,κ is positively stable.

Main result. Consider the Winfree model (1) and assume that hypothesis (H) is satisfied. Then for every parameter (γ, κ) ∈ U , for every n ≥ 1 and every choice of natural frequencies

(ω i ) n i=1 as in (2), the Winfree model (1) is Φ t -positively stable on C n γ,κ .
Using a more refined version of Theorem 2 in Saito, see [START_REF] Fink | Almost periodic functions[END_REF][START_REF] Saito | On dynamical systems in n-dimensional torus[END_REF], one can prove the existence of a uniform rotation vector ρ ∈ R n such that for every initial condition x ∈ C n γ,κ Φ t (x) = ρt + p x (t), ∀t ≥ 0 where p x (t) is an almost periodic function.

Invariant cone and stability

We study in this section the stability of a system of the form (4) using the positive invariant cone method for the linearized equation. Propositions 5 and 7 are the two main ingredients that guarantee the stability of the Winfree model. We actually consider more generally a parametrized linear system of the form,

ẏ = A(x, t)y, t ≥ 0, x ∈ C, ( 9
)
where C is an open set and A(x, t) is a continuous n × n matrix function on C × R + . Let Ψ t x be the fundamental matrix of (9) parametrized by x ∈ C. The fundamental matrix cocycle of the system (9) is denoted by

Ψ t,t ′ x (z) := Ψ t x Ψ t ′ x -1 (z), ∀z ∈ R n , ∀t ≥ t ′ ≥ 0.
Let V + be the positive cone defined by

V + := {(z 1 , . . . , z n ) ∈ R n : z i ≥ 0, ∀i = 1, . . . , n}. (10)
Definition 3. Consider the linear system (9). We say that the cone

V + is Ψ t x -positively invariant uniformly in x ∈ C if ∃δ > 0, ∀x ∈ C, ∃t x ∈ [0, δ] : Ψ t,tx x (V + ) ⊂ V + , ∀t ≥ t x .
Definition 4. Consider the linear system (9). Let Ψ t x be its fundamental matrix. We say that (9) is Ψ t

x -positively stable uniformly in C if

∃λ > 0, ∀x ∈ C, ∀t ≥ 0, Ψ t x ≤ λ.
We study in the next proposition the stability of some classes of nonlinear systems using the positive invariant cone method. Proposition 5. Consider the system (4). Let be F := (f 1 , . . . , f n ). Suppose that there exists a φ t -positively invariant open set C ⊂ R n and there exists α > 0 such that

f i (φ t (x)) ≥ α, ∀x ∈ C, ∀t ≥ 0, ∀i ∈ {1, . . . , n}. Let x ∈ C and Ψ t
x be the fundamental matrix of the linearized system

ẏ = DF (φ t (x))y, t ≥ 0. ( 11
)
Suppose that V + as in (10) is Ψ t x -positively invariant uniformly in C, then (4) is φ t -positively stable on C.

To prove Proposition 5 we use the next Lemma which gives a sufficient condition of the stability of the system (4) as defined in Definition 2. Lemma 6. Consider the system (4). Suppose that there exists a φ tpositively invariant open set C ⊂ R n such that the linear system

ẏ = DF (φ t (x))y, ∀t ≥ 0, ∀x ∈ C, ( 12
)
is Ψ t x -positively stable uniformly in C. Then (4) is φ t -positively stable on C.
Proof. The system (4) can be written as

d dt Dφ t (x) = DF (φ t (x))Dφ t (x) with Ψ t x = Dφ t (x).
Since the system (12) is Ψ t x -positively stable uniformly in C, we have

∃λ > 0, ∀x ∈ C, ∀t ≥ 0, Dφ t (x) ≤ λ. (13) Let (z 1 , z 2 ) ∈ C × C such that z(s) := (1 -s)z 2 + sz 1 ∈ C for all s ∈ [0, 1]. Then φ t (z 1 ) -φ t (z 2 ) = 1 0 d ds φ t (z(s))ds = 1 0 Dφ t (z(s)) dz(s) ds ds , = 1 0 Dφ t (z(s))(z 1 -z 2 ) ds ≤ sup s∈[0,1] Dφ t (z(s))(z 1 -z 2 ) .
Finally, use the fact z(s) ∈ C and use equation ( 13) to obtain

