Competition Between Regulation-Providing and Fixed-Power Charging Stations for Electric Vehicles Wenjing Shuai, Patrick Maillé, Alexander Pelov Institut Mines-Telecom/Telecom Bretagne Abstract-This paper models a non-cooperative game between two EV charging stations. One is a fixed-power charging station purchasing electricity from the grid at wholesale price and reselling the energy to EV owners at a higher retail price; the other is regulation-providing and varies the recharging power level of its clients to provide regulation services to the grid, so its profit comes from both EV owners (who buy energy) and the grid (which pays for regulation services). Users are reluctant to charging power variations and prefer shorter overall charging times, hence regulation-providing charging has to be cheaper than fixed-power charging.

We analyze the competition among those charging providers, and examine the performance at the equilibrium in terms of user welfare, station revenue and electricity prices. As expected, competing stations provide users with lower charging prices than when both charging solutions are offered by a monopolistic provider. Moreover, while competition benefits users, it also benefits the grid in that the amount of regulation services increases significantly with respect to the monopolistic case.

I. INTRODUCTION

Among the main difficulties of the penetration of Electric Vehicles (EVs) in the smart city is the associated energy equation: how can the power grid accommodate the corresponding demand? [START_REF] Trigg | Global EV outlook-understanding the electric vehicle landscape to 2020[END_REF]. And together with the technical limitations, the question of economic incentives to elicit the most efficient use of resources needs also to be considered (see [START_REF] Shuai | Charging electric vehicles in the smart city: A survey of economy-driven approaches[END_REF] and references therein). But EVs do not use the energy in real time, they just store it in their batteries until leaving the charging station. This particularity of EV charging demand can be leveraged, in particular for regulation purposes. In practice, when there is extra (resp. a lack of) energy production with respect to demand, the grid can send a "down" (resp., "up") regulation signal so that the production side-or here, the consumption side-reacts accordingly. In this paper, we consider the economic aspects of such an option, from the point of view of EV owners and charging stations.

Previous work focuses on fairness issues among users in terms of final state-of-charge [START_REF] Cover | Elements of information theory[END_REF]; on incentivizing EV owners to contribute to regulation [START_REF] Wu | Vehicle-to-aggregator interaction game[END_REF], [START_REF]PEV-based reactive power compensation for wind DG units: A Stackelberg game approach[END_REF]; or on the resulting user welfare [START_REF] Sun | Real-time welfare-maximizing regulation allocation in aggregator-EVs systems[END_REF], [START_REF]Real-time welfare-maximizing regulation allocation in dynamic aggregator-EVs system[END_REF]. The closest works to ours are [START_REF] Gao | A contract-based approach for ancillary services in V2G networks: Optimality and learning[END_REF], [START_REF] Shuai | Incentivizing electric vehicles to provide regulation while recharging[END_REF], where the focus is on the pricing strategies of the charging stations: in [START_REF] Gao | A contract-based approach for ancillary services in V2G networks: Optimality and learning[END_REF], Gao et al. consider a regulator designing contracts to incentivize EVs to participate so that the station profit

The three authors are with the Networks, Security, Multimedia Department, Institut Mines-Telecom/Telecom Bretagne, 2 rue de la Chataigneraie CS 17607, 35576 Cesson-Sévigné, FRANCE e-mail: {first}.{last}@telecombretagne.eu is maximized. On the other hand, in [START_REF] Shuai | Incentivizing electric vehicles to provide regulation while recharging[END_REF] we considered a charging station offering two simple options, namely a fixedpower charging (no regulation) and a varying-power charging (following regulation signals). Another difference is that [START_REF] Gao | A contract-based approach for ancillary services in V2G networks: Optimality and learning[END_REF] allows vehicle-to-grid energy exchanges while in [START_REF] Shuai | Incentivizing electric vehicles to provide regulation while recharging[END_REF] the regulation services are just provided by varying the current charging power. But both of those works, as well as [START_REF] Sortomme | Optimal charging strategies for unidirectional Vehicle-to-Grid[END_REF], assume a monopolistic revenue-maximizing charging station.

In this paper, we focus on the effect of competition, by considering two competing charging stations, one implementing only regulation-based charging and the other only fixedpower charging. When compared to monopolistic situations, we expect competition to benefit to users, through lower recharging prices. But also, we investigate the viability of each competitor: indeed, regulation is rewarded through financial incentives, and providing regulation during charging may not yield sufficient revenues if those incentives are not large enough. Hence some regions of reward values where EVcharging-based regulation can occur; this was investigated in the monopolistic case in [START_REF] Shuai | Incentivizing electric vehicles to provide regulation while recharging[END_REF], here we study the effect of competition on that aspect as well.

Our results indicate that, as expected, competition is beneficial to users, through lower recharging prices. Also, competition appears to be better from the grid perspective, since both the region of rewards for which regulation is viable and the amount of regulation offered are larger in the competition setting. The remainder of the paper is organized as follows.

Section II presents our model; Section III analyzes the price competition and the resulting Nash equilibrium; In Section IV we compare the performance of the competition with the monopolistic case and Section V concludes the paper.

II. MODEL DESCRIPTION

According to a national household travel survey of the United States [START_REF] Krumm | How people use their vehicles: Statistics from the 2009 national household travel survey[END_REF], [START_REF] Santos | Summary of travel trends: 2009 national household travel survey[END_REF], a passenger vehicle spends on average 75 minutes a day on journey, hence is parked most of the time. We assume this to remain true for EVs. So this is interesting, at least for some EV drivers, to accept longer charging durations for cheaper energy. This opens an opportunity for charging stations to increase revenue through the rewards offered to regulation-contributing entities, as well as for EV owners to save on their energy bill.

Conventional recharging services are provided by what we will call an S-charging station, purchasing electricity at the low wholesale unit price te/kWh and reselling it to EV owners at a higher price; whereas in a R-charging, charging power is not guaranteed but subject to variations over time, as a response to regulation requests issued by grid operators. We model the interactions among both stations (or sets of stations, each set controlled by a separate entity) as a noncooperative game since they compete over prices to attract EV owners. User preferences between price and charging power variations are assumed heterogeneous, so each station seeks the best tradeoff between market shares and per-client profit in order to maximize its expected revenue.

A. Regulation mechanism

Frequency regulation, depending on the response time, is mainly divided into: primary, secondary, and tertiary control, with the response time increasing from seconds, to minutes and finally to half an hour respectively [START_REF]Balancing and frequency control[END_REF]. In our proposal, the R-charging station modulates the EV charging power to provide the secondary control: one regulation time slot lasts for ∆ hours, with ∆ typically between 0.1 (6 minutes) and 0.25 (15 minutes). Periodically, the grid operator, buyer of the regulation service, sends a regulation request to the R-charging station specifying its demand, which can be regulation-up,down or -null. Receiving the signal, the R-charging station sets the EV recharging power to be 0 kW 1 , P d kW, or P n kW respectively: P d is the maximum acceptable power level allowed by the EV supply equipment in the station, and P n is the default recharging power (0 ≤ P n ≤ P d ) defined by the Rcharging station itself, when no regulation is needed, namely regulation null. Note that this mechanism increases (decreases) the EV consumption responding to regulation-down (-up). This counter-intuitive naming stems from conventional regulation services, where providers are generation units whereas the task is given to consumers here. For later convenience we will use the notation x := Pn P d , so that x ∈ [0, 1]. At the S-charging station, EVs are always charged at full speed P d kW. Figure 1 illustrates the charging power profiles for the two stations as well as the energy accumulated in an EV battery being charged for a given scenario of regulation requests. We denote by C B the energy demand of an EV, and by ρ u (ρ d ) the probability of occurrence of regulationup (-down) at each time slot, those signals being assumed independent at each regulation period in this paper.

