# Stop-and-Go Waves in Car-Following Models

Guy Fayolle, Jean-Marc Lasgouttes

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Guy Fayolle, Jean-Marc Lasgouttes. Stop-and-Go Waves in Car-Following Models. INRIA. 2025, pp.31. hal-04854057v3

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# Stop-and-Go Waves in Car-Following Models

Guy Fayolle*

Jean-Marc Lasgouttes*

May 6, 2025

## Abstract

Waves, known as *stop-and-go waves* or *phantom jams*, can appear spontaneously in dense traffic. This causes a situation where drivers are faced with consecutive phases of acceleration and braking. Although waves are well understood in the setting of macroscopic models, the results for car-following models are not so numerous. Assuming string instability, this paper gives asymptotic estimates of the velocity and shape of these waves. It relies on the well-known saddle-point method in order to describe the trajectory of a vehicle caught in such a wave. Numerical experiments show that this method yields remarkably good estimates of the linearized model, even with a small number of vehicles, as well as a good estimate of the wave velocity.

**Keywords:** Stop-and-go waves, saddle-point method, transfer function, time scaling, intelligent driver model (IDM), optimal velocity model (OVM)

## 1 Introduction

The phenomenon of *stop-and-go waves*, also known as *phantom jams* or *jamitons*, has been documented in the second part of the XXth century [12]. More recently, field experiments [15, 19, 20] have shown that vehicles in a highway-type lane can suffer from stop-and-go traffic conditions without any external intervention other than a small perturbation. These waves usually travel backward with respect to traffic. They should not be confused with *shock waves*, which typically occur when the number of lanes is reduced. Shock waves are a well-studied feature of first-order macroscopic traffic models, such as the popular Lighthill-Whitham model [16], and gave rise to an elaborate theory (see [12] for more details).

Among macroscopic models, second-order models and especially the Aw-Rascle-Zhang (ARZ) models [1, 27] are well-suited to exhibit stop-and-go

*ASTRA Project-team, INRIA, 48 rue Barrault, 75013 Paris, France.
*firstname.lastname@inria.fr*
waves. A comprehensive classification of what “stable” waves (jamitons) can actually occur in an ARZ model is provided in [17].

In this paper, we rather consider microscopic models, and among them car-
following models (also known as follow-the-leader models), which describe
the acceleration of a human-driven vehicle as a function of its speed, its
distance to the previous vehicle, and (optionally) the relative speed between
the two. They are also used today to describe the control loop of autonomous
vehicles or advanced driver-assistance systems (ADAS), such as adaptive
cruise control. Although these models have a long history [12, 21], two of
them stand out: the Optimal Velocity Model [2, 3], which only depends on
the vehicle’s speed and the inter-vehicle distance, and the more elaborate
Intelligent Driver Model [23] and its variants [18, 22].

We will be interested in this study in situations of traffic string instability
conditions [25], in which deviations from equilibrium will tend to be amplified,
giving rise to stop-and-go traffic. This instability propagates in traffic as
a wave [12]. Several authors proposed solutions for dissipating these waves
using automated vehicles (AVs), most of them based on microscopic models [4,
6, 9, 11, 13, 26]. Another fruitful approach has been the combination of a
macroscopic model for bulk traffic with microscopic models for AVs [5, 10].

In this paper, our aim is to provide methodological tools for a better un-
derstanding of stop-and-go waves in car-following models. Unlike [8], which
considered the effect due to driver reaction time, we focus on characterizing
waves at a sufficiently long distance from the disturbance that triggered them.
In [24, Section 15.4], a Fourier-Ansatz was used to analyze possible waves
in the frequency domain. We propose here a direct method, which relies on
the *saddle-point method* and yields explicit estimations of the trajectory and
speed profile of a given vehicle, as described by the underlying linear model.
A parameter of particular interest is the velocity of the front of the wave, in
a sense that will be made more precise.

The organization of this paper is as follows. Section 2 describes the lineariza-
tion of our car-following model, states the problem properly, and establishes
some basic properties of interest. In Section 3, some preliminary results
are obtained on wave propagation in finite time. The main substance is in
Section 4, when time scales linearly with N, with a quantitative description
of the wave in Theorems 4.5 and 4.6. Numerical experiments on the Intel-
ligent driver model (IDM) are presented in Section 5. Possible extensions
are discussed in Section 6. Finally, Appendix A gives a concise description
of the saddle point method, and Appendix B gives hints on the difficulty of
obtaining precise asymptotics for finite time, see Section 3.
## 2 The model

We consider a sequence of vehicles numbered $j = 0, 1, \dots, N$ in a single lane, where vehicle 0 is the first. Each vehicle $j > 0$ is characterized by its position $\hat{x}_j(t)$, its speed $\hat{v}_j(t) \stackrel{\text{def}}{=} \hat{x}'_j(t)$ and finally its acceleration $\hat{a}_j(t) \stackrel{\text{def}}{=} \hat{x}''_j(t)$. The latter is computed in car-following models as a function of what happens directly in front of it, through three values:

* the inter-vehicle gap $\Delta\hat{x}_j \stackrel{\text{def}}{=} \hat{x}_{j-1}(t) - \hat{x}_j(t)$,

* the inter-vehicle relative speed $\Delta\hat{v}_j \stackrel{\text{def}}{=} \hat{v}_{j-1}(t) - \hat{v}_j(t)$,

* the vehicle's speed $\hat{v}_j(t)$.

The general form used in this paper is

$$ \hat{a}_j(t) = f(\Delta\hat{x}_j(t), \Delta\hat{v}_j(t), \hat{v}_j(t)), \quad (2.1) $$

where $f$ is a continuously differentiable function on $\mathbb{R}^3$. Note that in the case of the optimal velocity model, the inter-vehicle relative speed is not taken into account and $f$ will be simpler.

Alas, such a very general non-linear formulation does not lend itself to explicit solutions. A classical way to study it (see e.g. [4, 26]) is to determine the equilibrium defined by the equation

$$ 0 = f(\Delta x^*, 0, v^*). $$

Here, $\Delta x^*$ and $v^*$ are, respectively, the stationary heading and the stationary speed.

Assume that the vehicles are organized at $t = 0$ such that $x_j(0) = -j\Delta x^*$. If the system were strictly in equilibrium, then the trajectory of vehicle $j$ would be

$$ x_j^*(t) \stackrel{\text{def}}{=} v^*t - j\Delta x^*. $$

In the following, we assume that a local disturbance in the trajectory $x_0(t)$ of the first vehicle does occur, for example a speed variation during a few seconds after $t = 0$. We want to understand how this disturbance propagates to the $N$-th vehicle, when $N$ is large. In practice, we will assume that, for some finite value of $t_0 > 0$:

$$ v_0(t) = v^*, \text{ for all } t \ge t_0. $$

This implies in particular that the perturbation of the trajectory has a finite effect:

$$ x_0(t) - x_0^*(t) = \int_0^{t_0} (v_0(s) - v^*)ds, \text{ for all } t > t_0. \quad (2.2) $$
Let us now linearize (2.1) around a given equilibrium point $(\Delta x^*, 0, v^*)$. Dropping the hat on the variables to denote the solutions $x_j$, $v_j$ and $a_j$ of the linearized model, the new equation of interest is

$$a_j(t) = \alpha(\Delta x_j(t) - \Delta x^*) + \beta \Delta v_j(t) - \gamma(v_j(t) - v^*), \quad (2.3)$$

where the parameters $\alpha$, $\beta$ and $\gamma$ represent the following partial derivatives at equilibrium

$$\alpha \stackrel{\text{def}}{=} \frac{\partial f}{\partial \Delta x}(\Delta x^*, 0, v^*), \quad \beta \stackrel{\text{def}}{=} \frac{\partial f}{\partial \Delta v}(\Delta x^*, 0, v^*), \quad \gamma \stackrel{\text{def}}{=} -\frac{\partial f}{\partial v}(\Delta x^*, 0, v^*).$$

In what follows, we will assume that $\alpha > 0$, $\beta \ge 0$ and $\gamma > 0$. The signs of the parameters correspond to a reduced acceleration as the danger increases. The fact that $\beta$ can be zero allows one to analyze models such as OVM, which do not depend on the relative speed between vehicles (see Section 4.4).