φ t (z 1 ) -φ t (z 2 ) ≤ λ z 1 -z 2 , ∀t ≥ 0, which implies that the (4) is φ t -positively stable on C. Proof of Proposition 5. Since V + is Ψ t x -positively invariant uniformly in C, ∃δ > 0, ∀x ∈ C, ∃t x ∈ [0, δ] : z ∈ V + =⇒ Ψ t,tx x (z) ∈ V + , ∀t ≥ t x .
Let be η := α -1 , x ∈ C, y ∈ R n , and denote z + := η y F (φ tx (x)) + y and z -:= η y F (φ tx (x)) -y.

On the one hand, z

+ := (z + 1 , . . . , z + n ) ∈ V + and z -:= (z - 1 , . . . , z - n ) ∈ V + , min 1≤i≤n {z - i , z + i } ≥ η y min 1≤i≤n inf x∈C f i (φ tx (x)) -y ≥ (ηα -1) y = 0.
On the other hand, F (φ t (x)) is solution of the linearized system (11)

d ds F (φ s (x)) = DF (φ s (x))F (φ s (x)), F (φ t (x)) = Ψ t,t * x F (φ t * (x)).
Since V + is Ψ t x -positively invariant uniformly in C, we obtain

η y F (φ t (x)) + Ψ t,tx x (y) = Ψ t,tx x (z + ) ∈ V + , ∀t ≥ t x , η y F (φ t (x)) -Ψ t,tx x (y) = Ψ t,tx x (z -) ∈ V + , ∀t ≥ t x . Put r := max{ F (φ t (x)) , DF (φ t (x)) } < +∞ we obtain Ψ t,tx x (y) ≤ ηr y , ∀t ≥ δ, Ψ t x ≤ exp(rδ), ∀t ∈ [0, δ], Ψ t x (y) ≤ λ y , ∀x ∈ C, ∀t ≥ 0,
where λ := ηr exp(rδ). The linearized system (11) is Ψ t x -positively stable uniformly in C. Lemma 6 implies that (4) is φ t -positively stable on C.

We give in the following proposition a sufficient condition for the invariance of the cone V + . Proposition 7. Let p, q : R → R be continuous 2π-periodic functions, and g i , h i,j : [0, +∞) → R, 1 ≤ i, j ≤ n, be continuous functions. Consider the linear non-autonomous ODE

dz i ds = g i (s)z i + 1 n n j=1 p(s) + h i,j (s) z j , ∀s ≥ 0, ∀1 ≤ i ≤ n. ( 14
)
Assume there exists a constant D > 0 and a continuous 2π-periodic function δ : R → (0, D) such that

• 2π 0 p(s) ds > 0,
• dδ ds = -p(s)δ + q(s), ∀s ≥ 0,

• 0 ≤ g i (s) ≤ q(s) 4D , |h i,j (s)| ≤ q(s) 8D , ∀s ≥ 0.
Let Ψ s,s ′ be the fundamental matrix of (14). Then there exists s * ∈ [0, 2π] such that Ψ s,s * (V + ) ⊂ V + for all s ≥ s * .