There may be concerns that varying the charging power for all EVs in R-charging station(s) simultaneously and drastically following this "ping-pong" policy can lead to an oversupply of regulation, i.e., the aggregated increase or decrease in power is larger than that actually needed by the grid operator. This is hardly possible since in the scale of a grid operator, the disposable regulation capacity scattered in EVs is nondominant if not negligible given the current penetration levels. For example data from RTE (Réseau de transport d'electricité), the biggest independent system operator in France, show that the regulation-down demand in 30 minutes 2 can easily go over 1 We do not allow here EVs to deliver energy to the grid (the so-called vehicle-to-grid transfer). [START_REF] Sae | Sae charging configurations and ratings terminology[END_REF]) at the same time. Since the whole country has an EV population of 30 thousand, sharing 8600 public charging facilities, the regulation oversupply problem is not of concern so far. But it can rapidly become one if EV penetration increases; nevertheless we expect that in this case, the incentives to provide regulation will be adjusted (regulation being rewarded less) so that market mechanisms will reduce supply. In addition, demand for regulation is likely to increase in the next future, with the development of renewable energy production which cannot be controlled like fossil-based electricity plants can: the overall supply-demand balance will be more difficult to maintain, hence a probable larger need for ancillary services such as regulation.

B. Regulation incentives

In return for providing regulation, the R-charging station receives monetary incentives, with respect to the default wholesale price t.

• In regulation-null periods, it charges each plugged EV with power P n kW, and pays ∆tP n monetary units (no compensation) over such a duration-∆; • in regulation-up periods, the grid operator "re-buys" the energy saved at a unit price r u t, hence the station pays ∆t(1 -r u )P n monetary units over such a period (note that we can expect to have r u ≥ 1, although it is not always the case in practice); • similarly, in a regulation-down, the R-charging station pays for the extra energy it consumes at a discount price t(1 -r d ) monetary units per kWh, thus a total price paid ∆ P n t + (P d -P n )t(1 -r d ) monetary units per EV. Together with the probabilities of regulation-up (ρ u ) and down (ρ d ), the expected net revenue (possibly negative) over one regulation slot is:

E∆ = t∆(ρuruPn -ρ d (1 -r d )(P d -Pn) -Pn) (1)

C. User preferences

We assume that each EV owner needs C B kWh of energy, say, per day, the owner can choose to charge at the constant power P d in the S-charging station, or to charge at a variable power in the R-charging station. They can also choose neither solution (a no charging choice) if they consider both too expensive. Naturally, users are assumed to:

• prefer to recharge faster, i.e., at higher power rate;

• be reluctant to uncertainty in the recharging power caused by regulations. Additionally, batteries can be sensitive to power variations in the recharging process, another reason for EV owners to be reluctant to contributing to regulation. Following these criteria, we define the user utility (willingness-to-pay minus price paid) for a recharging option as being of the form

V = θ( P -γδ(P )) -T CB
where P is the expected charging power, and δ(P ) its standard deviation. θ is user-specific: we assume it to be exponentially distributed, with mean θ, over the EV owner population. The parameter γ is the reluctance toward power fluctuations, and is assumed the same for all users. Finally, T represents the unit energy price set by the charging station chosen by the user. (We take T = 0 for users who choose no charging).

Let us define P A as the value of P -γδ(P ) for the Rcharging option, which can easily be expressed from P d , P n , ρ d and ρ u . The parameter (γ) we choose always guarantees P A ≥ 0 which means that this proposal does not target users with a too high sensitivity to power fluctuation. The probability α r (resp., α s ) that a user chooses the R-charging (resp., Scharging) station can then be expressed as

αr =      1 -exp(-C B (Ts-Tr ) θ(P d -P A ) ) if Tr < 0 exp(-C B Tr θP A ) -exp(-C B (Ts-Tr ) θ(P d -P A ) ) if 0 ≤ Tr ≤ P A P d Ts 0 otherwise αs = exp(-C B Ts θP d ) if Ts ≤ P d P A Tr exp(-C B (Ts-Tr ) θ(P d -P A ) ) otherwise. (2)
(3)

Note that we allow negative charging prices with the Rcharging station: indeed, since that station can make money from the grid thanks to EV owners, the corresponding rewards could be so large that the station would be willing to attract a large number of EVs, even by paying them. This case is for completeness of the model, we think it is not very likely to occur but we cover it in this paper.

Following the classical backward induction method, we first assume P n (or equivalently x) fixed and analyze the pricing game (defined bellow). The outcome is dependent on x so the R-charging station can maximize its profit through altering its value. We examine the first part analytically while the second numerically due to complexity. Definition 1. The pricing game between the S-charging station and the R-charging station as a collection: N , T , (R i ) , where the player set N consists of the two stations, the price profile T is a vector (T s , T r ) on the semi-plane R ≥0 × R, and the payoff function R i : T → R gives each station's expected revenue obtained from one EV.

III. ANALYSIS OF THE GAME

In this section, we analyze the non-cooperative strategic game defined in 1. We derive their respective best-response prices, to characterize the Nash equilibria.

A. Best-response prices 1) S-charging station revenue and best-response price T br s : For the S-charging station owner, its average income R s depends on the market share α s , and the unit price T s it offers:

Rs = CB(Ts -t)αs = CB(Ts -t) exp(-C B Ts θP d ) Ts ≤ P d P A Tr CB(Ts -t) exp(-C B (Ts-Tr ) θ(P d -P A ) ) Ts > P d P A Tr. (4)
Depending on its opponent's strategy T r , the price T s that maximizes R s provides the best-response price. Proposition 1. The S-charging station has a unique bestresponse price as follows:

T br s (Tr) =                    t + (P d -P A ) θ C B if Tr < (t + (P d -P A ) θ C B ) P A P d t + P d θ C B if Tr > (t + P d θ C B ) P A P d Tr P d P A otherwise (5a) (5b) (5c)
Proof: The proof is provided in the Appendix of the full paper version [START_REF] Shuai | Competition between regulationproviding and fixed-power charging stations for electric vehicles[END_REF].

Figure 2 illustrates the S-charging station revenue as a function of T r , and the best-response T br s (T r ). 2) R-charging station revenue and best-response price T br r (T s ): Let us now consider the R-charging station owner, having to decide its price T r .

To estimate how much net renumeration the R-charging station gets from recharging EVs through regulation, we multiply the regulation revenue per slot, i.e. E ∆ in (1), by the average number of slots a regulating EV remains pluggedin before its battery is fully recharged, i.e. C B /(∆ P ). To facilitate the writing we further divide the product, which has a unit of e, by the EV energy demand (C B kWh), so that its final unit is e/kWh and has a form of The average R-charging station revenue consists of renumeration from providing regulation and income from charging EVs:

Er := t(ρurux -ρ d (1 -r d )(1 -x) -x) P d P .
Rr =        C B (Tr + Er)[1 -exp(- C B (Ts-Tr ) θ(P d -P A ) )]
Tr < 0

C B (Tr + Er)[exp(-C B Tr θP A ) -exp(- C B (Ts-Tr ) θ(P d -P A ) )] 0 ≤ Tr ≤ P A P d Ts 0 otherwise (6)
The following result summarizes the optimal R-charging station reaction to its competitor. Proposition 2. The R-charging station has a unique bestresponse price as follows:

T br r (Ts) =            Ts P A P d if Ts ≤ -Er P d P A 0 if Ts ∈ {Ts : E r,1 (Ts) ≤ Er ≤ E r,2 (Ts)} {Tr ∈ R : ∂Rr ∂Tr = 0} ⊂ min{0, -Er}, max{0, min{ θP A C B -Er, P A P d Ts}} otherwise (7) where E r,1 (Ts) = θ(P d -P A ) 1 -exp(-C B Ts θ(P d -P A ) ) C B P d P A -1 + exp( C B Ts θ(P d -P A ) ) E r,2 (Ts) = E r,1 (Ts) 1 + ( P d P A -1) exp( C B Ts θ(P d -P A )
) .

