M Chligui

G Guimbretiere

A Canizares

G Matzen

Y Vaills

Patrick Simon

G Guimbretière

A Canizarès

New features in the

published or not. The documents may come

This paper reports the evidence of new signatures in the low-frequency Raman spectrum of silica, obtained by very high signal-to-noise experiments. Two new Raman lines are observed at 295 and 380 cm -1 , and the presence of a third one at 245 cm -1 is needed to account for the experimental spectra. The origin of these supplementary lines is discussed in terms of n-membered rings of [SiO 4/2 ] tetrahedral entities inside the glassy network. Amorphous silica (SiO 2 ) is one beyond the mostwell known materials, with interests going from a model compound in glass physics up to numerous industrial applications for optics and electronics. A huge number of papers devoted to it stands in the literature, in such a point that its structure and dynamics look well-known and are then used as reference for the study of more complex systems. Raman scattering investigations for instance are numerous and it is then difficult today to identify the first report on silica. The Raman spectrum of silica is today rather well understood since the works by Galeener and Lucovsky thirty years ago 1-2 and confirmed by subsequent ab-initio calculations 3 . The low-wavenumbers region of the silica Raman spectrum in the VV configuration polarization is usually decomposed as made of the boson peak at low wavenumbers (pointing around 50 cm -1 ), a broad and asymmetric band at 450 cm -1 assigned to δ bend (SiOSi) bending vibrations (frequently referred as R-band), plus the D 1 and D 2 narrow peaks closed to 495 and 600 cm -1 . One widely accepted view of the structure of silica is a disordered more or less open network of corner-linked interconnected [SiO 4/2 ] tetrahedral entities. Such view intrinsically considers the existence of [SiO 4/2 ] membered rings (hereafter denoted MR), and as an evidence, the socalled D 1 and D 2 peaks are now unambiguously assigned to respectively four-tetrahedral (4-MR) and threetetrahedra membered rings (3-MR). The specific dynamics of these MR was recently observed 4 by picosecond spectroscopy, that confirms the strong decoupling of these vibrations from the whole silica network. Nevertheless these assignments remain partially discussed, 5 with possible mixed influence of 3-MR and 4-MR in the D 1 band 6 and the absence of consensus in the estimation of rings population 7,6,[8][9] . Raman spectroscopy offers then a simple way to probe the statistics for 3 and 4 -MR, and the greatest part of the literature devoted to rings statistics has focused on these small rings. However, the population of 3-and 4-MR is not dominant in silica compared to the whole MR distribution that can be obtained by molecular dynamics 7 . Unfortunately, larger MR are rather hardly experimentally accessible, and very few knowledge is available on them apart from numerical approaches. In the literature one can found also some descriptions of the (SiOSi) bridges dynamic by force constant models 10,11 . These rather simple models link vibrational spectroscopic data with the average angle θ and effective forces constant of (SiOSi). From considerations of these models, the Raman signatures of 5-MR and 6-MR larger rings are expected to be at lower wavenumbers than for small rings, i.e., in the R-band region. Besides, independently of any specific ring patterns consideration, three different spectral features were suggested 12 to compose part of the isotropic R-band (symmetric stretching involving (SiOSi) bridges, δ scis scissoring of [SiO 4/2 ], and Δ scis scissoring of the [SiO 4/2 ]-[Si 4/4 ] extended tetrahedron). Therefore, it is not surprising that the R-band is sometimes described as the superposition of several individual symmetric bands (see for instance 13 ), but the proposed decompositions remain a little bit ambiguous as different band combinations can reproduce the broad experimental asymmetric shape, due to lack of any pointed inflexion point on the rising edge between 200 and 400 cm -1 .

-2-This short paper reports very high signal-to-noise Raman spectra of silica that highlights new structuring in the broad asymmetric 450 cm -1 R-band. These new features closed to 295 and 380 cm -1 are observed here for the first time in the nonetheless well know spectrum of silica. They arise owing to increase of the signal-to-noise ratio and have to be considered after a rigorous and exhaustive study, in order to exclude any instrumental origin. So, in the following, we present first the experimental condition leading to the new spectral features observation. Next, we introduce and illustrate in a short discussion, the relevant consequence of such observations in the study of vitreous silicates structure by mean of Raman spectroscopy.