Proof. Let be

s * ∈ [0, 2π] satisfying max s∈[0,2π] δ(s) = δ(s * ). ( 15
)
Let Ψ s,s ′ = (Ψ s,s ′ 1 , . . . , Ψ s,s ′ n ) be the fundamental matrix cocycle of ( 14). Let V+ be the interior of the set V + , z * ∈ V+ fixed, and z(s) = Ψ s,s * (z * ). By continuity,

∃s 1 > s * : z(s) ∈ V+ , ∀s ∈ [s * , s 1 ). Define S := sup s > s * : z(s ′ ) ∈ V+ , ∀s ′ ∈ [s * , s) .
The proposition is proved if we show S = +∞. By contradiction, suppose that S < +∞, then z(S) / ∈ V+ .

dz i ds = g i (s)z i + (p(s) + h i (s))µ(s),
where

h i (s) := n j=1 h i,j (s)z j (s) n j=1 z j (s) . Define g(s) := n i=1 g i (s)z i (s) n i=1 z i (s)
, and h(s

) := 1 n n i=1 h i (s). Then 0 ≤ g(s) ≤ q(s) 4D , |h i (s)| ≤ q(s) 8D , |h(s)| ≤ q(s) 8D .

Since |p(s)

+ h i (s) -a(s)| = | -g(s) + h i (s) -h(s)| ≤ q(s)/(2D), we have dz i ds ≥ (p(s) + h i (s))µ(s), ≥ (p(s) + h i (s))µ(s * ) exp s s * a(ζ) dζ , z i (s) -z i (s * ) µ(s * ) ≥ s s * p(s ′ ) + h i (s ′ ) -a(s ′ ) exp s ′ s * a(ζ) dζ ds ′ + s s * a(s ′ ) exp s ′ s * a(ζ) dζ ds ′ ≥ exp s s * a(ζ) dζ -1 - s s * q(s ′ ) 2D exp s ′ s * a(ζ) dζ ds ′ .
Multiplying by δ(s * ) exp -s s * a(ζ) dζ and using δ(s * ) < D, we get

δ(s * )z i (s) µ(s) ≥ δ(s * ) -δ(s * ) exp - s s * a(ζ) dζ - s s * q(s ′ ) 2 exp - s s ′ a(ζ) dζ ds ′ .
Let δ(s) be the unique solution of

d δ ds = -a(s) δ + q(s) 2 , ∀s ∈ [s * , S], δ(s * ) = δ(s * ). Then δ(s) =δ(s * ) exp - s s * a(ζ) dζ s s * q(s ′ ) 2 exp - s s ′ a(ζ) dζ ds ′ , δ(s * )z i (s) µ(s) ≥ δ(s * ) -δ(s), ∀s ∈ [s * , S]. ( 17
)
Notice that

a(s) = g(s) + p(s) + h(s) ≥ p(s) - q(s) 2D , ∀s ∈ [s * , S]. Then d δ ds ≤ -p(s) δ + q(s) 2 1 + δ D , ∀s ∈ [s * , S].
To obtain z(S) ∈ V+ and get a contradiction with (16), it is sufficient to prove that δ(s) ≤ δ(s), ∀s ∈ (s * , S]. For that, we use the comparison principle of differential equations. Since 0 < δ(s * ) = δ(s * ) < D and

d δ ds (s * ) < -p(s * ) δ(s * ) + q(s * ) = dδ ds (s * )
there exists ǫ > 0 such that δ(s) < δ(s) for all s ∈ (s * , s * + ǫ). Define

S := sup s ∈ [s * , S] : δ(s ′ ) < δ(s ′ ), ∀s ′ ∈ (s * , s] .
We show that that S = S. By contradiction, if S < S, then δ( S) = δ( S),

d δ ds ( S) < -p( S) δ( S) + q( S) = dδ ds ( S),
and we could find s < S close enough to S such that δ(s) > δ(s). We have obtained a contradiction. Then S = S and δ( S) < δ( S) ≤ δ(s * ). Equation (17) implies z(S) ∈ V+ , which is a contradiction with (16). We have obtained z(s) ∈ V+ for all s ≥ s * . By continuity of the fundamental matrix cocycle, we have proved that z(s) ∈ V + for all z(s * ) ∈ V + and all s ≥ s * .