Proof: The proof is provided in the Appendix of the full paper version [START_REF] Shuai | Competition between regulationproviding and fixed-power charging stations for electric vehicles[END_REF].

Figure 3 shows the R-charging station revenue as well as the best-response price T br r (T s ), as a function of T s .

B. Nash equilibrium

Proposition 3. The pricing game defined in 1 has either a unique Nash equilibrium or a unique Pareto-dominant one when there exist infinite number of Nash equilibria. The equilibria prices N E in different circumstances are:

                                                                 Tr = -Er; Ts = - P d P A Er if Er ≤ - P A P d [t + (P d -P A ) θ C B Tr ∈ (0, min{ θP A C B -Er, P A P d Ts)}; Ts = t + (P d -P A ) θ C B if - P A P d [t + (P d -P A ) θ C B ] < Er < E r,1 t + (P d -P A ) θ C B Tr = 0; Ts = t + (P d -P A ) θ C B if E r,1 t + (P d -P A ) θ C B ≤ Er ≤ E r,2 t + (P d -P A ) θ C B
Tr ∈ (-Er, 0);

Ts = t + (P d -P A ) θ C B if E r,2 t + (P d -P A ) θ C B < Er (8a) (8b) (8c) (8d)
Proof: The proof is provided in the Appendix of the full paper version [START_REF] Shuai | Competition between regulationproviding and fixed-power charging stations for electric vehicles[END_REF].

Note that N E (8a) which occurs when E r ≤ -

P A P d [t + (P d - P A ) θ C B ]
is not profitable for the R-charging stations since zero revenue is obtained, and that the condition for a positive Rcharging station revenue is -

P A P d [t + (P d -P A ) θ C B ] < E r .
We will refer to this condition in Section IV-B.

Figure 4 illustrates best-response prices and resulting Nash equilibria in four different circumstances. Figure on the left hand side shows that when regulation remuneration is not significant (eg. r u = 1 and r d = 0 in the figure), it is likely to encounter equilibrium N E (8a) or N E (8b), due to small regulation profit E r . On the right hand side, equilibrium N E (8c) or N E (8d) happens when regulation revenue is very attractive. Note that we choose very high regulation prices just to depict the case where free recharging or even negative price recharging is offered. To our knowledge, such high prices rarely exist on the market.

1) Optimization of P n : The pricing game defined in 1 is played given a fixed P n , set by the R-charging station, who Fig. 4: Four Nash equilibria in different cases can afterwards modify its value to pursue a higher equilibrium revenue. Due to the complexity of the equilibrium price profile we resort to numerical search for the optimal P n .

IV. COMPARISON BETWEEN THE NASH EQUILIBRIUM AND THE MONOPOLISTIC CASE

In this section we compare the competition model with the monopolistic case where a single manager sets both the Scharging price as well as the R-charging price to maximize its overall revenue. We do not repeat the results in [START_REF] Shuai | Incentivizing electric vehicles to provide regulation while recharging[END_REF] for the monopolistic case due to space limit, but simply compare their performances under the same parameters.

A. User welfare

User welfare is the average user utility over the distribution of user preference parameter, i.e. θ. The following formula works for both the monopolistic case and the Nash equilibrium. The first column in Figure 5 shows a significant increase of user welfare (U M for monopoly and U E for equilibrium) after breaking a monopolistic station into two competing ones. Although the total station revenue decreases, the social welfare which is the user utility plus station revenue has a net increase of over 20%. The second column illustrates an increase of EVs being served, thanks to a decrease of energy prices depicted in the third column.

B. Application in a real world market

Figure 6 compares the regions for rewards {r d , r u } where offering R-charging is profitable. At equilibria (second row), the black zones where R-charging is not preferred is remarkably smaller than those in the monopolistic case. This is because that in a monopoly, feasible region for rewards {r d , r u } is composed of those that make the following equation of x solvable in the interval of [0, 1] [9]:

ρurux -ρ d (1 -r d )(1 -x) -x -P (x)PA(x)P -2 d > 0
Whereas in competition, (8a) and (8b) give the condition of:

t(ρurux -ρ d (1 -r d )(1 -x) -x -P PAP -2 d ) + 1 t P PAP -2 d [P d -PA] θ CB > 0
Comparing these two we find that the competition enlarges the viable region of {r d , r u }. The blue and red areas in Figure 6 are referring to the optimal default recharging power P n in these regions, i.e. the optimal x after exhaustive search. In most combinations of {r d , r u }, this optimal x is either 0 or 1, except for a few {r d , r u } observed in the gap between the blue region and the red, in the figures on the third column where average user preference on power is smaller: θ = 0.1 and user sensitivity to variation is greater: γ = 0.5. We also plot the actual {r d , r u } offered by a French operator RTE on these figures. The blue circles correspond to the 48 {r d , r u } pairs on the day of 20/07/2015 and the red rectangles are showing the daily averages during the week from 20/07/2015 to 26/07/2015.

V. CONCLUSION AND PERSPECTIVES

This paper considers a competition between two selfinterested charging stations. At the Nash equilibrium of this non-cooperative game, both stations tends to offered lower prices to EV owners than a monopolistic controller would do, thus more clients are attracted and greater regulation services is provided to the grid operator. This work can be extended in several ways including: bring in more actors such as charging stations with private renewable energy sources; considering the actor of a "Grid" who can play with the wholesale electricity price imposed on both R-charging and S-charging station; or differentiate two charging stations by their locations, which effect users' preferences among them.

Fig. 2 :

 2 Fig. 2: S-charging station revenue as a function of Tr and Ts (t = 0.03, θ = 0.3, C B = 50kW h, x = 0.8). The red, yellow and blue areas are separated by Tr = (t + (P d -P A ) θ C B ) P A P d and Tr = (t + P d θ C B ) P A P d ,

TsFig. 3 :

 3 Fig. 3: R-charging Station revenue as a function of Tr and Ts (t = 0.03, r d = 0.7, ru = 2.1, C B = 50kW h, ρ d = 0.48, ρu = 0.48, γ = 0.05, θ = 0.3, x = 0.8). Red region corresponds to non-negative revenue, i.e. Tr P A < Ts P d

4 • 10 -2 6 • 10 -2 8 • 10

 10610810 Tr (r d = 0.8; ru = 5) Ts (r d = 0.8; ru = 5) Tr (r d = 0.99; ru = 20) Ts (r d = 0.99; ru = 20)

  =αr θPA + αs θP d

Fig. 5 :Fig. 6 :

 56 Fig. 5: Comparison between Monopoly (first row) and Nash equilibrium (second row), with t = 0.03, θ = 0.3, C B = 50, ρ d = ρ u = 0.48, γ = 0.05, r d = 0.4, r u = 1.6 (r d and r u are the daily average of 20/07/2015). θ = 0.3, γ = 0.05 θ = 0.1, γ = 0.05 θ = 0.1, γ = 0.5

  Table I summarizes the notations used in our model.

TABLE I :

 I Model notations + ρ d (P d -P ) 2 + (1 -ρu -ρ d )(Pn -P ) 2 ) P A (x), or P

	t	unit price of energy paid by stations (unit: e/kWh)
	ru	remuneration ratio for regulation-up (no unit)
	r d	discount ratio for regulation-down (no unit)
	ρu (resp. ρ d )	probability of an "up" (resp. "down") regulation signal
	C B	average energy recharged per EV per day
	θ	user sensitivity to recharging power (including variabil-
		ity)
	θ	average value of θ among users
	γ	user reluctance to power variation
	Pn (resp. P d )	default (resp. "regulation-down") recharging power
	x P	Pn P d ρ d P d + (1 -ρu -ρ d )Pn
	δ(P )	ρu P 2

A P -γδ(P ) ( > 0 )

ACKNOWLEDGEMENT

This work has been partially funded by the Fondation Telecom through the Futur&Ruptures program and by the French FUI program through the GreenFeed project.