To evaluate the variation around this equilibrium value, for $j \ge 0$, let $X_j(z)$ and $V_j(z)$ denote the respective Laplace transforms of $x_j(t) - x_j^*(t)$ and $v_j(t) - v^*$:

$$X_j(z) \stackrel{\text{def}}{=} \int_0^\infty (x_j(t) - x_j^*(t))e^{-zt} dt,$$

$$V_j(z) \stackrel{\text{def}}{=} \int_0^\infty (v_j(t) - v^*)e^{-zt} dt.$$

Note that $V_j(z) = zX_j(z)$ and that, as a consequence of (2.2), $X_0(z)$ has in general a pole of multiplicity 1 at $z = 0$:

$$\lim_{z \to 0} zX_0(z) = V_0(0) = \int_0^{t_0} (v_0(s) - v^*)ds.$$

Using the properties of Laplace transforms, (2.3) can be rewritten as

$$z^2 X_j(z) = \alpha(X_{j-1}(z) - X_j(z)) + \beta(zX_{j-1}(z) - zX_j(z)) - \gamma z X_j(z).$$

Setting now $\delta \stackrel{\text{def}}{=} \beta + \gamma$, the transfer function associated with (2.3) is equal to

$$Q(z) \stackrel{\text{def}}{=} \frac{X_j(z)}{X_{j-1}(z)} = \frac{\beta z + \alpha}{D(z)}, \quad \text{where } D(z) \stackrel{\text{def}}{=} z^2 + \delta z + \alpha. \quad (2.4)$$

Then, for any integer $N$, $X_N(z) = X_0(z)Q(z)^N$ and the inverse Laplace transform formula yields the relations, for any positive $u$,

$$x_N(t) = x_N^*(t) + \frac{1}{2i\pi} \int_{u-i\infty}^{u+i\infty} e^{tz} X_0(z) Q^N(z) dz, \quad (2.5)$$

$$v_N(t) = v^* + \frac{1}{2i\pi} \int_{u-i\infty}^{u+i\infty} e^{tz} V_0(z) Q^N(z) dz. \quad (2.6)$$
The above calculations are a classical way to address car-following models [4, 8, 26]. Starting from these results, we shall focus here on the behavior of $x_N(t)$ when $N$ and possibly $t$ are large, in order to understand how waves propagate over long distances.

**The core problem** Estimate, as $N \to \infty$, the following integral, where $\phi(z)$ is a Laplace transform analytic for $\Re(z) > 0$:

$$I(t_N; N) \stackrel{\text{def}}{=} \frac{1}{2i\pi} \int_{u-i\infty}^{u+i\infty} e^{t_N z} \phi(z) Q^N(z) dz, \quad u > 0. \quad (2.7)$$

where $t_N$ may depend on $N$, and $\phi(z)$ is sufficiently smooth and bounded at infinity on the contours of interest.

It is well known that the dynamical system associated with $Q$ is always stable, since the poles of $Q$ (the two zeros of $D$) are located in the left half complex plane, and will be denoted by $d_0, d_1$, with

$$\Re(d_1) \le \Re(d_0) \le 0. \quad (2.8)$$

This notion of stability is quite classical in automatic control theory. It implies that a finite perturbation should never have an unbounded effect on the system. Note that it would not always hold in a more complicated setting, for instance, when taking into account reaction-time delays [8].

It will also be necessary to consider the zeros of

$$Q'(z) = -\frac{\beta z^2 + 2\alpha z + \alpha \gamma}{D^2(z)} \stackrel{\text{def}}{=} -\frac{R(z)}{D^2(z)}, \quad (2.9)$$

i.e., the zeros of $R(z) = \beta z^2 + 2\alpha z + \alpha \gamma$, denoted by $z_0, z_1$, with $\Re(z_1) \le \Re(z_0)$.

A further important concept is *string stability*, which says that perturbations tend to vanish rather than amplify along a string of vehicles.

**Definition 2.1.** The system is string stable when one of the two equivalent properties holds [4, 8]:

$$1. |Q(z)| \le 1 \text{ for all } \Re(z) \ge 0,$$

$$2. 2\alpha \le \mu \stackrel{\text{def}}{=} \delta^2 - \beta^2.$$

As briefly enlightened in the Introduction, stop-and-go waves can clearly not occur when the system is string stable. We will therefore need the following hypothesis.

**Hypothesis.** Throughout this study, we suppose that the system is string unstable, which is tantamount to

$$2\alpha > \mu \stackrel{\text{def}}{=} \delta^2 - \beta^2. \quad (2.10)$$
We begin by establishing some preliminary important inequalities, the sub-stance of which will be commented on below.

**Lemma 2.2.** Assume (2.10) holds. Then

$$\alpha > \beta\gamma \quad \text{and} \quad z_1 < z_0 < 0. \tag{2.11}$$

In addition:

1. If
$$\delta^2 \geq 4\alpha, \tag{2.12}$$

then
$$\gamma \leq \beta(\sqrt{2} - 1) < \beta, \tag{2.13}$$

the poles $(d_0, d_1)$ of $Q$ are real and satisfy

$$d_1 < z_1 < d_0 < -\frac{\alpha}{\beta} < z_0 < 0. \tag{2.14}$$

2. If
$$\delta^2 < 4\alpha, \tag{2.15}$$

then $d_0, d_1$ are complex conjugate and

$$z_1 \leq \Re(d_i) = -\frac{\delta}{2} < z_0 < 0. \tag{2.16}$$

*Proof.* From definition (2.10), we have

$$\mu - 2\alpha = \gamma(\gamma + 2\beta) - 2\alpha = \gamma^2 + 2(\beta\gamma - \alpha),$$

and (2.11) follows immediately.

When (2.10) and (2.12) simultaneously hold, we can write

$$\delta^2 - \beta^2 < 2\alpha \leq \frac{\delta^2}{2},$$

which implies $\delta^2 = (\beta + \gamma)^2 < 2\beta^2$ and (2.13).

Still assuming (2.12), we have $D(d_0) = 0$ with

$$d_0 = \frac{-\delta + \sqrt{\delta^2 - 4\alpha}}{2},$$

and it suffices to show $R(d_0) < 0$. From the definition (2.9),

$$\begin{aligned}
R(d_0) &= \beta d_0^2 + 2\alpha d_0 + \alpha\gamma = d_0(2\alpha - \beta\delta) + \alpha(\gamma - \beta) \\
&= \frac{\sqrt{\delta^2 - 4\alpha}}{2}(2\alpha - \beta\delta) + \frac{\beta(\delta^2 - 4\alpha)}{2},
\end{aligned}$$
and it follows that $R(d_0)$ has the same sign as the quantity

$$2\alpha - \beta\delta + \beta\sqrt{\delta^2 - 4\alpha},$$

which is negative by (2.13), since

$$2\alpha - \beta\delta < \frac{\delta^2}{2} - \beta\delta = \frac{\delta(\gamma - \beta)}{2} < 0,$$

and

$$\beta^2(\delta^2 - 4\alpha) - (2\alpha - \beta\delta)^2 = 4\alpha(\beta\delta - \beta^2 - \alpha) = 4\alpha(\beta\gamma - \alpha) < 0.$$

Hence, $R(d_0) < 0$, which implies $z_1 < d_0 < z_0$. On the other hand,

$$R(d_1) = \frac{\sqrt{\delta^2 - 4\alpha}}{2}(\beta\delta - 2\alpha) + \frac{\beta(\delta^2 - 4\alpha)}{2} > 0,$$

which shows $d_1 < z_1$. As we also have the inequalities

$$R(-\frac{\alpha}{\beta}) = \frac{\alpha(\beta\gamma - \alpha)}{\beta} < 0, \quad D(-\frac{\alpha}{\beta}) = \frac{\alpha(\alpha - \beta\gamma)}{\beta^2} > 0,$$

(2.14) is proved.

Assuming now that (2.15) holds, we get

$$R(-\frac{\delta}{2}) = \beta(\frac{\delta^2}{4} - \alpha) < 0,$$

which leads to (2.16) and concludes the proof of the lemma. ■

### 3 Asymptotic estimates in finite time, as $N \to \infty$

As shown in Appendix B, the calculation of a precise estimate for $I(t; N)$ defined by (2.7) would require intricate technicalities. In fact, the presence of the poles $d_0, d_1$ in the left half-plane, see (2.8), renders the use of the saddle-point method practically out of question here. Therefore, we only exhibit a bound on the modulus $|I(t; N)|$, as $N \to \infty$.