Proof of the Main result

We prove in this Section the Main result of Section 1. We consider the Winfree model (1) and its associated flow Φ t . We recall that the Winfree model satisfies the hypothesis (H). The linearized Winfree model is given by

           dy dt = DW(Φ t (x))y, t ≥ 0, y = (y 1 , . . . , y n ), W i (x) := ω i -κσ(x)R(x i ), x = (x 1 , . . . , x n ) ∈ R n , ∂W i ∂x j = -κ σ(x)R ′ (x i )δ i,j + R(x i )P ′ (x j ) n . (18)
We fix (γ, κ) ∈ U and an initial condition x * ∈ C n γ,κ defined in [START_REF] Winfree | Biological rhythms and the behavior of populations of coupled oscillators[END_REF]. We denote by Ψ t

x * the fundamental matrix of (18). Let x(t) = Φ t (x * ) be the solution of (1) starting at x * , and

µ(t) := 1 n n i=1 x i (t), ∀t ≥ 0.
The main idea of the proof is to rewrite the linearized Winfree model by making a change of time t ↔ s and a linear change of the tangent vectors y ↔ z. We first notice that the velocity of µ is strictly positive,

dµ dt = 1 n n i=1 ω i -κσ(x) 1 n n i=1 R(x i ), ≥ 1 -κσ(µ)R(µ) -γ + κM D ≥ 1 -κ/κ * -γ -κM D > 0.
The first inequality uses the definition of the constant M in (3), the estimates (2) on the natural frequencies, the fact that C n γ,κ is positively invariant, that ∆ γ,κ defined in ( 7) is bounded from above by D, and the simple estimate,

|x i -µ| ≤ ∆ γ,κ (µ) ≤ D, ∀1 ≤ i ≤ n, σ(µ)R(µ) - 1 n n i=1 σ(x)R(x i ) ≤ M n n i=1 |x i -µ| ≤ M D.
The second inequality uses the definition of κ * in ( 5) and the third inequality uses the bound from below (6). Let be s * := µ(0). The map

t ∈ [0, +∞) → µ(t) ∈ [s * , +∞)
f i (s) := κσ(x(t))R ′ (x i (t)) v(s) f (s) := max 1≤i≤n f i (s), z i (s) := y i (t) exp s s * f (u) du , g i (s) := f (s) -f i (s), p(s) := - κP ′ (s)R(s) 1 -κP (s)R(s) , q(s) := (1 -κ/κ * )α(γ, κ, D) 1 -κP (s)R(s) , h i,j (s) := - κR(x i (t))P ′ (x j (t)) v(s) + κP ′ (s)R(s) 1 -κP (s)R(s) . Lemma 8. Then 1. dz i ds = g i (s)z i + 1 n n j=1 p(s) + h i,j (s) z j , ∀s ≥ 0, ∀1 ≤ i ≤ n, 2.
2π 0 p(s) ds > 0, 3. p, q : R → R are continuous and 2π-periodic,

4. 0 ≤ g i (s) ≤ q(s) 4D , |h i,j (s)| ≤ q(s) 8D . Proof. Using the change of variable zi (s) = y i • τ (s), xi (s) = x i • τ (s), equation (18) becomes, v(s) := d dt µ • τ (s), dz i ds (s) = 1 v(s) dy i dt (t) = - κ v(s) σ(x)R ′ (x i )z i + 1 n n j=1 R(x i )P ′ (x j )z j , = -f i (s)z i + 1 n n j=1 p(s) + h i,j (s) zj
Making the scaling z i (s) := zi (s) exp( s s * f (u) du), one obtains item 1. Item 2 is a consequence of hypothesis (H) and

d ds log 1 1 -κP (s)R(s) = p(s) - κP (s)R ′ (s) 1 -κP (s)R(s) , 2π 0 p(s) ds = 2π 0 κP (s)R ′ (s) 1 -κP (s)R(s) ds > 0.
Item 3 is true by definition of p and q. Using |x i (s) -s| ≤ D, the estimate on h i,j is given by