Keywords: Eigenvalue perturbation, finite element method, quantum chaos, reverberation chamber

Wave chaos theory is used to study a modeled reverberation chamber (RC). The first 200 modes of an RC at a given stirrer position are determined by the finite element method, and the Weyl formula is checked for various RC geometries, from integrable to chaotic. The eigenfrequency spacing distribution varies according to the degree of ray chaos in the RC related to its geometry. The eigenmode distributions are also analyzed and compared to the theoretical Gaussian distribution: close to the lower useable frequency, the modes of the studied chaotic RC fairly respect this asymptotic property. A general result of chaotic systems is illustrated: when perturbed by the stirrer rotation, the resonant frequencies of an RC avoid crossing. This means that the frequency sweeps tend to vanish at high frequency.

I. INTRODUCTION n the last decade, reverberation chambers (RCs) have been increasingly used for electromagnetic compatibility tests. The behavior of an electrically large RC can be easily explained by the ray propagation approach: on the basis that the electric field is a superposition of an infinite number of uncorrelated and isotropic plane waves reflected by the walls, the field is isotropic and uniform [START_REF] Hill | Plane wave integral representation for fields in reverberation chambers[END_REF].

Using this geometric optic point of view, Cappetta [START_REF] Cappetta | Electromagnetic chaos in mode-stirred reverberation enclosures[END_REF] has illustrated the chaotic behavior of an RC with a moving wall. In this paper, the chaotic behavior of a cavity equipped with a metallic stirrer will be discussed making use of quantum chaos theory.

Quantum chaos (or wave chaos) theory is nowadays frequently validated by microwave experiments in metallic cavities: the analogy between the Schrödinger and Helmholtz This work was partially supported by the Brazilian agency CNPq. G. Orjubin is with Laboratório de Telecomunicações e Ciência e Engenharia dos Materiais (LOCEM), Universidade Federal do Ceará, Fortaleza, 60455-760, Ceará, Brazil (e-mail: orjubin@univ-mlv.fr). E. Richalot and O. Picon are with Equipe Systèmes de Communications et Microsystèmes (ESYCOM) Laboratory, Université de Marne-la-Vallée, 77454 Champs sur Marne, France (e-mail: elodie.richalot@univ-mlv.fr; odile.picon@univ-mlv.fr). O. Legrand is with Laboratoire de Physique de la Matière Condensée, Université de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice cedex 2 France (email: olivier.legrand@unice.fr) equations is total in the case of a flat metallic cavity [START_REF] Stöckmann | Quantum chaos in billiards studied by microwave absorption[END_REF][START_REF] Sridhar | Experimental observation of scarred eigenfunctions of chaotic microwaves cavities[END_REF][START_REF] Haake | Manifestation of wave chaos in pseudointegrable microwave resonators[END_REF], and the results [START_REF] Deus | Statistical properties of the eigenfrequency distribution of the three-dimensional microwave cavities[END_REF][START_REF] Dörr | Scarred and chaotic field distributions in a three-dimensional Sinaimicrowave resonator[END_REF] prove that 3-D electromagnetic problems are compatible with the universal laws of quantum chaos theory. A law is said to be universal if it depends only on the system symmetry, but not on the geometrical details.

This paper presents an application of the quantum chaos theory to a modeled RC with the objective to study the pertinence of the universal laws for this particular 3-D electromagnetic problem. Another particularity lies in the restricted number of studied modes, whereas the theory applies to a large number of modes.

The RC chaoticity can be assessed from the statistics of the eigenfrequencies that are numerically determined by the finite element method (FEM). The considerable computational effort, which had been pointed out by Bunting in [START_REF] Bunting | A two-dimensional finite-element analysis of reverberation chambers[END_REF], can now be overcome [START_REF] Orjubin | On the FEM modal approach for a reverberation chamber analysing[END_REF] due to progresses in solver algorithms and hardware. It is worth noting that this work only concerns the modes, with neither consideration about the total electric field nor the quality factor, and that the main part of the paper is devoted to the RC analysis for a given stirrer position.

The outline of this paper is the following. First, we present the rationale for the use of quantum chaos theory [START_REF] Stöckmann | Quantum Chaos. An Introduction[END_REF] and we explain how the frequency distribution can be used to assess the RC chaoticity. This technique is then applied to three different kinds of RC, from integrable to chaotic ones. As well the eigenfield distributions are compared to the theoretical predictions in the integrable and chaotic cases.

In the last section, an important property of chaotic systems is presented, relative to the perturbation of resonant frequencies by the stirrer displacement. This study is conducted using a specific mode tracking procedure (MTP) [START_REF] Orjubin | Mode Perturbation Induced by the Stirrer Rotation in a Reverberating Chamber[END_REF].

II. THEORETICAL BACKGROUND

A. Quantum Mechanics and Microwave Studies

The analogy between Schrödinger and Helmholtz equations justifies the use of quantum mechanics theory to analyze the electromagnetic field inside a flat cavity. To show it, we recall the time-independent Schrödinger equation of a quantum

0 2 2 2 = + ∇ ψ ψ h E m , ( 1
)
where h is the Planck constant. When the particle is inside a box, the boundary conditions are ψ = 0 on the frontiers. Quantum theory is validated from numerical simulations or experiments. In recent years, microwave analog experiments have become a well-established alternative to theoretical studies. The first 2D experiments were made with a flat cavity parallel to the plane Oxy for which the Helmholtz equation for the component

E z is 0 2 2 2 =       + ∇ z z E E λ π , ( 2
)
where λ is the wavelength of the harmonic wave. Equations (1-2) reveal the one-to-one correspondence between the two applications.

The validation technique consists in finding the resonant frequencies of a cavity from the transmission coefficient between two antennas [3-5][12]. Then the eigenfrequency distributions are analyzed with regard to the universal laws predicted by quantum chaos theory.

It is important to note that the extension of quantum techniques to 3-D electromagnetic cavities is not obvious, since the Helmholtz equation is vectorial, and is not equivalent to the Schrödinger equation. However, recent studies [START_REF] Deus | Statistical properties of the eigenfrequency distribution of the three-dimensional microwave cavities[END_REF][START_REF] Dörr | Scarred and chaotic field distributions in a three-dimensional Sinaimicrowave resonator[END_REF] prove that the universal laws are still valid for 3D electromagnetic cavities.

B. Semiclassical Mechanics and Geometric Optics

The properties of quantum systems that will be considered in Sections II-E and F are related to the chaotic dynamics of a particle free to move inside a box, in the classical limit where the de Broglie's wavelength is vanishing small compared to the typical size of the box. In the electromagnetic context, the analogue limit is the geometrical imit of rays that are reflected by the metallic walls of the cavity. The reader is referred to [START_REF] Galdi | Wave propagation in ray-chaotic enclosures: Paradigms, oddities and examples[END_REF] for a review of results on time-harmonic wave propagation in a ray chaotic enclosure. It is explained that a system with separable geometry (such as a rectangular cavity) is said to be integrable and has a regular behavior, to be opposed to chaotic systems.

On the basis of the close connection existing between ray and modal spectral properties, this paper will show how to use the modes of an RC to determine its degree of chaos.