To this end, we will work in the right half-plane. By Cauchy's Theorem, we can write

$$I(t; N) = \frac{1}{2i\pi} \int_{u-i\infty}^{u+i\infty} e^{tz}\phi(z)Q^N(z)dz, \quad (3.1)$$

where $u$ is a positive real number and $\phi(z)$ is chosen to ensure the existence of the integral. The main substance of this section is the following proposition.
**Proposition 3.1.** There exists $u_0 > 0$ such that, with the classical Landau notation (e.g. [7, III.3]),

$$x_N(t) - x_N^*(t) = \mathcal{O}\left[e^{tu_0}|X_0(u_0)|Q^N(u_0)(cN)^{-\frac{1}{4}}\right], \quad \text{as } N \to \infty, \quad (3.2)$$

with

$$Q(u_0) < 1, \quad \text{and} \quad c = \frac{1}{2} \frac{(\alpha + \beta u_0)^2}{D^4(u_0)}, \qquad (3.3)$$

which means in particular that $\lim_{N \to \infty} x_N(t) - x_N^*(t) = 0$ exponentially fast. In other words, at any fixed time $t$, the wave has not yet reached vehicle $N$, for large $N$.

*Proof.* In order to prove (3.2), it is necessary to show the existence of $u > 0$, such that on the vertical line of equation $z = u + iy$, we have $|Q(z)| < 1$, uniformly for all real $y$.

Setting $Y = y^2$, we obtain after a direct algebra

$$F(Y, u) \stackrel{\text{def}}{=} |Q(z)|^2 = \frac{A(u) + \beta^2 Y}{(D(u) - Y)^2 + B(u)Y}, \qquad (3.4)$$

together with

$$\frac{\partial F(Y, u)}{\partial Y} = \frac{P_{F'}(Y, u)}{[(D(u) - Y)^2 + B(u)Y]^2}, \qquad (3.5)$$

where

$$A(u) = (\alpha + \beta u)^2, \quad B(u) = (\delta + 2u)^2.$$

and

$$P_{F'}(Y, u) \stackrel{\text{def}}{=} -\beta^2 Y^2 - 2A(u)Y + A(u)(2D(u) - B(u)) + \beta^2 D^2(u). \quad (3.6)$$

By first taking $u > 0$ such that $|F(Y, u)| < 1, \forall Y \ge 0$, we obtain from (3.4) the equivalence

$$|F(Y, u)| < 1 \iff Y^2 + (B(u) - 2D(u) - \beta^2)Y + D^2(u) - A(u) > 0, \quad (3.7)$$

keeping in mind that $Y = y^2$ is a positive variable.

We also note that

$$D^2(u) - A(u) = u(u + \gamma)(D(u) + \beta u + \alpha) > 0, \forall u > 0.$$

On the other hand, the coefficient of $Y$ in (3.7), which is equal to

$$C(u) \stackrel{\text{def}}{=} B(u) - 2D(u) - \beta^2 = 2u^2 + 2\delta u + \mu - 2\alpha,$$
has a positive root at $u = u_m$ and remains positive for $u > u_m$, thus ensuring the validity of the inequality (3.7) on the real axis $u \ge u_m$.

The next step is to find $u_0 \ge u_m$, such that $F(Y, u_0) \le F(0, u_0), \forall Y \ge 0$. For this purpose, using (3.5), it suffices to take $u_0$ as the first positive root greater than $u_m$ of the 4-th degree polynomial

$$P(u) \stackrel{\text{def}}{=} A(u)[2D(u) - B(u)] + \beta^2 D^2(u),$$

since in this case the numerator of (3.5) is strictly negative, so that $F(Y, u_0)$ is an increasing function of $Y \ge 0$. To check the existence of this root, we note that, since $C(u_m) = 0$,

$$P(u_m) = \beta^2(D^2(u_m) - A(u_m)) > 0,$$

while $P(u)$ is of order $-\beta^2 u^4 < 0$ as $u \to +\infty$. Hence, $P(u)$ vanishes at least once for $u \ge u_m$.

It is also worth to emphasize that, by (2.10),

$$P(0) = \alpha^2(2\alpha - \mu) > 0,$$

which implies that $\frac{\partial F(Y, u)}{\partial Y}$, viewed as a function of $Y$, might be positive for some $u \in [0, u_0]$, in which case the saddle point would not be real.

Setting

$$J(t; N) \stackrel{\text{def}}{=} \frac{1}{2\pi} \int_{-\infty}^{\infty} |\phi(u_0 + iy)| e^{tu_0} e^{N \log |Q(u_0 + iy)|} dy,$$

we have $|I(t; N)| < J(t; N)$ and the change of variable $y^2 = Y$ yields

$$J(t; N) = \frac{e^{tu_0}}{4\pi} \int_0^\infty \frac{1}{\sqrt{Y}} |\phi(u_0 + i\sqrt{Y})| e^{N \frac{\log F(Y, u_0)}{2}} dY. \quad (3.8)$$

We are now in a position to estimate (3.8) by using Laplace's method (briefly presented in Appendix A) as follows.

Since $P(u_0) = 0$, we get from (3.5)

$$\frac{\partial F(Y, u_0)}{\partial Y} \Big|_{Y=0} = 0 \quad \text{and} \quad \frac{\partial^2 F(Y, u_0)}{\partial Y^2} \Big|_{Y=0} = -\frac{2A(u_0)}{D^4(u_0)},$$

whence

$$\log F(Y, u_0) = \log \frac{A(u_0)}{D^2(u_0)} - \frac{A(u_0)}{D^4(u_0)} Y^2 + \mathcal{O}(Y^3).$$

Then a direct application of the formula (A.1) with $p = -\frac{1}{2}$ and $q = 2$ yields the asymptotic expansion

$$J(t; N) \sim \frac{e^{tu_0}|\phi(u_0)|}{4\pi} \Gamma\left(\frac{1}{4}\right) e^{aN(cN)^{-\frac{1}{4}}}, \quad \text{as } N \to \infty,$$
where c is given by (3.3) and

$$a = \frac{1}{2} \log \frac{A(u_0)}{D^2(u_0)} = \log Q(u_0), \quad \text{whence } e^{aN} = Q^N(u_0),$$

remarking that, since (3.7) holds, a is negative, as expected. The proof of the proposition is concluded. ■

For any fixed $t$, $x_N(t) - x_N^*(t)$ is exponentially small as $N$ becomes large, which means that the wave has not yet reached vehicle $N$. In order to obtain more useful information, we propose in the following section an analysis of the system after a linear scaling of time.

# 4 Asymptotic estimates with a linear time scaling, as $N \to \infty$

In this section, using the saddle-point method (see Appendix A), we analyze the behavior of the system on a time scale proportional to its size $N$. Therefore, for any $\tau > 0$, the integral of interest becomes

$$I(\tau N; N) = \frac{1}{2i\pi} \int_{u-i\infty}^{u+i\infty} \phi(z) [e^{\tau z} Q(z)]^N dz, \quad u > 0,$$

where we assume that the function $\phi(z)$, which will take into account the disturbance of the trajectory of vehicle 0, is properly chosen to guarantee the existence of the integral.

We will say that $z$ is a candidate saddle point for this integral if, for any $\tau \ge 0$, $\frac{d}{dz}[e^{\tau z} Q(z)] = 0$, that is,

$$\tau = T(z) \stackrel{\text{def}}{=} -\frac{Q'(z)}{Q(z)} = \frac{\beta z^2 + 2\alpha z + \alpha\gamma}{(\beta z + \alpha)D(z)}, \quad (4.1)$$

Note that the zeros $z_0, z_1$ of $R(z)$ defined by (2.9) are candidate saddle points when $\tau = 0$. In the sequel, we will be more interested in admissible saddle points, at which $|e^{\tau z} Q(z)|$ reaches its maximum value on the integration path.

Although any saddle point $z$ is obviously a function of $\tau$, it will be more convenient in the sequel to reverse the dependence and consider $\tau$ as a function of $z$.

As we shall see later, the variations of $T$ do matter, especially its derivative
given by

$$ T'(z) = \frac{P_{T'}(z)}{(z^2 + \delta z + \alpha)^2(\beta z + \alpha)^2}, $$
with $P_{T'}(z) \stackrel{\text{def}}{=} -\beta^2 z^4 - 4\alpha\beta z^3 - 2\alpha(2\beta\gamma + \alpha)z^2 + 2\alpha(\alpha(\beta - \gamma) - \beta\gamma(\beta + \gamma))z + \alpha^2(2\alpha - \mu). \quad (4.2) $$

Note that when $z$ is real, $T'(z)$ has the same sign as its numerator $P_{T'}(z)$.