|h i,j (s)| ≤ κ|R(s)P ′ (s) -R(x i )P ′ (x i )| 1 -κP (s)R(s) + κ|R(x i )| |P ′ (x i )| |v(s) -(1 -κP (s)R(s))| v(s)(1 -κP (s)R(s)) ≤ κM 8(1 -κP (s)R(s)) D + γ + κM D 1 -κ/κ * -γ -κM D ≤ α(γ, κ, D) 8D 1 -κ/κ * 1 -κP (s)R(s) = q(s) 8D .
The estimate on g i is given by

g i (s) ≤ max 1≤i,j≤n κ|σ(x)| |R ′ (x i ) -R ′ (x j )| v(s) ≤ κM D 4(1 -κP (s)R(s)) + κM D(1 -κP (s)R(s) -v(s)) 4v(s)(1 -κP (s)R(s)) ≤ κM D 4(1 -κP (s)R(s)) 1 + γ + κM D 1 -κ/κ * -γ -κM D ≤ q(s) 4D .
We now conclude the proof of the main results: we will show that V + is Ψ t

x -positively invariant uniformly in C n γ,κ ; proposition 5 will imply that the Winfree model is Φ t -positively stable uniformly on C n γ,κ . The fact that V + is positively invariant is a direct consequence of proposition 7 applied to the linearized Winfree model written in terms of the new variables z(s) = (z 1 (s), . . . , z n (s)). Part of the hypotheses of proposition 7 have been proved in lemma 8. We prove in the following lemma the remaining hypothesis.

Lemma 9. There exists a continuous 2π-periodic function δ : R → (0, D) such that dδ ds = -p(s)δ + q(s), ∀s ≥ 0.

Proof. Let ∆ γ,κ (s) be the positive 2π-periodic function as in [START_REF] Quinn | Singular unlocking transition in the Winfree model of coupled oscillators[END_REF]. Define

Conclusion

We studied the stability of the Winfree model in its synchronized state. The proof is based on the positive invariant cone method. The main synchronization hypothesis used in [START_REF] Oukil | Synchronization hypothesis in the Winfree model[END_REF] is again a critical hypothesis for the linear stability.

  solutions µ(s) > 0, ∀s ∈ [s * , S]. Then for all s ∈ [s * , S],

  Definea(s) := g(s) + p(s) + h(s), ∀s ≥ s * .Then dµ ds = a(s)µ, µ(s) = µ(s * ) exp

  is a smooth diffeomorphism admitting as inverse maps ∈ [s * , +∞) → τ (s) ∈ [0, +∞). Define for t = τ (s) ⇔ s = µ(t), v(s) := dµ dt (t),

δ(s) := 1 -

 1 κ/κ * 1 -κP (s)R(s) ∆ γ,κ (s).Then δ ≤ ∆ γ,κ < D, anddδ ds = (1 -κ/κ * )κ(P ′ (s)R(s) + P (s)R ′ (s)) (1 -κP (s)R(s)) 2 ∆ γ,κ + 1 -κ/κ * 1 -κP (s)R(s) d∆ γ,κ ds , = κP ′ (s)R(s) 1 -κP (s)R(s) δ + (1 -κ/κ * )α(γ, κ, D) 1 -κP (s)R(s)= -p(s)δ + q(s).

Acknowledgements

We would like to thank the referee for his/her precise remarks and corrections that helped us to improve the final text.


Etienne Parizet

Ryan Robart

Perceval Pondrom

Jean-Christophe C Chamard

Guillaume Baudet

David Quinn

Karl Janssens

Manfred Haider

Additional Efficient Warning Sounds for Electric and Hybrid Vehicles

Keywords: electric vehicles, noise, warning sounds, safety, noise environment véhicules électriques, bruit, signal d'alarme, sécurité, environnement sonore