C. Eigenfield Distribution in a Chaotic RC

As indicated in [START_REF] Galdi | Wave propagation in ray-chaotic enclosures: Paradigms, oddities and examples[END_REF], the scalar eigenfunction ψ of a quantum system can be written as a superposition of plane waves (rays) with fixed wave vector amplitude. If the system is integrable, the number of these waves is limited (e.g. 8 plane waves for a 3-D rectangular cavity). For a chaotic system, the number is large and the propagation directions are uniformly distributed: this is the random plane wave omdel introduced by Berry [START_REF] Berry | Regular and irregular semiclassical wavefunctions[END_REF] for quatum mechanics and by hill [START_REF] Hill | Plane wave integral representation for fields in reverberation chambers[END_REF] in the RC context. Applying the central limit theorem to this conjecture, one can deduce that ψ follows a Gaussian distribution for almost 2 all modes. As the eigenfields are real, their squared amplitudes fallow a Porter-Thomas distribution [START_REF] Stöckmann | Quantum Chaos. An Introduction[END_REF].

The experimental verification using 3D electromagnetic cavities shows that the first few hundred eigenfields seem to conform to that normal distribution [START_REF] Deus | Statistical properties of the eigenfrequency distribution of the three-dimensional microwave cavities[END_REF][START_REF] Dörr | Scarred and chaotic field distributions in a three-dimensional Sinaimicrowave resonator[END_REF]. It must be noted that this result is valid only for points distant from the walls by at least roughly a quarter of wavelength [START_REF] Dörr | Scarred and chaotic field distributions in a three-dimensional Sinaimicrowave resonator[END_REF].

In the RC case, the normality of the eigenfield distributions implies that each squared rectangular component of the complex 2 field is exponentially distributed, when evaluated over a stirrer rotation. This is one of the so-called asymptotic laws valid for chaotic RCs. The RC chaotic behavior may come from the stirrer presence, the corrugated walls, the cavity irregularities, or the cavity shape. As an illustration, a generalization of the 2-D Bunimovich stadium to a 3-D structure with moving corrugated walls has been proposed as an RC geometry [START_REF] Arnaut | Evaluation of the NPL untuned stadium reverberation chamber using mechanical and electronic stirring techniques[END_REF].

D. Weyl Formula

Much of the techniques used in quantum chaos are related to the study of the eigenvalues, i. e. the resonant frequencies of the cavity. Balian and Bloch [START_REF] Balian | Distribution of eigenfrequencies for the wave equation in a finite domain: III. Eigenfrequency density oscillations[END_REF] have treated theoretically the distribution of electromagnetic eigenmodes in a cavity with perfectly conducting walls. Using the notation of Arnaut [START_REF] Arnaut | Operation of electromagnetic reverberation chambers with wave diffractors at relatively low frequencies[END_REF], the expression for the averaged modal density as a function of the frequency f is

              + -       = -2 2 8 c f o c c f c V df dN avg κ π , ( 3
)
where c is the light speed in vacuum, V is the cavity volume and the geometric dependent constant κ is defined by

( ) ( ) ( ) ( ) ∫ ∫ - - - = S L dl r r r r d r r r r ϕ ϕ π ϕ π π ρ σ π κ ) 5 )( ( 6 1 3 4 , ( 4
)
where S is the cavity surface, L is the edges length, and the We can emphasize two main properties. First, at high frequency, the dominant term of (3) does not depend on κ: the average modal density is then independent of the RC surface. The integration of (3) leads to the approximate Weyl formula 2 Deviations from this universal law may concern localized modes associated to short unstable ("scarred eigenfunctions") [START_REF] Sridhar | Experimental observation of scarred eigenfunctions of chaotic microwaves cavities[END_REF] or to non-isolated marginally-unstable periodic orbits ("bouncing-ball modes") [START_REF] Kudrolli | Experimental studies of chaos and localization in quantum wave billiards[END_REF]. 2 Applying a perturbation technique to low losses enclosures leads to a complex field, even if this one is expanded on real irrotational modes [START_REF] Orjubin | On the FEM modal approach for a reverberation chamber analysing[END_REF].

( ) [ ] 1 1 1 2 - - -+ = ϕ θ ρ ρ ρ r
of the cumulative number of modes N(f) below a given frequency f:

3 3 8 ) (       ≅ c f V f N avg π . (5)
As a second property, we note that the formula (3) is the same for chaotic or non chaotic systems. The difference between these two kinds of system is embedded within the local fluctuations of the modal density. This point is presented in §IIF.

E. Random Matrix Theory

According to accepted conjectures 3 , generic chaotic systems follow the universal laws predicted by the random matrix theory (RMT) [START_REF] Stöckmann | Quantum Chaos. An Introduction[END_REF]. In RMT the linear wave equation ( 1) is replaced by a linear matrix equation where the elements of the matrix are random variables. The matrix statistics are those that would result if the matrix were drawn from different types of ensembles, where the relevant ensemble type depends only on the underlying symmetry of the system. The ensemble that is relevant to lossless electromagnetic problems is the Gaussian orthogonal ensemble (GOE) as GOE is generally suited to describe chaotic system with time reversal symmetry. To describe a lossless integrable system, the Poissonian orthogonal ensemble (POE) must be used.

We will see in the next paragraph that, depending on the RC general property (chaotic or not), the distribution of the resonant frequencies is obtained from the GOE or POE hypothesis. Although RMT supposes that the number of modes to be analyzed should be infinite, we will see that some hundreds of them allow us to assess the chaotic behavior of an RC.

F. Eigenvalue Spacing Distribution

Derived from RMT, the distribution of the resonant frequency spacings is characteristic of the type of system (regular or not, chaotic or not). In order to deal with universal properties of the spectrum, independently from the cavity size, the so-called unfolding procedure has to be employed [START_REF] Galdi | Wave propagation in ray-chaotic enclosures: Paradigms, oddities and examples[END_REF][START_REF] Warne | Statistical properties of a linear antenna impedance in an electrically large cavity[END_REF].

From the sequence of ordered eigenvalues (f 1 , f 2 ,.., f n ), we first determine the fitted curve N avg to the empirical values N(f i ): in our case a third order polynomial approximation is sufficient. Then the nearest neighbor frequency spacing is defined as

) ( ) ( 1 i avg i avg i f N f N s - = + . ( 6
)
The frequency spacing distribution s is analyzed by its histogram. Normalizing by the total number of eigenfrequencies, one can estimate the probability distribution P(s), if the number n of studied modes is large. The general results of RMT are that i) regular systems show 3 It is worth nothing that the RMT predictions have found a strong support not only experimentally, but also theoretically. For instance, Berry [START_REF] Berry | Semiclassical theory of spectral rigidity[END_REF] has obtained the so-called spectral rigidity using a semiclassical approach.

a Poisson distribution (in fact an exponential),

s e s P - = ) ( , s ≥ 0 (7)
and ii) chaotic systems show a Wigner (or Rayleigh) distribution. One can see from ( 8) that for chaotic systems, the probability for zero spacing is null. This phenomenon is called level repulsion by analogy with the repulsion between neighboring energy levels in the case of a quantum system. Note that ( 8) is the exact result for 2x2 GOE matrices but only an excellent approximation for large GOE matrices, which should be more suited to the systems under study.

It follows from (7-8) that the eigenvalues spacing statistics reflect the degree of chaos of the system. In order to model this frequency spacing in a unified way, we can use the semi empirical Brody4 distribution [6][10 p.93] which depends on a unique parameter ν:

( ) [ ] . 1 1 1 where 0 exp 1 ) ( 1 1 + +             + + Γ = ≥ - + = ν ν ν ν α α α ν s s s s P (9)
For ν = 0 and 1, this distribution reduces to the Poisson and the Wigner distribution, respectively. Therefore the closer the parameter ν is to unity, the harder the chaos of the underlying system.