## 4.1 Asymptotic expansions

For purely algebraic reasons, it will appear that the characteristic values of interest can be parameterized by the real part $u$ of the relevant saddle point $z$, which satisfies $T(z) = \tau$. The question is to know whether we can construct a meaningful bijection between the $\tau$'s and the $u$'s, keeping in mind that $\tau$ must be a positive real number, which implies in particular $\Im(T(z)) = 0$. The results of this subsection do not depend on the sign of $\Re(T(z))$.

It turns out that it is difficult to consider all positive values of $\tau$ from the outset, and, consequently, we will first discuss in terms of $u$.

In fact, it is possible to classify saddle points by looking at the sign of $T'(u)$, as explained in the following lemma.

**Lemma 4.1.** Assume $u \in \mathbb{R}$.

1. When $T'(u) < 0$, the function $y \mapsto |Q(u + iy)|$, $y \in \mathbb{R}$, has a global maximum at $y = 0$.

2. When $T'(u) > 0$, then $y_u$, defined below, is real and $T(u + iy_u) \in \mathbb{R}$.

$$ y_u \stackrel{\text{def}}{=} \frac{1}{\beta} \sqrt{-(\beta u + \alpha)^2 + \sqrt{\Delta'}}, \quad (4.3) $$

$$ \Delta' = \alpha(\alpha - \beta\gamma) [4\beta^2 u^2 + 2\beta(\beta\delta + 2\alpha)u + 2\alpha\beta^2 + \alpha\beta\gamma + \alpha^2]. \quad (4.4) $$

Furthermore, $y_u$ is also the maximum of $y \mapsto |Q(u + iy)|$ for $y > 0$.

*Proof.* Before using the explicit form of $Q$, it is useful to establish a few generic relationships. If we choose an arbitrary complex point $z = u + iy$, then the variation of the modulus $|Q|$ along a vertical line is given by

$$ \begin{align*} \frac{\partial}{\partial y} |Q(u + iy)|^2 &= \frac{\partial}{\partial y} Q(u + iy) Q(u - iy) = iQ'(z)Q(\bar{z}) - iQ(z)Q'(\bar{z}) \\ &= 2|Q(z)|^2 \Im(T(z)). \end{align*} $$

Thus, the extrema of $y \mapsto |Q(u + iy)|$ are reached when $\Im(T(z)) = 0$, and therefore might correspond to admissible real saddle points.
To analyze the behavior of $F(Y, u) = |Q(u + iy)|^2$ viewed as a function of $Y = y^2$ (see (3.4)), we study its derivative, already computed and defined in (3.5), namely

$$ \frac{\partial F(Y, u)}{\partial Y} = \frac{P_{F'}(Y, u)}{[(D(u) - Y)^2 + B(u)Y]^2} $$

Although this can be easily verified from the expression above, it is interesting to note that, for any $Q$,

$$ \frac{\partial F(0, u)}{\partial Y} = \lim_{y \to 0} \frac{1}{2y} \frac{\partial}{\partial y} |Q(u + iy)|^2 = Q(u)^2 T'(u). $$

On the other hand, according to (3.6), $T'(u) < 0$ immediately implies that $P_{F'}(Y, u)$ is negative, for all $Y \ge 0$. Therefore, $F(Y, u)$, viewed as a function of $Y$, decreases for all $Y \ge 0$ and $|Q(u + iy)|$ is maximal for $y = 0$, which proves part 1. of the lemma.

Assume now $T'(u) > 0$. Then $P_{F'}(Y, u)$, seen as a quadratic polynomial in $Y$, has two roots of opposite signs, the positive one being $Y_u = y_u^2$ as defined in (4.3)-(4.4). Note that some algebraic simplification takes place, which yields a polynomial $\Delta'$ quadratic in $u$. ■

When $T'(u) < 0$, then $u$ is an admissible saddle point associated with $\tau = T(u)$, which allows us to compute the principal part of the integral $I(\tau N; N)$, as we did in Section 3 for $|I(t; N)|$, using the saddle-point method (also known as steepest descent method).

**Lemma 4.2.** When $u > \Re(d_0)$ (see (2.8)) and $T'(u) < 0$, the following asymptotic formula holds, as $N \to \infty$:

$$ I(T(u)N; N) \sim \frac{\phi(u)[e^{uT(u)}Q(u)]^N}{\sqrt{2\pi N|T'(u)|}} + \mathbb{1}_{\{u<0\}} \mathrm{Res}(\phi; z=0), $$

where $\mathrm{Res}(\phi; z=0)$ denotes the residue of $\phi$ at $z=0$.

*Proof.* The proof is a direct application of the theory presented in [7, IX.1.] or in [14, V,78]. Since $Q(z)$ has no pole in the right half-plane (see Section 2), by Cauchy's theorem, $I(T(u)N; N)$ can be rewritten as

$$ I(T(u)N; N) = \frac{1}{2i\pi} \int_{u-i\infty}^{u+i\infty} \phi(z) [e^{zT(u)}Q(z)]^N dz + \mathbb{1}_{\{u<0\}} \mathrm{Res}(\phi; z=0), \quad (4.5) $$

where the integral is taken along the vertical line going through the real saddle point of abscissa $u > \Re(d_0)$.
In a small neighborhood of $u$, we have the following Taylor expansion

$$zT(u) + \log Q(z) = uT(u) + \log Q(u) - \frac{T'(u)}{2}(z-u)^2 + \mathcal{O}(z-u)^3,$$

where $T'(u) < 0$. Then the classical approach starts by showing that the principal part of the integral (4.5) is equivalent to:

$$I_r(T(u)N; N) = \frac{1}{2i\pi} \int_{u-ir}^{u+ir} \phi(z) [e^{T(u)z} Q(z)]^N dz, \quad (4.6)$$

where $r > 0$ is sufficiently small. In our case, this fact follows directly from Lemma 4.1.

The second step is to find an appropriate *thalweg line* allowing us to use Laplace's method for real functions (see Appendix A). Here, the Taylor expansion above shows that the vertical segment $[u - ir, u + ir]$ is perfectly suitable since, in that segment,

$$-\frac{T'(u)(z-u)^2}{2} = -\frac{T'(u)(iy)^2}{2} = \frac{T'(u)y^2}{2} < 0.$$

Consequently, we can apply equation (A.2) from Appendix A, with $\alpha = 0$. ■

Now, when $T'(u) > 0$, $u$ cannot be a usable saddle point. However, in this case, $z = u + iy_u$, with $y_u$ defined by (4.3)-(4.4), is an admissible saddle point associated with $\tau = T(z) \in \mathbb{R}$. This yields the following result.

**Lemma 4.3.** When $u > \Re(d_0)$ (see (2.8)) and $T'(u) > 0$, the following asymptotic formula holds, as $N \to \infty$:

$$I(T(z_u)N; N) \sim \sqrt{\frac{2}{\pi N}} \Re \left[ \frac{\phi(z_u) [e^{z_u T(z_u)} Q(z_u)]^N}{T'(z_u)^{1/2}} \right] + \mathbb{1}_{\{u<0\}} \mathrm{Res}(\phi; z=0),$$

with $z_u \stackrel{\text{def}}{=} u + iy_u$ and $y_u$ defined by (4.3)-(4.4).

*Proof.* Here we have to take into account the two conjugate complex saddle points $u + iy_u$ and $u - iy_u$ located on the vertical line of abscissa $u$. Splitting the analogous of (4.5) into two parts

$$I(T(z_u)N; N) = \frac{1}{2i\pi} \int_{u-i\infty}^{u-i0} \phi(z) [e^{zT(z_u)} Q(z)]^N dz + \frac{1}{2i\pi} \int_{u+i0}^{u+i\infty} \phi(z) [e^{zT(z_u)} Q(z)]^N dz + \mathbb{1}_{\{u<0\}} \mathrm{Res}(\phi; z=0),$$

we use Lemma 4.1 to estimate each part separately, as done in Lemma 4.2. Then, by adding these two estimates (which are complex conjugates), we obtain the announced real principal part, which involves sinusoidal oscillations, since a priori $T'(u + iy_u)$ is not a real number, unlike $T(u + iy_u)$. ■
<figure><img src="image_2.png" /><figcaption>Figure 4.1: $T(u)$ (red) as a function of $u > -\alpha/\beta$. The line is dashed when the corresponding saddle points are not admissible. The blue line will be explained in Section 4.3 (parameters are $\alpha = 5$, $\beta = 3.6$, and $\gamma = 1$).</figcaption></figure>

## 4.2 Main results

The following lemma shows how to invert the dependency between $\tau$ and $u$, to determine the sign of $T'(u)$ as a function of the corresponding $\tau$ in an interval. It will be useful for the reader to refer to Figure 4.1.