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Electric and hybrid vehicles are very quiet at low speeds (typically below 30 km/h). The level of the noise emitted by such vehicles is 5 to 7 dB(A) lower than the one of conventional Diesel cars. This is very beneficial to people living in an urban area, because transportation noise is a major source of annoyance in cities. On the other hand, this represents a hazard for pedestrian, who may not hear an approaching car. Vulnerable people as visually-impaired ones have a very strong concern about such cars and manufacturers use additional warning sounds in order to prevent this risk. Though several studies have already been conducted in that field (e.g. [START_REF] Yamauchi | Cross-cultural study on feasible sound levels of possible warning sounds for quiet vehicles[END_REF][START_REF] Emerson | Blind pedestrians and quieter vehicles : how adding artificial sounds impacts travel decision[END_REF], the easiest way to solve this safety issue is to use loud warning sounds, which cancels the noise reduction advantage of electric cars. Regulations, either already decided (NHTSA) or under preparation (in Europe), go this way. As an example, the regulation currently prepared in Europe states that "the (warning) sound level may not exceed the sound level of a similar internal combustion engine vehicle". It should be possible to use more efficient warning sounds. The eVADER project brings together partners from universities, research centers, car manufacturers and a supplier. Last but not least, the European Blind Union is a partner, so that end users are represented in the consortium. The goal of this project is to develop a prototype vehicle combining a high safety for pedestrian and a low noise annoyance for city residents. Several technical solutions are developed :

• An automatic pedestrian detection device;

• A set of loudspeakers focusing the sound in the direction of the detected pedestrian;

• A warning sound designed so that it can be detected in an urban environment at a low level. This paper will describe studies focusing on this third objective. It will present three listening test experiments : two of them aimed at evaluating the influence of some timbre parameters on the detectability and annoyance of warning sounds. A third one intended to determine whether some information about the speed of the car can be evoked by the warning sound, without any training of the listener. These three experiments were simultaneously conducted by various partners of the project, which allowed to use large subjects samples.

Detectability

The first part of the study was related to the detectability of warning sounds. Given the background traffic noise of an urban environment, is it possible to make a warning sound easily detectable in spite of a low level ? It was decided to limit the study to multi-tone sounds, in a middle frequency range (300 -1500 Hz). The lowest frequency was due to technical limitation of the loudspeakers to be used in the prototype (for a high radiation efficiency at low frequencies, very large loudspeakers would have been necessary). And the high frequency limit was selected because people suffering from presbyacousis have high hearing thresholds at higher frequencies.

Procedure

In this experiment, three timbre factors were investigated : the number of tones, the frequency variation and the temporal variation. Each factor could have three levels : as an example, the number of tones could be 3, 6 or 9. A fractional factorial design was used, so that 9 combinations were used (instead of 27 in a full factorial design). More details about the stimuli definition can be found in [START_REF] Robart | eVADER : a perceptual approach to finding minimum warning sound requirements for quiet cars[END_REF]. Therefore, 9 stimuli were synthesized; they all had the same A-weighted level. Then they were modified in order to represent a moving source, passing in front of a listener at the speed of 20 km/h. Finally, each modified stimulus was added to the recording of an electric vehicle recorded at 20 km/h by a dummy head located close to the road. This way, it was possible to simulate the situation of a pedestrian facing the road, waiting to cross this road and paying attention to any approaching car (figure 1).

Fig. 1. "Waiting to cross" scenario

The recording of the electric vehicle alone as well as the one of a similar diesel car were added to this set of stimuli. The level of warning sounds were adjusted so that they increase the level of the electric vehicle very slightly (less than 2 dB(A) , see figure 2). The level of the diesel car was higher by more than 5 dB(A).

Fig. 2. Peak level (A-weighted SPL) of each stimulus used in experiment 1

During the experiment, the listener was hearing a background traffic noise through headphones. Rain noise was added, as this represents a very difficult situation for blind pedestrian. The level of this noise was 69 dB(A). At randomly selected times, one of the car stimuli was added to the noise. This car could arrive from the left or from the right of the listener. The task of the listener was to detect the approaching car as soon as possible and its direction. He gave his answer by pressing a key of a computer keyboard. Two keys were used : the <Enter> key in the case of a "right" answer, and the <Space> bar for the "left" answer. The response time (from the starting of the stimulus) was measured and stored by the computer. Each sound was presented 8 times (4 times from each direction) so that a listener was presented 88 stimuli (in a random order). 110 subjects participated to the experiment; among them, 33 were visually-impaired people.