III. REGULAR STIRRERLESS CAVITY

In this paragraph we present the case of a regular system that will serve as a reference when studying stirrer-equipped cavities (cf. Sections IV-V). The integrable cavity_1 considered here is a parallelepipedic lossless enclosure of dimensions (L x , L y , L z ) = (3.10 m × 2.47 m × 3.07 m).

A. Weyl Formula

For the simple geometry of cavity_1, the averaged radius of curvature ρ is infinite and the dihedral angle is ( )

2 π ϕ = r r .
The linear integration of (4) leads to )

2 1 3 8 ) ( 3 + + + -       = c f L L L c f V f N z y x avg π (10)
This expression has been shown [START_REF] Liu | Eigenmodes and the composite quality factor of a reverberating chamber[END_REF] to be equal to the mean number of modes of the regular cavity, whose eigenfrequencies have well known analytical expressions. This result is easily illustrated in Fig. 1, where both the actual cumulative number of modes N(f), found analytically, and N avg , obtained from [START_REF] Stöckmann | Quantum Chaos. An Introduction[END_REF] are plotted versus the frequency, for the first 200 modes. Equation ( 10) is more than an approximation: it gives the mean value of the number of modes.

B. Eigenvalue Spacing Distribution

As explained in Section II-F, the eigenvalue spacing distribution can be used to check if a system is chaotic or not. The resonant frequency spacing is unfolded following the procedure described in Section II-F, leading to a normalized s spacing. As TE/TM degeneracies occur, s is calculated from the sorted eigenfrequencies without degeneracy [START_REF] Deus | Statistical properties of the eigenfrequency distribution of the three-dimensional microwave cavities[END_REF]. Figure 2 gives the probability density function (pdf) of exponential, Wigner and s empirical distributions. Figure 2 shows clearly that the empirical data do not follow a Wigner distribution, characteristic of chaotic systems. Instead of binning the data to obtain the histogram of frequencies, we prefer to compare in the empirical cumulative density functions (cdf) to the theoretical cdfs, as shown in Fig. 3. A last fact must be brought to light: although no antenna is modelled in our modal approach, the mere insertion of a wire into the cavity for excitation purpose transforms the integrable system into a pseudointegrable one. This phenomenon has been studied theoretically and experimentally in [START_REF] Haake | Manifestation of wave chaos in pseudointegrable microwave resonators[END_REF] and [START_REF] Laurent | Diffractive orbits in the length spectrum of a two-dimensional microwave cavity with a small scatter[END_REF].

IV. REGULAR CAVITY WITH STIRRER_1

The RC_1 contains a metallic stirrer, which is a 2D rectangular plate parallel to the cavity bottom and which rotates around a vertical axis placed at the stirrer center at the height h = 2.2 m. The stirrer dimensions in meters are (l a , l b ) = (1.5, 0.75). The stirrer orientation used in this Section (28° between the larger stirrer dimension and Ox axis) is sketched in Fig. 4.

A. Mode Determination

Using FEM, the discretization of the electromagnetic problem leads to a generalized eigenproblem that is solved using Jacobi-Davidson algorithm [START_REF] Orjubin | On the FEM modal approach for a reverberation chamber analysing[END_REF]. We are looking for the first N = 200 modes: the subspace dimension is taken to be 2N, and the amount of memory required for such task does not permit to solve a very large eigenproblem. In order to reduce the number of DoFs, first order shape elements (instead of second order) are used to model the field on the unstructured mesh which is hand made reffined on the stirrer. This results in 60 103 DoFs for the stirrer position indicated in Fig. 4, and the run time is 4.5 10 3 s (Pentium IV, 3.2 GHz). It must be pointed out that the solver determines all the modes, even if they degenerate. Thus there is no risk of missing some modes, as happens in experimental procedures using the spectrum peaks. This is an important issue when the cumulative number of modes is estimated [START_REF] Deus | Statistical properties of the eigenfrequency distribution of the three-dimensional microwave cavities[END_REF].

Another important issue is the existence of a systematic error in the eigenfrequency determination, due to FEM discretization. This is particularly true for the higher rank modes.

As a result we notice that the stirrer presence suppresses mode degeneracy: for instance, the perturbation by the stirrer of the TE 111 and TM 111 modes (resonant frequency 91.6 MHz) leads to two distinct resonance frequencies (90.2 and 92.3 MHz). This important fact means that RC designers do not need to choose cavity dimensions such that degeneracy is avoided. For instance, it has been experimentally and numerically shown by Bruns and Vahldieck in [START_REF] Bruns | A closer look at reverberation chambers-3-D simulation and experimental verification[END_REF] that cubic cavities can yield the same field uniformity as rectangular cavities.

B. Weyl Formula

As the stirrer modifies the RC geometry, the mean number of modes must be evaluated using equations [START_REF] Stöckmann | Quantum chaos in billiards studied by microwave absorption[END_REF][START_REF] Sridhar | Experimental observation of scarred eigenfunctions of chaotic microwaves cavities[END_REF]. The dihedral angle is still ( ) Figure 5 proves that the insertion of the stirrer in Cavity_1 leads to a negligible increase of the mode number. The slight difference between the Weyl formula and the averaged curve deduced from mode counting can be attributed to discretization errors inherent to the FEM: this makes impossible a precise determination of the resonant frequencies. In order to estimate the fluctuations of the density of modes, a polynomial curve fitting of order 3 is used to determine N avg (f).

C. Eigenvalue Spacing Distribution

From the first 200 eigenvalues of RC_1 and the numerical expression of N avg , the normalized spacings s are calculated then classed into bins of width 0.2. The empirical pdf and the theoretical predictions are given in Fig. 6. A slight level repulsion, predicted in §IIF for chaotic systems, can be observed in Fig. 6, as confirm the cdfs plotted in Fig. 7. In spite of a little discrepancy for small s values, the general behavior deduced from the first 200 modes of RC_1 is characteristic of an integrable system. The Brody parameter is ν = 0.15.

D. Eigenfields

The solver of the modal FEM problem provides the eigenfrequencies and the eigenfields inside the RC. These fields are projected onto a grid whose steps in the x, y and z directions are respectively Lx/10, Ly/10 and Lz/10, thus roughly 0.2 m. The x, y, and z components of the electric field are then collected; we only retained the values corresponding to points distanced from the walls by at least 0.5 m, as indicated in Fig. 4. In order to process uncorrelated data, we adopt 100 as the minimum mode rank, or f = 242 MHz: this way, the spatial step is larger than λ/4. The E x component of mode #106 does not seem to follow a Gaussian distribution, unlike mode #190.

In order to quantify the goodness-of-fit (GoF) of the empiric data to a Gaussian distribution with unknown parameters, the Shapiro-Wilks test is used. The result of this test is 0 if the hypothesis (data follow a normal distribution) is retained and 1 if rejected. The result of the GoF for the E x component is shown in Fig. 9, at a 0.05 significance level. Figure 9 proves that E x is not normally distributed for many modes of RC_1. To get a global idea, the mean values of GoF tests for the mode ranks between 100 and 200 (i.e. 242 and 304 MHz) are respectively 0.22, 0.29, and 0.44 for the components E x , E y and E z 6 . This discrepancy can be explained by the poorly chaotic behavior of RC_1.

V. NON REGULAR CAVITY_2 WITH STIRRER_2

A cavity corner is replaced by one eighth of a spherical surface of radius r, transforming the integrable cavity_1 into a nonintegrable cavity_2 of Sinai type. This technique is derived from the work of Dorr [START_REF] Dörr | Scarred and chaotic field distributions in a three-dimensional Sinaimicrowave resonator[END_REF]. A V-folded Stirrer_2 with centered vertical rotating axis is oriented by 28° angle as shown in Fig. 10.