**Lemma 4.4.** There exists a unique local maximum $u_c$ of $T(u)$ for $u > 0$. Letting $\tau_c \stackrel{\text{def}}{=} T(u_c) > 0$, $Z \stackrel{\text{def}}{=} T^{-1}$ is a decreasing bijection from $[0, \tau_c]$ to $]u_c, \infty[$.

*Proof.* A real extremum of $T(u)$ is given by a zero of $P_{T'}(u)$, whose second derivative is the quadratic polynomial

$$P_{T'}''(u) = -4(3\beta^2u^2 + 6\alpha\beta u + 2\alpha\beta\gamma + \alpha^2),$$

with a reduced discriminant equal to $24\alpha\beta^2(\alpha - \beta\gamma)$, which is always positive due to (2.11). One sees at once that $P_{T'}''$ has two negative roots and $P_{T'}''(u) < 0$ for $u > 0$. Hence, $P_{T'}$ is concave on $\mathbb{R}^+$. Since $P_{T'}(0) = \alpha^2(2\alpha - \mu) > 0$ and $\lim_{u \to \infty} P_{T'}(u) = -\infty$, there exists a unique $u_c > 0$ such that $T'(u_c) = 0$. Consequently, $T(u)$ strictly increases (resp. decreases) in $[0, u_c]$ (resp. $]u_c, \infty[$), and the lemma is proved. ■

Combining this result with Lemma 4.2 and equation (2.5), we obtain the following theorem.
**Theorem 4.5.** When $\tau \in ]0, \tau_c[$, the deviations of $x_N(\tau N)$ and $v_N(\tau N)$ from equilibrium have the following asymptotic expansion, as $N \to \infty$,

$$x_N(\tau N) - x_N^*(\tau N) \sim \frac{X_0(Z(\tau))}{\sqrt{2\pi N |T'(Z(\tau))|}} \left[e^{\tau Z(\tau)} Q(Z(\tau))\right]^N, \quad (4.7)$$

$$v_N(\tau N) - v^* \sim \frac{Z(\tau) X_0(Z(\tau))}{\sqrt{2\pi N |T'(Z(\tau))|}} \left[e^{\tau Z(\tau)} Q(Z(\tau))\right]^N. \quad (4.8)$$

The range $]0, \tau_c[$ is the time interval that contains the onset of the stop-and-go wave. Specifically, when $|e^{\tau Z(\tau)} Q(Z(\tau))| < 1$, the wave is not yet present, and eventually there is a $\tau_w$ after which $|e^{\tau Z(\tau)} Q(Z(\tau))| > 1$, signaling a phase transition in equations (4.7)–(4.8). This is made more precise in the following theorem.

**Theorem 4.6.** There exists a unique real $u_w > u_c$ satisfying

$$T(u_w) = -\frac{\log Q(u_w)}{u_w} \stackrel{\text{def}}{=} \tau_w, \quad (4.9)$$

and, as $N \to \infty$, the following asymptotic expansion holds:

$$x_N(\tau_w N) - x_N^*(\tau_w N) \sim \frac{X_0(u_w)}{\sqrt{2\pi |T'(u_w)| N}}.$$

When $N$ is large, the $N$-th vehicle is affected at time $t = \tau_w N$ by the wave generated at $t = 0$ at the point $x_0(0) = 0$. The velocity of this wave is

$$\frac{x_N(\tau_w N)}{\tau_w N} = v^* - \frac{\Delta x^*}{\tau_w} + \mathcal{O}(N^{-\frac{3}{2}}), \quad (4.10)$$

which shows that the wave is moving upstream.

*Proof.* Let us first define the function

$$f(u) \stackrel{\text{def}}{=} -\frac{\log Q(u)}{u}.$$

Equation (4.9) expresses the fact that the pair $(\tau_w, u_w)$ corresponds to a saddle point in the sense of Lemma 4.4 and that $|e^{\tau_w u} Q(u)| = 1$. Let us prove its existence. In the vicinity of $u = 0+$, the following series approximations hold.

$$T(u) = \frac{\gamma}{\alpha} + \frac{2\alpha - \mu}{\alpha^2} u + \mathcal{O}(u^2), \quad f(u) = \frac{\gamma}{\alpha} + \frac{2\alpha - \mu}{2\alpha^2} u + \mathcal{O}(u^2),$$
so that, using (2.10), locally $T(u) > f(u)$. In contrast, as $u \to \infty$, $f(u) > T(u)$ since

$$T(u) = \frac{1}{u} + \mathcal{O}(u^{-2}) \qquad f(u) = \frac{\log u}{u} + \mathcal{O}(u^{-2}).$$

Therefore, the two functions intersect, which proves the existence of $u_w$ such that $T(u_w) = f(u_w)$.

The next easy thing to check is that any real $u_w$ that satisfies (4.9) is necessarily an extremum of $f$. In fact, its derivative vanishes at $u_w$, since

$$f'(u_w) = -\frac{1}{u_w^2} \left[ \frac{Q'(u_w)}{Q(u_w)} u - \log Q(u_w) \right] = \frac{1}{u_w} [T(u_w) - f(u_w)] = 0.$$

The sign of the second derivative of $f$ at $u_w$ is given by $T'(u_w)$:

$$f''(u_w) = \frac{1}{u_w^2} \left[ (T'(u_w) - f'(u_w))u_w - T(u_w) + f(u_w) \right] = \frac{T'(u_w)}{u_w}.$$

Since $f(u)$ increases for small values of $u$, its first extremum cannot be a local minimum. Therefore, $T'(u_w) \le 0$ and $u_w \ge u_c$. The case $u_w > u_c$ leads to a local maximum which is unique, because the next extremum of $f$ would have to be a local minimum, which is not possible, since in this region $T'(u) < 0$. The case $T'(u_w) = 0$ is also possible, but the uniqueness of $u_w$ still holds by continuity arguments.

The other claims of the theorem are simple consequences of the identity

$$e^{\tau_w u_w} Q(u_w) = 1.$$

<figure><img src="image_3.png" /></figure>

It is interesting to compare the wave velocity in (4.10) with the one provided in [24, equation (15.47)] for the so-called propagation velocity. With the notation of the present paper, this velocity satisfies

$$\tilde{c}(k) = v^* - \Delta x^* \frac{\alpha}{\gamma} + \mathcal{O}(k),$$

where $k$ is the wavenumber of the Fourier-Ansatz, such that $|k| \ll \pi$. Comparing the two formulas, one sees that the $\tau_w$ coming in (4.10) is replaced by $\frac{\gamma}{\alpha}$. It turns out that

$$\frac{\gamma}{\alpha} = T(0) < T(u_w) = \tau_w.$$

Thus, the estimate of $\tilde{c}(k)$ is likely a lower bound on the actual velocity. We will return to this question in Section 5.
<figure><img src="image_4.png" /><figcaption>Figure 4.2: $T(u)$ (red) and $T(z_u)$ (blue) as functions of $u < u_c$. The line is dashed when the corresponding saddle points are not admissible. Note that the right part of this figure corresponds to Figure 4.1, up to a factor 40 in the scale of the vertical axis (parameters are $\alpha = 5$, $\beta = 3.6$, and $\gamma = 1$).</figcaption></figure>

## 4.3 The case $\delta^2 > 4\alpha$

In the previous subsection, we have described the wave front (corresponding to $\tau < \tau_c$) created by string-unstable traffic in the context of a fairly general car-following model. Establishing the regime for $\tau > \tau_c$ is more complicated, but can be done in a general way when the poles of $Q$ are real. Here again, the reader may refer to Figure 4.1 and Figure 4.2.

**Lemma 4.7.** When (2.12) holds, so that $Q$ has two real poles $d_1 < d_0 < 0$, there exists a unique local minimum $u_d$ of $T(u)$ in the interval $]d_0, -\frac{\alpha}{\beta}[$. Letting $\tau_d \stackrel{\text{def}}{=} T(u_d) > 0$, $Z = T^{-1}$ is also a decreasing bijection from $]\tau_d, \infty[$ to $]d_0, u_d[$.

Finally, the function $Z$ can be extended to the interval $]\tau_c, \tau_d[$ with $Z(\tau)$ of the form $u + iy_u$, $u \in ]u_d, u_c[$, and $y_u$ defined by (4.3)-(4.4), in such a way that $T(Z(\tau)) = \tau$.