Results

The averaged response time was converted to distance from the listener at detection. These distances are shown in figure 3. In this figure, the "risk area" is represented in red. If the pedestrian starts crossing the road while the car is closer than 5 meters away, he may be hit by the car, given the averaged reaction time needed by the driver to start breaking (Kerber, 2011). The electric vehicle is detected in this area, which confirms that EVs can be dangerous for pedestrians [START_REF] Glaeser | Sound dectection of electric vehicles by blind or visually impaired persons[END_REF].

Warning sounds had quite different efficiencies, depending on their timbre. Some warning sounds are nearly inefficient (s6, s8 and s9). Some other ones make the EV as easy to detect as the diesel car (s3 and s7). And figure 2 clearly shows that such differences are not related to the level of these sounds -timbre features are the only reason for such differences. Further analysis showed that some controlled features were favorable to detection :

• A low number of tones. The overall sound level was kept constant, so that the difference between each tone level and its detection threshold was greater when the number of tones was low. • Amplitude modulation. Fluctuations in the signal amplitude helps the listener to detect it in the background of traffic noise.

Sound meaning

In a second experiment, the question was raised whether it is possible to design the warning sound so as to provide the pedestrian with some information about the distance and the speed of the car. The hypothesis was that a higher pitch and a quicker temporal variation of the warning sound would evoke a faster car.

Procedure

To test this hypothesis, a full experimental design was used. Sounds had three components and a pseudo-random fluctuation of the amplitude, as these characteristics were determined as favorable for detection in the previous experiment. Each factor could have three level, denoted as "low", "mid" or "high". As an exemple, one factor was pitch; the "low" value means 275 Hz, the "mid" one 300 Hz and the "high" level was 325 Hz. Similarly, the speed of amplitude modulation could have three values. These sounds were synthesized and added to two recordings of a passing-by electric vehicle. One recording was made at 20 km/h as in the first experiment and the other one at 30 km/h. This way, 18 stimuli were created. Two recordings of a diesel car were also used, recorded at the same speed. These sounds were presented to listeners in a background noise. The level of this noise was lowered as compared to the first experiment (64 dB(A) instead of 69 dB(A)) so that each approaching car could be easily detected. The task of the listener was to imagine he was going to cross the road in spite of this approaching car. He had to indicated when he thought it was no longer safe to do it, given his appreciation of the speed of the car. Once again, he gave his answer by pressing a key and the response time was stored. Each sound was repeated four times, giving a number of 80 responses from each listener. 116 people participated to the experiment (77 sighted and 39 visually impaired).

Results

The effect of each factor at each level was computed as the averaged response time for this level subtracted by the grand mean of all data. These effects are shown in figure 4. The speed effect is clear and natural : people feel unsafe sooner when the car is approaching at 30 km/h compared to 20 km/h. It should be noted that this 10km/h difference is greater than the velocity threshold as mentioned in a previous study from [START_REF] Lutfi | Correlational analysis of acoustic cues for the discrimination of auditory motion[END_REF], who measured a threshold of 8 km/h at 36 km/h. But the response time difference between the two speeds is small : 0.4 s. If the response times are converted to distances from the car, averaged values are 14 m at 20 km/h and 10 m at 30 km/h. This indicates that listeners clearly detected speeder cars, but underestimated the speed difference.

But results also shown that varying the speed of the temporal fluctuation of the additional sound gives no information to listeners : the effect of this factor is very close to zero, for any of the three levels. And, very surprisingly, the effect of the pitch is that the highest value gives a later response time. This was totally unexpected.

The main conclusion of this experiment is that it seems very unlikely to use the warning sound for conveying some information about the speed of the car which can be understood without any training. The speed of the car is certainly evaluated from the temporal increase of loudness as the car is approaching. Any warning sound will participate to this temporal loudness variation, whatever its characteristics. However, varying the pitch of this sound with the speed of the car may indicate an accelerating (or decelerating car); this was not evaluated by this experiment, as the speed was constant for each stimulus.