A. Weyl Formula

As explained by Arnaut in [START_REF] Arnaut | Operation of electromagnetic reverberation chambers with wave diffractors at relatively low frequencies[END_REF], the corner modifies the cavity geometry but (3-4) can still be used. Despite the small reduction of the cavity volume, the negative κ parameter is increased, resulting from the increase of the edge length and the curved surface. The overall consequence is that the mode density for cavity_2 can be larger than for cavity_1. Because of the complex geometry of the stirrer_2, we will not try to predict the accurate Weyl formula for RC_2, but we consider [START_REF] Orjubin | Mode Perturbation Induced by the Stirrer Rotation in a Reverberating Chamber[END_REF], obtained for cavity_1, as a good approximation of the cumulative number of modes for RC_2. To check this point, we compare in Fig. 12 the number of modes [START_REF] Stöckmann | Quantum Chaos. An Introduction[END_REF] of the cavity_1 to the one determined after FEM solving for RC_2. As seen in Fig. 11, Eq. ( 10) provides a good approximation of the number of modes of RC_2 even if it does not strictly apply to this enclosure. This satisfactory concordance justifies the use by practitioners of [START_REF] Stöckmann | Quantum Chaos. An Introduction[END_REF] or [START_REF] Haake | Manifestation of wave chaos in pseudointegrable microwave resonators[END_REF].

B. Eigenvalue Spacing Distribution

N avg (f) is obtained by a third order curve fitting of Fig. 11. Then the variate s is calculated using [START_REF] Deus | Statistical properties of the eigenfrequency distribution of the three-dimensional microwave cavities[END_REF] and the pdf of s is plotted in Fig. 12. The level repulsion at s = 0 can be clearly seen in Figure 12 for empirical data. A clear difference exists between Fig. 13 and Figs. 3 and7. The Brody parameter is ν = 0.61. The Wigner distribution acceptably 7 fits the empirical data issued from solely 200 eigenfrequencies. Small spacings are less probable (level repulsion) and large spacings are also improbable, as the maximum value is s Max = 3.16.

This tends to prove that the level fluctuation laws, derived from RMT, are also valid for the vectorial solution of the Helmholtz equation. Finally, the RC chaoticity is easily assessed from a relatively small set of eigenvalues, as indicates Table I. It is worth noting that despite the complex geometry of RC_2, the frequency spacing does not follow exactly the Wigner distribution. This is consistent with the fact that RMT is supposed to strictly apply in the limit λ/L <<1, where L is a geometric dimension.

C. Eigenfields

The result of the GoF for the E x component is shown in Fig. 14. Comparing Fig. 14 to Fig. 9, one can see that the Gaussian hypothesis is reasonable for more modes with RC_2 than with RC_1. For instance, the mean values of the GoF tests for ranks between 100 and 200 (i.e. 242 and 304 MHz) are respectively 0.14, 0.23 and 0.19 for the Ex, Ey and Ez component 8 . Moreoever, the small 9 values show that most of the modes close to the Lowest Useable Frequency 10 (LUF) correspond to a Gaussian distribution.

Nevertheless, the strict interpretation of Figs 13 and 14 leads to the following conclusion: as the eigenfield distribution is not perfectly Gaussian, RC_2 presents a non universal behavior at low frequencies. This means that the geometry details, such as the shape of the stirrer, play a crucial role to obtain the desired uniformity and isotropy of the field. Because of this non universality, the performance of two RCs can be compared: it is usually done by a uniformity test on the overall field, but could also be deduced from comparisons of the distributions of the eigenfrequency spacings or of the 8 The isotropy of the modes is better for RC_2 rather for RC1: RC_2 has no particular symmetry. 9 If the data are normally distributed, as for high rank modes, the GoF test with 5% significance level rejects the Gaussian hypothesis with a probability of 0.05. 10 Here we use the common estimation that gives LUF = 3 fmin = 206 MHz.. This is close to the 60 th mode criterion, as shows Fig. 11. eigenfield. In this case, ensemble statistics should be used for various RC realizations associated to independent stirrer positions.

VI. FREQUENCY PERTURBATION IN A CHAOTIC RC

A. Presentation

In the previous sections, two different RCs (regular cavity and simple stirrer, non regular cavity and more sophisticated stirrer) were analyzed at a given stirrer position.

In this section the effect of the stirrer rotation on the resonant frequencies is investigated, following the work of Wu and Chang [START_REF] Wu | The effect of an electrically large stirrer in a mode-stirred chamber[END_REF] who modelled a 2-D RC by TLM. Employing this technique, the resonant frequencies are determined indirectly: they are the maxima of the frequency response obtained by Fourier transform of an impulse time response. The implicit time windowing makes it difficult to separate close modes, and the determination of the frequency sweeps is uneasy.

With a modal approach, the eigenfrequency perturbations (or evolutions with the stirrer position) are determined directly and can be characterized more precisely. We will first check in this section that the modal density is not affected by the stirrer displacement. Then an illustration will be given of an important property: for a chaotic system, each eigenvalue curve proceeds without crossing any other. This general property does not come from RMT, and it can apply to a restricted number of modes.

In order to minimize the numerical noise due to FEM, the RC is discretized by second order elements, yielding 2 10 5 DoF. It takes 1500 s to find the first 20 modes. This determination is iterated for K stirrer positions evenly spaced by a 2° angle. For Stirrer_1, K = 90 using symmetry, and K = 180 for Stirrer_2.

B. Avoided Crossings

The values of the first 20 eigenfrequencies of RC_1 are plotted in Fig. 15. Figure 15 shows that the stirrer rotation does not increase the number of modes and produces frequency sweeps. Due to the special configuration of Stirrer_1, some modes are unperturbed, such as the third one that is similar to TM 110 of the cavity_1.

As RC_1 does not present a chaotic behavior, crossings situations can be observed: a detail of Fig. 15 is given in Fig. 16, concerning the modes of ranks 4~6 and 14~17 at the first stirrer position. 15 for the rank sorted eigenfrequencies (without tracking), and evolution of each of these modes after using the Mode Tracking Procedure [START_REF] Orjubin | Mode Perturbation Induced by the Stirrer Rotation in a Reverberating Chamber[END_REF].

The resonant frequencies being sorted by the solver, a given frequency rank is not related to a same mode when crossing situations occur: this is the case of the mode starting at 93.7 MHz. To detect automatically the crossings and assign the correct rank to a given mode, a mode tracking procedure (MTP) has been designed [START_REF] Orjubin | Mode Perturbation Induced by the Stirrer Rotation in a Reverberating Chamber[END_REF]. For the sake of completeness, the method is briefly presented in the Appendix. Applying the MTP, each frequency rank is related to a single mode, as shows Fig. 16.

In order to illustrate the level repulsion related to avoided crossings for RC_2, Fig. 17 For many modes (e.g. ranks 2, 3 and 7, 8, 9), the curves get close but no apparent crossing occurs. This fact has been confirmed by the MTP for the first 20 modes: the rank of each mode is the same during the stirrer rotation. Finally, we can retain that a chaotic system presents avoided crossings. Therefore, the eigenfrequency perturbation induced by the stirrer is bounded by the mean spacing between nearest neighbor eigenfrequencies. As this one is proportionnal to the inverse of the modal density (3) that increases with the frequency, the eigenvalues perturbation tends to vanish at high frequencies. Since the resulting field is expanded on the same set of modes, it is most likely that the eigenfield perturbation plays a key role in the stirring process at high frequency. This phenomenon has been discussed in [START_REF] Orjubin | Mode Perturbation Induced by the Stirrer Rotation in a Reverberating Chamber[END_REF] for low frequencies.

VII. CONCLUSION

This paper presents a study of the chaos in a Reverberation Chamber (RC) derived from the analyze of its modal distributions and making use of quantum chaos theory. This theory is often validated by electromagnetic experiments, usually with empty 3-D cavities. We have extended the application to RCs, cavities containing a stirrer.