Experiments were done on two spectrometers: a Renishaw Invia Reflex with a Leica DM 2500 microscope (simple spectrometer configuration), and a Jobin-Yvon T64000 (triple subtractive configuration), in macro configuration. Both systems are multichannel ones with CCD detectors. Different excitation laser wavelengths were used: 457, 514 (Ar + lines) and 633 nm (He-Ne line). Measurements with the Leica microscope were done with two objectives in order to check the effect of numerical aperture and the subsequent changes in light polarization. The specificities of the experiments reported here are the very long acquisition times (up to 20 hours), in order to substantially increase the signal-to-noise ratio. As in this case the saturation limit of the CCD detector can be attained, measurements were performed in accumulation mode (several subsequent spectra of lower time, to keep beyond the saturation limit, which are added at the end of experiment). Silica samples were of Corning 7980 type. A test was done on a 7980 sample, thermally treated to change the fictive temperature (Corning 7980 T f is 1040°C). Figure 1 (a) displays the 25-650 cm -1 frequency range of the VV polarized Raman spectrum of SiO 2 , obtained on the Jobin-Yvon spectrometer, with the Ar + 514.5 nm laser line, in macro-right angle configuration. This spectrum was acquired with a very long acquisition time (17h) in order to improve the signal-to-noise ratio. One can see, added to the well-known boson peak, R-band, D 1 and D 2 bands, two supplementary bumps, near 300 and 400 cm -1 . These bumps are rather broad and then they cannot give any detectable inflexion point on the rising edge of R-band if measurement is done with smaller acquisition times. As these bumps remain very small, possibility of any artifact in the experiment have been checked exhaustively: a first test to perform when recording such small and broad features is to check if there is no effect on non-uniform lightening on the CCD detector, by shifting the spectrograph of a small value: a non-uniform lightening stays at the same pixels whereas a real spectral feature shifts of pixels, to stay at the same wavenumbers. It was checked here that the two bumps are constant in wavenumbers. Other checks were done, to unambiguously establish that these bumps are intrinsic to the Raman response of silica. (i) Measurements were also done on the Renishaw Invia spectrometer (backscattering geometry and micro-Raman configuration) (ii) two other laser excitation lines were tried, i.e., 457 nm and 633 nm. (iv) Measurements with the Leica microscope -3-(Invia spectrometer) were done with two objectives (x20 and x50) in order to check the effect of numerical aperture and the possible subsequent changes in light polarization (v) changes of size of sample (from thickness of 1cm to 1mm), to check any interference or reflection effect between opposite faces. All these check measurements (all done with very long acquisition times) have confirmed the spectrum of figure 1(a) with its bumps. To determine more precisely the bump positions, it is useful to see the derivative of the spectrum (Figure 1(b)), much more sensitive to small features. The derivation was performed by the algorithm comprised in Microcal OriginPro8.1 software, with a Savitzky-Golay smoothing. The maximum of a peak in the original curve corresponds in the derivative curve to a zero value, or crossing through the baseline in the case of a peak superimposed to a broader and more intense feature. The bumps positions can then be determined rather precisely from figure 1(b): ≈ 295 and ≈ 380 cm -1 , the uncertainty being about 5 cm -1 . Hereafter they will be called respectively D 3 and D 4 bands by analogy with the D 1 and D 2 bands. Now that the presence of both D 3 ≈ 295 and D 4 ≈ 380 cm -1 bands are established and even if to our knowledge, they were never reported explicitly before, one can remark that in some papers, the reported spectra have presumably exhibited them ( [13][14] for instance, among the most recent ones). Nevertheless, we underline that these features are very smooth compared to the underlying band and it was absolutely necessary to take specific acquisition times and to check for any possible artifact due to the experimental setup, as done in the present study. Above we have introduced that existence of numerous contributions to the Raman activity are expected in the R-band frequency region. Consequently, the possibility to fix the frequency of two of them has got considerable interest in the discussion of silicates structure. In Figure 2, we present a new classic deconvolution of the spectrum. Generally, deconvolution results are dependent of the chosen band shapes and we have performed several tries: Figure 2 pictures one of these tries, where we have used a log normal shape for the boson peak, Lorentzian shapes for the D 1 and D 2 bands and Gaussian shapes for others. As one can see, additionally of the D 2 ≈ 605 cm -1 , D 1 ≈ 495 cm -1 , , the ≈ 465 cm -1 band (maximum of the Rband), the D 4 ≈ 380 cm -1 new feature, the D 3 ≈ 295 cm -1 new feature and the boson peak, a band closed to 245 cm -1 is necessary in order to fit the full Raman activity in this wavenumbers region. In fact, this latter additional contribution is a common result of all the deconvolutions that we have performed (not shown here).