*Proof.* The discriminant of the quartic equation $P_{T'}(u) = 0$, thanks to Maple symbolic computing tool, has the factorized form

$$ -16\alpha^4\beta^2(\alpha - \beta\gamma)^4(\delta^2 - 4\alpha)[27\beta^2(\delta^2 - 4\alpha) + 8\alpha(\alpha - \beta\gamma)]. $$

Thus, owing to (2.11) and (2.12), this discriminant is negative. Hence, $P_{T'}$
has two distinct real zeros and two complex conjugate zeros. One of these
real zeros is actually $u_c > 0$.

To find the second real zero, one examines the behavior of $T$ at $d_0 + \varepsilon$, for $\varepsilon$
sufficiently small. It follows from (2.14) that $T$ decreases from $+\infty$. Similarly,
$T$ increases to $+\infty$ in the left neighborhood of $-\alpha/\beta$, and the second real
zero $u_d$ of $P_{T'}$ lies necessarily on $]d_0, -\frac{\alpha}{\beta}[$. Therefore, $T$ decreases on $]d_0, u_d[$
and the first part of the lemma is proved.

We do not know at this point whether $\tau_c < \tau_d$ and whether it is possible to
extend $Z$ so that it decreases on $\mathbb{R}^+$. Let us consider again equation (4.1).
Solving it with respect to $z$ obviously amounts to solving a cubic polynomial,
the exact form of which is not relevant here. The only important information
is that it has 3 real or complex solutions.

One of these solutions can be found in the $]d_1, d_0[$ interval. Indeed, $T$ always
decreases for $u < u_d$, which implies in particular that it decreases from $+\infty$
to $-\infty$ over $]d_1, d_0[$, and yields a unique solution to (4.1). Thus, there are at
most two other solutions for $u > d_0$.

Assume for a moment that $\tau_d < \tau_c$. Then, for any $\tau \in ]\tau_d, \tau_c[$, the equation
$T(z) = \tau$ has one real solution in $]u_c, \infty[$, another one in $]-\alpha/\beta, u_c[$ (remember
that $T$ is increasing in this interval), and similarly two other solutions in
$]d_0, -\alpha/\beta[$. This would amount to the existence of four distinct solutions,
which is clearly impossible. Therefore, $\tau_d > \tau_c$.

Likewise, for any $u \in ]u_d, u_c[$, $T(z_u)$ must fall into $]t_c, \tau_d[$, in order to avoid
an excessive number of solutions, and the complex conjugates $z_u$ and $\bar{z}_u$ are
necessarily the two remaining solutions of equation $\tau = T(z)$. Finally, the
surjectiveness of $Z$ comes from the continuity of $T$ at points $u_d$ and $u_c$. ■

The first consequence of this lemma is that (4.8) also holds for $\tau \in ]\tau_d, \infty[$,
but equation (4.7) must be corrected to take into account a possible residue:

$$
x_N(\tau N) - x_N^*(\tau N) \sim \frac{X_0(Z(\tau))}{\sqrt{2\pi N |T'(Z(\tau))|}} \left[e^{\tau Z(\tau)} Q(Z(\tau))\right]^N + \mathbb{1}_{\{\tau > \tau_0\}},
$$

where $\tau_0$ satisfies $Z(\tau_0) = 0$ and is given by (4.3)–(4.4)

$$
\tau_0 \stackrel{\text{def}}{=} T(0 + iy_0) = T \left[ \frac{i}{\beta} \sqrt{\alpha \sqrt{(\alpha - \beta\gamma)(2\beta^2 + \beta\gamma + \alpha)} - \alpha^2} \right].
$$
For $\tau \in ]\tau_c, \tau_d[$, the result is slightly different and is based on Lemma 4.3:

$$x_N(\tau N) - x_N^*(\tau N) \sim \frac{2}{\sqrt{2\pi N|T'(Z(\tau))|}} \Re[X_0(Z(\tau)) (e^{\tau Z(\tau)} Q(Z(\tau)))^N] + \mathbb{1}_{\{\tau > \tau_0\}}, \quad (4.11)$$

$$v_N(\tau N) - v^* \sim \frac{2}{\sqrt{2\pi N|T'(Z(\tau))|}} \Re[Z(\tau)X_0(Z(\tau)) (e^{\tau Z(\tau)} Q(Z(\tau)))^N]. \quad (4.12)$$

The above analysis leads to a classification of the wave into three phases.

1. The front of the wave ($t < \tau_c N$), which sees the appearance of a growing wave.

2. The body of the wave ($\tau_c N < t < \tau_d N$), where traffic becomes stop-and-go with pseudo-sinusoidal variations.

3. The tail of the wave ($t > \tau_d N$), where the wave gradually dampens.

## 4.4 The case $\delta^2 < 4\alpha$

The situation becomes more complicated when $\delta^2 < 4\alpha$, since the poles of $Q$ (which are the roots of $D$) are now complex conjugates. The complete classification then depends on the sign of $T(\Re(d_0))$. Instead of proposing a tedious analysis with many subcases, here we propose to analyze a special case of interest: $\beta = 0$. However, as demonstrated in Section 5 the variations of $T$ are known for any fixed set of parameters and thus Lemma 4.2 or Lemma 4.3 will apply. What we are not going to provide here is the classification of possible cases.

The case $\beta = 0$ is interesting, not only for technical simplification reasons but more importantly because it corresponds to the classical optimal velocity model (OVM) [3], where the acceleration does not depend on $\Delta\hat{v}_j(t)$ and is of the form

$$\hat{a}_j(t) = \hat{a}_0 \times (V(\Delta\hat{x}_j(t)) - v_j(t)),$$

In this case, (2.10) becomes simply $2\alpha > \gamma^2$, which clearly implies $\delta^2 = \gamma^2 < 4\alpha$.

Then, for any fixed $\tau > 0$, the candidate saddle points satisfy (4.1), which now reduces to the simple second-degree equation

$$\tau z^2 + (\gamma\tau - 2)z + \alpha\tau - \gamma. \quad (4.13)$$

We provide a direct analysis of this equation that can be recast within the general framework of the previous Sections 4.1 and 4.2.
<table><caption>Table 1: Parameters for IDM [26]</caption><thead><tr><td>Parameter</td><td>value</td></tr></thead><tbody><tr><td>Desired speed v<sub>0</sub></td><td>33.3 m s<sup>-1</sup></td></tr><tr><td>Safe time headway T</td><td>1.5 s</td></tr><tr><td>Acceleration a</td><td>0.3 m s<sup>-2</sup></td></tr><tr><td>Comfortable deceleration b</td><td>3 m s<sup>-2</sup></td></tr><tr><td>Acceleration exponent δ</td><td>4</td></tr><tr><td>Jam distance s<sub>0</sub></td><td>2 m</td></tr><tr><td>Vehicle length ℓ</td><td>5 m</td></tr></tbody></table>

**Lemma 4.8.** When $\beta = 0$, the variable $\tau_c$ defined in Lemma 4.4 is equal to

$$ \tau_c = \frac{2}{\sqrt{4\alpha - \gamma^2}}. $$

The admissible saddle point associated with $\tau$ is given as:

$$ Z(\tau) = \begin{cases} \frac{2 - \gamma\tau + \sqrt{4 - (4\alpha - \gamma^2)\tau^2}}{\tau}, & \text{if } 0 < \tau < \tau_c \\ \frac{2 - \gamma\tau + i\sqrt{(4\alpha - \gamma^2)\tau^2 - 4}}{\tau}, & \text{if } \tau > \tau_c. \end{cases} $$

*Proof.* The above formulas come from the resolution of (4.13), taking into account the inequality (2.10), which, since $\beta = 0$, becomes $\gamma^2 < 2\alpha$. The classification of solutions of (4.13) as saddle points is based on the equivalence between the sign of $T'(u)$ and the nature of the saddle point given in Lemma 4.1. Here

$$ P_{T'}(u) = -\alpha^2(2u^2 + 2\gamma u + \gamma^2 - 2\alpha), $$

and the discriminant of (4.13) vanishes for $\tau = \tau_c$.

<figure><img src="image_5.png" /></figure>

As a consequence, the asymptotic expansions (4.7)–(4.8) and (4.11)–(4.12) still hold with the definition of $Z$ given above and $\tau_d = \infty$. The reader will notice that here the wave has only two phases, unlike the classification obtained in Section 4.3, which included three phases.