Unpleasantness

Procedure

The third experiment was devoted to the evaluation of the unpleasantness of all warning sounds used in the previous parts of the study. 20 sounds were presented to listeners : the 11 stimuli of experiment 1 and the 9 representing an EV with a warning sound at 20 km/h in experiment 2. The experiment was conducted in two conditions : some listeners were presented the stimuli without any background noise and for some other ones, a low-level traffic noise (57 dBA) was added to each stimulus. In both cases, the task of the listener was to evaluate the unpleasantness of the sound. He gave his answer by moving a cursor on a continuous scale, labeled from "not at all unpleasant" to "extremely unpleasant". The position of the cursor was stored as a number between 0 (for "not at all unpleasant") to 1000 ("extremely unpleasant"). Each sound was presented twice, the order of presentation being randomly selected. 145 people participated to this last experiment, which was conducted in four laboratories.

Results

The repeatability of each listener was evaluated by computing a mean squared difference between the two values he gave for each sound, namely :

( )

20 2 1 2 1 1 20 n n n C x x = = - ∑ (1)
where x n1 and x n2 represent the two evaluations of sound n.

Individual coefficients range between 50 and 450 (mean value : 179, standard deviation : 65). For 23 subjects, this coefficient is higher than 250, which represents a full category of the scale : such subjects can be considered as inconsistent. Such a high number of inconsistent subjects means that the task was difficult. So it was decided to select most reliable subjects for further analysis. The maximum value for C was fixed to 150, which allowed selecting 56 people. 26 of them did the experiment with the background traffic noise and 30 without this noise.

An analysis of variance was realized (repeated measures, background noise condition as an inter-individual factor and stimuli as intra-individual ones). The only influent factor was stimuli [ F(19, 988) = 48.5, p<0.0001 ]. The unpleasantness of each sound was averaged over this subpanel and results are shown in figure 5. Homogeneous groups of sounds are represented by thick horizontal lines (this groups have been determined using Scheffe's technique).

As it can be seen on figure 5, most warning signals strongly increase the unpleasantness of the sound. This can be due to the fact that people felt very unfamiliar with such warning sounds. Three of them can be considered as equally unpleasant as the diesel car. The particularity of these sounds is that no temporal fluctuation was applied to their amplitude. Amplitude fluctuation increased unpleasantness; the first experiment had shown that it also increased the detectability of sounds. The relation between efficiency and unpleasantness had already been proved for other kinds of warning sounds [START_REF] Tan | Multiple attribute evaluation of auditory warning signals for in-vehicle crash avoidance warning systems[END_REF], Richir, 2010or Fagerlönn, 2011).

Conclusion

This paper presented three experiments aiming at evaluating the influence of basic timbre parameters of a warning signal on its detectability and annoyance. It has been shown that some warning sounds can make an electric vehicle as detectable as a diesel car, for a much lower sound level. Nevertheless, people reported these signals to increase the unpleasantness of the car sound. It is hypothesized that this was due to the unnaturalness of such signals and to their novelty. Further studies during which subjects could get used to such sounds would be useful. Nevertheless, some signals seem to provide a correct compromise between the two objectives. Finally, it should be recalled that the goal of these studies was not to define a warning signal for a typical application, but to investigate the influence of timbre parameters. This way, it is expected that car manufacturers will have some guidelines when defining their own signal, which should also fulfill some brand image requirements.

Fig. 3 .

 3 Fig.3. Averaged distance from the listener at detection. The red area represents the "risk area" (see text).

Fig

  Fig. 4. Factors effects

Fig. 5 .

 5 Fig. 5. Unpleasantness of each sound

Acknowledgements

The eVADER project is funded by the European Commission, in the frame of the seventh framework program. And the first author is a member of the Labex CeLyA (Lyon Acoustics Center), funded by the French National Research Agency (ANR-10-LABX-60).