First, we note that these frequencies do not coincide with those of any of the natural SiO 2 crystalline polytypes (quartz, cristobalite, tridymite, stishovite, coesite) 15 . Second, frequency of the δ bend (SiOSi) bending dynamic was successfully connected to the value of (SiOSi) angle (θ) through simple effective force constant models 10,11 . Therefore, consideration of the D 1 and D 2 bands through these models assign them as δ bend dynamic of (SiOSi) bridges with different θ. In fact, in the specific case of 3-MR and 4-MR, the (SiOSi) bridges forming the rings are vibrating in phase leading to a "breathing" dynamic of the rings poorly coupled with their environment. The 3-and 4-MR rings appear then obviously in the Raman spectrum despite their very small amount obtained from different calculations 7,16 : around one ring of each specie for more than 500 SiO 4 tetrahedra according to Umari et al 7 . Then, if we consider, in one hand the (SiOSi) bridges in larger (n>=5) SiO 4 membered rings (hereafter denoted 5+MR) and, in other hand the widely observed and calculated θ distribution [17][18] , one can assign part of the R-band to δ bend dynamic of (SiOSi) bridges in 5+MR. Thus, the spectral signatures of larger rings are expected to be certainly broader than D 1 and D 2 , and lying at lower wavenumbers than these lines. In fact, in many of the amorphous SiO 2 calculated densities of states available in the literature ( see for instance 16,19,7 ), one can see peaks or clear shoulders lying in the wavenumbers range of those evidenced in the present paper. Obviously, calculated features of large structures such as 5+MR will exhibit lower linewidths than experimental ones, at least as the limited number of atoms in numerical approaches hardens a good description of coupling between different oscillators. Besides, we assume that the rest of the Raman activity comes from the scissoring modes introduced in ref 12. In order to illustrate the spectral consequence of the θ distribution, in the following, we consider the (SiOSi) angle distribution calculated by Yuan et al. 17 . This calculation is consistent with NMR characterization of silica 18 and gives (SiOSi) angle distribution centered to 147° with a full width at half maximum 23-30°. From the θ distribution, the induced frequency distribution in the Raman signature of (SiOSi) δ bend can be estimated by used of the Effective Force Constant model of Lucovsky 10 , with the frequency of the highest VV Raman doublet as frequency of the (SiOSi) asymmetric stretching that is also active in IR 12 . The distribution in frequency obtained is pictured by black crosses in Figure 2. In this calculation, the (SiOSi) δ bend frequency for θ = 147° is assumed to be at the 435 cm -1 maximum. These assumption give an angle of ≈ 128° at the 3-MR frequency, consistent with others recent demonstrations 20 . One can see that above estimated δ bend frequency distribution merges on both D 4 ≈ 380 and ≈ 465 cm -1 bands. Then, one of the most plausible structural elements that can be connected with them are effectively the 5+MR. In fact, one knows from molecular dynamics that the ring statistics in silica is dominated by such rings (between 5 and 10), whereas 3-MR and 4-MR, marginal in the statistics, appear obviously in the Raman spectrum due to their strong decoupling from the bulk of the density of states. One SiO 2 crystalline structure is known to exhibit a large palette of different rings, that is silicalite -1-F 14 (a SiO 2based zeolite, with 4-, 5-, 6-and 10-MRs). The Raman spectrum of silicalite is dominated by a sharp and intense line located at 380 cm -1 and a second one just below 300 cm -1 . These values coincide with those of D3 and D4 evidenced in the present results. Moreover, the remaining line at 445 cm -1 is presumably also visible in the silicalite 1F spectrum. Therefore, one can think to assign the full-set of lines used here to describe the silica Raman response (from 200 to 450 cm -1 ) to vibration of different MR rings (presumably by decreasing order of frequencies, 5-, 6-, 7, and more -MR). In an other hand, the "breathing" nature of the small rings dynamic at the origin of sharp welldefined D 1 and D 2 bands is difficult to be taken account in the case of larger rings that are undoubtfully more distorted and coupled with their environment. We postulate that reality of the Raman signature of δ bend dynamic of (SiOSi) bridges should be between these two extreme cases: broad R-band due to the distribution in θ, and discrimination of the D 4 ≈ 380 cm -1 and ≈ 465 cm -1 wavenumbers bands due to the presence of rings patterns. Therefore, this view allows an assignment of the two D 4 ≈ 380 cm -1 and ≈ 465 cm -1 bands to (SiOSi) δ bend in 5+MR. Besides, the lowest wavenumbers bands (≈ 245 cm -1 and D 3 ≈ 295 cm -1 ) can thus be assigned to δ scis and Δ scis .