# 5 Numerical experiments

In order to evaluate the precision of our asymptotic estimates and to see how useful they are with moderately large values of $N$, we will use here an
<table><caption>Table 2: Values of interest for the experiment</caption><thead><tr><th>Parameter</th><th>value</th></tr></thead><tbody><tr><td>Stationary speed <i>v</i>*</td><td>16.6 m s<sup>-1</sup></td></tr><tr><td>Stationary heading &Delta;<i>x</i>*</td><td>27.9 m</td></tr><tr><td>Heading coefficient &alpha;</td><td>0.02 s<sup>-2</sup></td></tr><tr><td>Relative speed coefficient &beta;</td><td>0.18 s<sup>-1</sup></td></tr><tr><td>Speed coefficient &gamma;</td><td>0.036 s<sup>-1</sup></td></tr><tr><td>2&alpha; &minus; &mu;</td><td>0.026 s<sup>-2</sup></td></tr><tr><td>&delta;<sup>2</sup> &minus; 4&alpha;</td><td>&minus;0.033 s<sup>-2</sup></td></tr><tr><td>scaling factor &tau;<sub>w</sub></td><td>3.07 s</td></tr><tr><td>scaling factor &tau;<sub>c</sub></td><td>3.34 s</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>Figure 5.1: *T(u)* (red) and *T(z<sub>u</sub>)* (blue) as functions of *u*. The line is dashed when the corresponding saddle points are not admissible (parameters given in Table 1).</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 5.2: Speeds of the vehicles $N = 5$ (left) and $N = 30$ (right) as functions of $t$: intelligent driver model (2.1)–(5.1) (blue), linear model (2.3) (red) and asymptotic approximation (4.8), (4.12) (thick dashed green) (parameters given in Table 1).</figcaption></figure>

Intelligent Driver Model (IDM) [23], where the function $f$ of (2.1) is

$$f(\Delta x, \Delta v, v) = a \left[ 1 - \left( \frac{v}{v_0} \right)^{\delta} - \left( \frac{s^*(v, \Delta v)}{\Delta x - \ell} \right)^2 \right], \quad (5.1)$$
with $s^*(v, \Delta v) = s_0 + vT - \frac{v \cdot \Delta v}{2\sqrt{ab}}$.

The descriptions and values of the parameters are given in Table 1. The equilibrium speed is set to half the desired speed, that is, $v^* = 16.6$ m/s. From there, one can compute the numerical values of the constants defined in the previous sections, as shown in Table 2.

It is interesting to note that we are in a situation where $\delta^2 - 4\alpha < 0$, which means that the poles of $Q$ are complex conjugate. Although no complete classification has been provided for the sign of $T'(u)$ in Section 4.4, plotting $T$ as in Figure 5.1 shows that it is increasing for $u \in ]\Re(d_0), u_c]$. Moreover, there is no extra pole of $T$ at $-\frac{\alpha}{\beta}$, since this point is located to the left of the real part of the poles $\Re(d_0) = -\frac{\delta}{2}$. Hence, the wave has only two phases in terms of saddle point decomposition, contrary to what happened in Section 4.3.

Let us now describe the initial perturbation of the trajectory of vehicle 0. We consider the scenario in which the vehicle decelerates at a rate $b$ (see Table 1) for 1 s and then accelerates at a rate $a$ for a long enough time to come back to the equilibrium speed $v^*$:

$$a_0(t) = -b(H(t) - H(t-1)) + a \left[ H(t-1) - H(t-1 - \frac{b}{a}) \right],$$
where $H$ stands for the Heaviside function. With this definition, $a_0(t) = 0$ for $t > 1 + \frac{b}{a}$, which fits the description of vehicle 0 perturbation described in Section 2.

The plot of the saddle point approximation of $x_N$ or $v_N$ is easily obtained: each point can be parameterized by $u$, first in the $]-\frac{\delta}{2}, u_c[$ interval, and then in the $]u_c, \infty[$ interval. Figure 5.2 shows the graphs of the speeds $v_5(t)$ and $v_{30}(t)$, as respective solutions of

* the IDM differential equation (2.1),

* the linear differential equation (2.3),

* the saddle-point approximations (4.8) and (4.12).

The first remark that can be made about these graphs is that even for $N = 5$ which corresponds to an equilibrium distance of $5\Delta x^* = 139.5$ m, the saddle point approximation is already very good (the dashed green curve often hides the red one). There is an area around $t = \tau_c N$ where the approximation is bad, since $T'(u_c) = 0$, which makes the expression (4.8) invalid and renders the classical saddle-point method inapplicable.

The comparison with the nonlinear IDM equation is less favorable, but this is a known limitation of equation linearization and out of the scope of this paper. However, note that even for $N = 30$, the duration of the wave and the wavelength appear to be adequate.

Although $\tau_c$ has been introduced as a purely mathematical notion related to the saddle point method, one sees on Figure 5.2 that it actually corresponds to a change of curvature of $v_i(t)$, which justify our assertion in Section 4.3 that $[0, \tau_c]$ is the onset of the wave.

Finally, $\tau_w = 3.07$ s, which corresponds for $N = 30$ to a wave beginning at $t = 92.1$ s. This fits well with the plot shown in Figure 5.1. By contrast, the method proposed in [24] leads to an estimate of $t = 53.2$ s, which is not satisfactory.

# 6 Conclusions and perspectives

We show how linearized car-following models lend themselves to asymptotic analysis via the saddle-point method. In the case of a string-unstable model, in which each vehicle amplifies the deviation from equilibrium, we provide a detailed description of the effect at time $t = \tau N$ of a stop-and-go wave on the trajectory of vehicle $N$. In particular, we explicitly obtain the velocity of the wave. Our numerical results confirm the validity of the linear model approximation, even for low values of $N$. This description yields more precise results than the Fourier-Ansatz approach proposed in [24, Section 15.4].
Our approach could also be applied to a car-following model in which the driver has a reaction time $r > 0$, as in [8]. In this case, the transfer function becomes

$$Q(z) = \frac{\beta z + \alpha}{e^{rz} z^2 + \delta z + \alpha},$$

which is a non-algebraic quantity and requires additional technicalities. In particular, the behavior of the function $e^{\tau z}Q(z)$ clearly depends on the sign of $r - \tau$.

Another interesting challenge would be to extend our approach to a heterogeneous setting, where vehicles can be of different types. This might require introducing generalized saddle-point methods that involve some difficult optimization problems.

## A Laplace's estimates and the saddle-point method

We take on the challenge of presenting the main ideas behind these methods on a single page!

Consider integrals of the form

$$\int_{0}^{\infty} g(x) e^{\lambda h(x)} dx$$

where $\lambda \to \infty$, $h(x)$ takes its maximum value (finite) at $x = 0$, with the following assumptions.

* The functions $g$ and $h$ are real, piecewise continuous in the interval $]0, \infty[$.

* The integral $\int_{0}^{\infty} |g(x)|e^{\lambda h(x)}$ is convergent

* In the neighborhood of 0, we have

$$g(x) \sim Ax^p, \quad h(x) \sim a - cx^q + o(x^q),$$

with $p > -1$, $q > 0$, $c > 0$.

Then, the first term of the expansion with respect to $\lambda$, also called the *principal part*, is given by

$$I(\lambda) \sim \frac{A}{q} \Gamma\left(\frac{p+1}{q}\right) e^{a\lambda} (c\lambda)^{-\frac{p+1}{q}}. \quad (A.1)$$

For more details, the reader can consult [7].

Assume now that $h(z)$ and $g(z)$ take complex values and we want to analyze the integral

$$J(\lambda) = \int_{\mathcal{L}} g(z) e^{\lambda h(z)} dz,$$
where $\mathcal{L}$ is contained in an open set of $\mathcal{C}$, in which $h(z)$ is analytic.

By definition, a saddle point $z_0$ satisfies $h'(z_0) = 0$ and $h''(z_0) \neq 0$. The so-called saddle-point method consists in returning to the real case, by examining the lines of steepest descent of the surface $\mathcal{S} = \{\Re(h(z)) = \Re(h(z_0))\}$. Assume that it is possible to properly deform the contour $\mathcal{L}$ into a contour $\mathcal{L}'$, passing through $z_0$ and called the *thalweg line*, on which $\Re(h(z)) \le \Re(h(z_0))$ and $\Im(h(z))$ is constant. It is important to note that such a contour does not always exist.