The Weyl formula is checked from the FEM determination of the first 200 modes of RCs with different shapes from integrable to chaotic.

The distribution function of the eigenfrequency spacing is in agreement with Random Matrix Theory (RMT) and reveals the RC chaoticity. The eigenmode distributions are also analyzed and compared to the theoretical Gaussian distribution: close to the lower useable frequency, the modes of the studied chaotic RC fairly respect this asymptotic property.

Although RMT assumes that a great number of modes (and consequently high frequency modes) must be analyzed, we have seen that our restricted number of modes allows an illustration of the universal properties of the RC, i.e. properties that are independent from the exact RC shape. However, some non universal behaviors have also been noticed. These deviations from the theoretical limit can be used as a metric to compare RC performances.

Using a Mode Tracking Procedure, we have illustrated the following property of a chaotic RC: during a stirrer displacement, the curves of the resonant frequencies do not cross. This means that at high frequency the frequency sweeps tend to vanish.

VIII. APPENDIX

The key to the mode tracking procedure [START_REF] Galdi | Wave propagation in ray-chaotic enclosures: Paradigms, oddities and examples[END_REF] is sketched here to make this paper self-explanatory. ( )

∫∫∫ Ω = = dV k E k E k E k E n m n m ) ( ). ( 0 ) ( ), ( r r r r ( 11
)
The idea is that ( 15) is quite respected for a small rotation step, considering two eigenvectors corresponding to adjacent positions, say ( )

) (k E m
) 1 ( ) ( , + = k U k U n m S n T m k r r , ( 12
)
where T designs the transpose.

After normalizing ) (k U m r , S k matrix is numerically close to a permutation matrix, thus enabling a crossing detection. For example, a S k view is given in Fig. 18 for the first 6 modes of RC_1. S k=1 can be assimilated to identity (no crossing) whereas S k=10 is a permutation matrix affecting the ranks 5 and 6 (mode 6 at position 10 will occupy the rank 5 at position 11).

The permutation vectors are easy to identify in most of the cases, but not all, as indicates the S 23 matrix (Fig. 19). According to S 23 values, the transition between k = 23 and k = 24 is uncertain. As tracking the modes from the first position till the position k is an iterative procedure, it can be dramatically affected by error propagation. To get a robust technique, we consider multiple possibilities (e.g. 2 choices at position k = 23). Then a tree is constructed for the solutions ensemble. The selection of the best (or actual) solution is based on the regularity of the eigenvector projection on the initial eigenvector (more details in [START_REF] Orjubin | Mode Perturbation Induced by the Stirrer Rotation in a Reverberating Chamber[END_REF]).

r

  is the averaged radius of curvature of surface element dσ at r for spherical angles θ and φ. Function ( ) r r ϕ is the local dihedral internal angle along edge dl.

  case, the Weyl formula becomes5

(

Fig. 1 .

 1 Fig. 1. Cumulative number of modes for cavity_1.

Fig. 2 .

 2 Fig.2. Probability density function of the eigenfrequency spacings s for cavity_1 and the theoretical exponential and Wigner pdfs, issued from (7) and (8).

Fig. 3 .

 3 Fig. 3. Cdf of s parameter, integrable cavity_1.

Figure 3

 3 Figure 3 indicates that the s parameter follows an exponential distribution, as expected for this regular system. This point is confirmed by the low value ν = 0.07 of the Brody parameter estimated by curve fitting.A last fact must be brought to light: although no antenna is modelled in our modal approach, the mere insertion of a wire into the cavity for excitation purpose transforms the integrable system into a pseudointegrable one. This phenomenon has been studied theoretically and experimentally in[START_REF] Haake | Manifestation of wave chaos in pseudointegrable microwave resonators[END_REF] and[START_REF] Laurent | Diffractive orbits in the length spectrum of a two-dimensional microwave cavity with a small scatter[END_REF].

Fig. 4 .

 4 Fig. 4. RC_1 with stirrer_1 into the integrable cavity_1.

Figure 4

 4 Figure 4 also represents the 140 points (*) at which the field is calculated (cf. Section IV-D).

  number of modes of RC_1 is determined numerically by FEM calculations, and is theoretically predicted by the Weyl formula. The result is plotted in Fig.5and can be compared to the case of cavity_1.

Fig. 5 .

 5 Fig. 5. Cumulative number of modes for RC_1 obtained by FEM and the Weyl formula.

Fig. 6 .

 6 Fig. 6. Probability density function of the eigenfrequency spacings s for RC_1 and the theoretical exponential and Wigner pdfs.

Fig. 7 .

 7 Fig. 7. Cdf of s parameter, stirrer_1 into the integrable cavity_1.

Figure 8 Fig. 8 .

 88 Figure 8 shows two kinds of histogram occurring for the E x component.

Fig. 9 .

 9 Fig. 9. Result of Shapiro-Wilks normality test for the Ex component, RC_1.Result is 0 when the data follow a normal law, at 0.05 significance level.

Fig. 10 .

 10 Fig. 10. Modeled RC_2 with stirrer_2 into the nonintegrable cavity_2. The 140 points at which the field is evaluated are indicated by (*).

Fig. 11 .

 11 Fig. 11. Cumulative number of modes for RC_2 obtained by FEM and Weyl formula for Cavity_1.

Fig. 12 .

 12 Fig. 12. Probability density function of the eigenfrequency spacings s for RC_2 and the theoretical exponential and Wigner pdfs.

Fig. 13 .

 13 Fig. 13. Cdf of s parameter, stirrer_2 into nonintegrable Cavity_2.

Fig. 14 .

 14 Fig. 14. Result of Shapiro-Wilks normality test for the Ex component, RC_2. Result is 0 when the data follow a normal law, at 0.05 significance level.

Fig. 15 .

 15 Fig. 15. First resonant frequencies of RC_1 versus the stirrer displacement.

Fig. 16 .

 16 Fig.[START_REF] Arnaut | Evaluation of the NPL untuned stadium reverberation chamber using mechanical and electronic stirring techniques[END_REF]. Details of Fig.15for the rank sorted eigenfrequencies (without tracking), and evolution of each of these modes after using the Mode Tracking Procedure[START_REF] Orjubin | Mode Perturbation Induced by the Stirrer Rotation in a Reverberating Chamber[END_REF].

Fig. 17 .

 17 Fig. 17. First resonant frequencies of RC_2 versus the stirrer displacement

  At stirrer position k (1 ≤ k ≤ K), the solver provides a set of track the modes from the first stirrer position till the position k. For this, we first look for an index permutation vector that models correctly the transition between the stirrer positions k and k+1.As a general result, the electromagnetic cavity eigenmodes are orthogonal: for the stirrer position k,

Fig. 18 .

 18 Fig. 18. Abs(Sk) matrices for k = 1 and k = 10, first 6 modes of RC_1.

Fig. 19 .

 19 Fig. 19. Abs(S23) matrix, first 6 modes of RC_1.

TABLE I BRODY

 I PARAMETER FOR DIFFERENT RC GEOMETRY.

	Cavity_1	RC_1	RC_2
	0.07	0.16	0.61

This a a Weibull distribution with mean value 1.

As indicated in[START_REF] Baltes | Asymptotic eigenvalue distribution for the wave equation in a cylinder of arbitrary cross section[END_REF], the constant is predicted only for parallelepipedshaped domain; for electromagnetism problem, its value is ½.

The difference between the 3 components is due to the shape of RC_1 where x and y play a rather similar role.

A better result is reported in[START_REF] Deus | Statistical properties of the eigenfrequency distribution of the three-dimensional microwave cavities[END_REF], where the cavity shape is very irregular, and the number of modes is higher (466).