In a neighborhood of $z_0$, $h(z)$ and $g(z)$ are supposed to satisfy the conditions

$$h(z) \sim h(z_0) + h''(z_0) \frac{(z - z_0)^2}{2} + \mathcal{O}(z - z_0)^3, \quad g(z) \sim g(z_0) + \mathcal{O}(z - z_0),$$

with $h''(z_0) = \rho e^{i\alpha}$. In this case, as $\lambda \to \infty$, we have (see e.g. formula (1.8.1) of [7, IX.1.]),

$$J(\lambda) \sim \sqrt{\frac{2\pi}{\lambda|h''(z_0)|}} e^{i\frac{\pi-\alpha}{2}} g(z_0) e^{\lambda h(z_0)}. \quad (A.2)$$

Although very powerful, this method can sometimes be difficult to implement, particularly with respect to the choice of $\mathcal{L}'$ or because of the existence of several saddle points.

## B About the estimate of $I(t; N)$ for fixed $t$

In this section, to avoid uninteresting technicalities, we take $\phi(z) = 1$ in the integral (2.7). Then, the integrand is a meromorphic function which, under the key assumption (2.10), has two poles of order $N$ corresponding to the roots of $D(z)$. So, why not use Cauchy's residue theorem directly? Simply, because it is partly impossible to manipulate, as will be seen below!

Throughout this section, we assume that (2.11) and (2.12) are in force. By means of Cauchy's theorem, we can rewrite the integral (2.7) in the form

$$I(t; N) = I_1(t; N) + I_2(t; N), \quad (B.1)$$

with

$$I_1(t; N) \stackrel{\text{def}}{=} \frac{1}{2i\pi} \int_{\mathcal{C}} e^{tz} Q^N(z) dz, \quad I_2(t; N) \stackrel{\text{def}}{=} \frac{1}{2i\pi} \int_{u-i\infty}^{u+i\infty} e^{tz} Q^N(z) dz,$$

where $u$ is a negative real number satisfying $d_1 < u < z_1$ and the integral $I_1(t; N)$ is taken counterclockwise along the circle $\mathcal{C}$ of diameter $[z_1, z_0]$. More precisely, the equality (B.1) follows from (2.14) and the fact that, in the negative half-plane,

$$e^{tz} Q^N(z) = \mathcal{O}(e^{t\Re(z)}|z|^{-N}), \quad \forall |z| \ge A \to \infty.$$
<figure><img src="image_8.png" /><figcaption>Figure B.1: The integration contours.</figcaption></figure>

**Proposition B.1.** In the complex plane, the following properties hold.

1. If $z = x + iy$ belongs to the following family of cubic plane curves

$$y[\beta(x^2 + y^2) + \alpha(2x + \gamma)] + KG(x, y) = 0, \quad (B.2)$$

where $K$ is an arbitrary finite real constant and

$$G(x, y) \stackrel{\text{def}}{=} (\beta x + \beta\delta - \alpha)y^2 + (\beta x + \alpha)D(x),$$

then $\Im(\log Q(z)) = K$.

2. The curve obtained for $K = 0$ is decomposed into the real axis $y = 0$ and a circle $C$ of radius $R$ shown in Figure B.1, given by the equation

$$\left(x + \frac{\alpha}{\beta}\right)^2 + y^2 = R^2, \quad \text{where} \quad R^2 = \frac{\alpha(\alpha - \beta\gamma)}{\beta^2}, \quad (B.3)$$

admitting the real segment $[z_1, z_0]$ as its diameter, see Figure B.1.

3. For $K \neq 0$, all the curves go through the zeros of $D(z)$, including $G(x, y) = 0$, which is the curve corresponding to $K = \infty$.

*Proof.* From (2.4), we have

$$\log Q(z) = \log(\alpha + \beta z) - \log D(z),$$

as well as

$$\left\{ \begin{array}{l} \arg(\alpha + \beta z) = \arctan\left(\frac{\beta y}{\alpha + \beta x}\right), \\ \\ \arg D(z) = \arctan\left(\frac{y(\delta + 2x)}{x^2 - y^2 + \delta x + \alpha}\right). \end{array} \right.$$
By using the elementary trigonometric formula

$$ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}, $$

and taking for the function arg the principal determination of the argument, we infer, after some direct algebra, that the equality

$$ \arg(\alpha + \beta z) = \arg D(z) + \ell, $$

where $0 \le \ell \le 2\pi$ is some constant, implies that $z$ belongs to the cubic curve defined in (B.2) together with $\Im(Q(z)) = \tan(\ell) \stackrel{\text{def}}{=} K$.

The rest of the proof is straightforward and details are omitted. ■

**Estimate of $I_1(t; N)$, for $t \ge 0$**

In polar coordinates, the equation of $C$ is given by

$$ \begin{cases} x = R \cos \theta - \frac{\alpha}{\beta}, \\ y = R \sin \theta. \end{cases} $$

Then, setting $p = z + \frac{\alpha}{\beta}$, we can write, for $z \in C$,

$$ Q(z) = \frac{\beta p}{p^2 + p\left(\delta - \frac{2\alpha}{\beta}\right) + R^2} = \frac{\beta}{2R \cos \theta + \delta - \frac{2\alpha}{\beta}}, $$

after using (B.3) and the identities $p\bar{p} = R^2$ and $p + \bar{p} = 2 \cos \theta$.

For $z \in C$, it will be convenient to note $\tilde{Q}(\theta) \stackrel{\text{def}}{=} Q(z)$. Then, by (2.12), (2.13) and (B.3), we have

$$ \beta\delta > \frac{\delta^2}{2} > 2\alpha, $$

together with

$$ \left(\delta - \frac{2\alpha}{\beta}\right)^2 - 4R^2 = \delta^2 - 4\alpha > 0, $$

which shows that $\tilde{Q}(\theta)$ is always finite and positive. Consequently, for all $z \in C$,

$$ \log Q(z) = \log|Q(z)| = \log \tilde{Q}(\theta) $$

is a real quantity, and we are in a position to estimate $I_1(t; N)$ by means of the classical Laplace method, which was our primary objective. It is immediate to check that $\tilde{Q}(\theta)$ has a unique maximum value

$$ \tilde{Q}(\pi) = \max_{0 \le \theta \le 2\pi} \tilde{Q}(\theta) = \frac{\beta}{\delta - 2R - \frac{2\alpha}{\beta}}. \qquad (B.4) $$
In addition, we have

$$ \tilde{Q}(\pi) > \tilde{Q}(0) > 1, $$

since inequality

$$ \beta > \delta - 2R - \frac{2\alpha}{\beta} $$

holds as soon as $\alpha > \beta\gamma$. Hence, in the neighborhood of $\theta = \pi$, since $Q'(z_1) = 0$, we can write the following Taylor's expansion

$$ \log \tilde{Q}(\theta) = \log \tilde{Q}(\pi) + m(\theta - \pi)^2 + \mathcal{O}((\pi - \theta)^3), $$

where

$$ m = \frac{-R \tilde{Q}^2(\pi)}{\beta} < 0. \qquad (B.5) $$

After an immediate change of variable, $I_1(t; N)$ takes the form

$$ \begin{aligned} I_1(t; N) &= \frac{R}{2\pi} \int_0^{2\pi} e^{t(Re^{i\theta} - \alpha/\beta) + i\theta} e^{N \log \tilde{Q}(\theta)} d\theta \\ &= \frac{R}{2\pi} \int_0^{2\pi} e^{t(R \cos \theta - \alpha/\beta)} \cos(Rt \sin \theta + \theta) e^{N \log \tilde{Q}(\theta)} d\theta. \end{aligned} \qquad (B.6) $$

On the circle $C$, the function $\tilde{Q}(\theta)$ does reach its maximum. Hence, by using the classical expansion derived from Laplace's method (see e.g. [14]), after having separated the real and imaginary parts in (B.6), we get, with the notation of equations (B.4), (B.5), and noting that $R + \alpha/\beta = -z_1$,

$$ I_1(t; N) = Re^{tz_1} \frac{\tilde{Q}^N(\pi)}{\sqrt{2\pi N}} \left( \frac{-1}{\sqrt{-m}} + \mathcal{O}\left(\frac{1}{N}\right) \right), \qquad (B.7) $$

which, by the way, is a negative quantity tending to $-\infty$, as $N \to \infty$.

Then, on the basis of the analysis carried out in Section 3, we get

$$ I_2(t; N) = -I_1(t; N) + o(1), $$

noting, by Cauchy's theorem, that $I_2(t; N)$ is exactly equal to the residue of $I(t; N)$ around the pole $d_1$ in the left half-plane.

Unfortunately, as the interested reader might verify, contrary to what was done in Sections 3 and 4, it is impossible here to find a convenient contour allowing to estimate integral $I_2(t; N)$ via the saddle-point method.
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