Above we have discussed a "classical deconvolution". A rigorous discussion of the bands assignment containing band-shapes consideration needs to take also account of the Raman coupling parameter 21 that was assumed to be mode and frequency dependent 22 . Such fine and complex discussion will be published elsewhere, being beyond the scope of this short paper where we essentially want to highlight the presence beyond doubt of two distinguishable new Raman spectral features in the R-band of silica: D 3 ≈ 295 cm -1 and D 4 ≈ 380 cm -1 .

  FIG.1. (a) Raman spectrum of silica (VV polarization configuration, 514.5 laser line, 90° configuration) with 17h of accumulation time. Arrows denote the position of the bumps visible on the rising edge of Rband. Dashed frame is the spectrum region derived in (b). (b) Derivative of the dash-enclosed part of the Raman spectrum of SiO 2 , allowing accurate determination of the position of the bumps (indicated by arrows).

FIG. 2 .

 2 FIG.2. Deconvolution of the Raman spectrum of Silica (more details are given in the text).


Mireille Bilger

Henry Tyne

Speaking Style and Oral Text Construction in Second Language French

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Introduction

We propose to look here at oral text construction by second language learners of French. In this exploratory paper1 , which seeks to apply some of the descriptive tools developed by the Aix group of linguists, GARS2 for the analysis of first language spoken French, we will essentially be focussing on two aspects: first, the use of the two axes of text construction; second, the make-up of the syntactic units produced by the speakers. The aim of this paper is to explore the types of elements that can be linked to the difference in level of acquisition and to compare findings with those of native speakers of French.

The data analysed here is taken from a corpus of spoken French produced by undergraduate learners at the University of Surrey in 2003. The productions of 8 learners are observed: 4 first years and 4 finalists3 . This represents around 17,000 words overall, though individual word counts vary considerably depending on the duration of the recording. The recordings were all carried out during the students' end of year oral examinations, in which the students first make an oral presentation (prepared) and then answer questions by the examiners. Generally, the oral examination lasts between fifteen and twenty minutes. Transcription was carried out using GARS/DELIC4 conventions.

The notion of oral text

It is often remarked that oral language does not resemble that accounted for in many textbooks or grammars. This does not mean all elements of speech are completely alien from the types of examples we find in such works. However, what it does mean is that spoken language contains certain features that are typically not accounted for in traditional works as well as many elements which, though not necessarily discarded, are often considered superfluous for the analysis of language content and structure.

The approach to the study of spoken language developed in the 1980s by GARS, led by Claire Blanche-Benveniste, has a double agenda: to provide a comprehensive description of spoken French 5 syntax; and also, importantly, to account for the manner in which the oral text takes shape through production. For GARS, if somebody says "d'abord je je je penj'es-j'espère que vous le savez", then this is not only what we must transcribe but also what we must work with: the "dysfluencies" become an integral part in looking at how productions take shape during the interaction. Whilst there is obviously no denial that planning processes are at work before and during speech production, and that these are unlikely to include features such as those highlighted in the above utterance, there is behind the GARS stance-which is also our own-a refusal of the idea that speech performance is somehow an imperfect, "degenerate" version of the "real thing". There is, in fact, very little evidence to suggest that the integral linguistic message (devoid of the type of "accidents" highlighted in the above example) actually exists conceptually prior to production [START_REF] Krötsch | Les ruptures syntaxiques en français parlé[END_REF]. Indeed, many linguists see syntax as being borne out of the wider discourse and communication framework. For example, A. Berrendonner (e.g. 1990), in his workings on the notion of "macro-syntax", which serves to relate smaller syntactic constructions to one another through social cognition, questions the totally mentalist approach to language construction:

[…] toute interaction verbale met en jeu un ensemble évolutif M (mémoire discursive, ou savoir partagé), contenant les informations qui, à chaque instant, sont validés pour les deux interlocuteurs et publiques entre eux. En parlant, ceux-ci opèrent tour à tour dans M des révisions, transformations, incrémentations, etc., qui visent à porter M jusqu'à un état stable, satisfaisant pour les deux parties. Le discours n'est donc que la trace de ce processus coopératif de retraitement de l'information. Et si l'on entend par fonction communicative le fait d'opérer une transformation quelconque dans M, on peut alors faire l'hypothèse qu'il existe un niveau auquel la chaîne parlée s'articule en unités minimales à fonction communicative. (Berrendonner 1990: 25-6) In a sense, then, speaking, rather like writing, can be seen to involve a number of elaboration procedures before the "final" version is produced [START_REF] Halliday | Spoken and Written Language[END_REF]. The fact that someone writes "Thank you for your very kind invitation" does not necessarily mean that the element "very" is an accident or error; rather it shows that this element was deemed by the writer not to be desirable in what he/she wanted as the end product. It does not matter whether we think this word was part of the original plan or not. So, by choosing not differentiate between that which is intentional (what we think the speaker "had in mind" to do in terms of linguistic structure, an area of some debate in psycholinguistics -Harley 2001: 358) and that which is not, we are not so much ignoring the role of psychological planning as simply agreeing to show the broader picture, i.e. how spoken language comes together on the virtual page: text construction rather than unit planning and production. We take the view here that language construction need not be viewed as merely the product of covert processes (whether conscious or not). Indeed, we see language use not only as a "doing" activity, but also as redoing, (re)working, editing activity.

Axes of construction

Representing speech over two dimensions

Oral texts can be accounted for by "grid representation", whereby the two axes of construction (syntagmatic versus paradigmatic) become apparent. The syntagmatic (horizontal) axis accounts for the forward nature of the text in terms of the different combinations that occur (e.g. S-V-O); the paradigmatic (vertical) axis accounts for similarity between elements from a lexico-syntactic point of view. Let us take an example6 :

1. j'avais une bourse au collège

In terms of the syntactic makeup, this can be termed as a wholly linear unit, i.e. it only makes use of the syntagmatic axis: j'avais une bourse au collège However, if we admitted that this was not really what was produced and that the actual production was in fact 1'. j'avais une une bourse euh au chose au lycée quoi au collège we could use a grid to sort out the linear from the vertical. So, this is what we in fact get: j'avais une une bourse euh au chose au lycée quoi au collège

In other words, from a syntactic point of view (i.e. in terms of the S-V-O structure), the production remains unchanged. However, we now see from our inclusion of the vertical dimension that certain paradigms have been used to refine the content of what is said: the speaker dwells in a given paradigm, for whatever reason, before moving on along the horizontal axis. Dwelling in a paradigm does not necessarily mean that the speaker is experiencing difficulty: it may be due to conscious listing (as in "I grew turnips, parsnips and beetroot") just as it may be due to repetition in order to bar the floor to other speakers ("I I I I I [said with increasing volume] grew turnips, parsnips and beetroot") or to redefine something following a look of incomprehension from the audience (as in "I grew turnips, parsnips [correction: parsnips are what I grew, not turnips] and beetroot ").

By accounting for all types of listing in the same way in terms of the grid representation, it may appear as though we are blindly cutting across the sacred divide operated by Chomskyans between "deep" and "surface" forms. Indeed, it does mean we are ruling out the option of looking at either purely performance-related or purely intentional forms of listing. However, given both the extreme difficulty (impossibility even in many cases) in differentiating between types of verticalisation [START_REF] Tannen | Repetition in Conversation: Toward a Poetics of Talk[END_REF], and the overly "modelistic" idea of trace insertion in intentional lists (unrealistic in actual production terms), we consider it more useful here to group within the same paradigm all syntactically like elements, whatever the actual motivation behind their physical presence. Moreover, it must be understood that paradigms are not "closed" series. In other words, is not because we have moved on to a new paradigm that the previous one is necessarily finished: existing paradigms can be added to, or even split apart by the insertion of new ones as can be seen in example 2. Here, if we draw a line along the bottom of the grid, we obtain the "end product", in other words what the speaker is effectively saying, i.e. "je revois ce joli petit lit rose": je revois le ce ce petit lit ce joli petit lit rose

The study

We looked at the following aspects of oral text construction, all typically found in native French productions: hesitation (production repetitions and false starts), left expansion, right expansion, redefinition or "paradigm work", normative corrections.

(a) Hesitation

Under hesitation we traditionally find a whole range of phenomena, from unfilled pauses (silence) to repetitions, false starts, etc. We concentrate here on two aspects of "linguistic" filled hesitation, namely production repetitions (bribes or "répétitions faits de parole" -Henry & Pallaud 2004) and false starts (amorces). We look at these features not simply as performance errors but rather as basic elements of oral text construction. We found that both groups of learners used bribes and amorces in their productions. In both groups, these features were generally less present in the initial stages of the oral presentation (although this did vary from one individual to another) and became more commonplace as the presentation became more interactive and as questions began to be asked. As in native speaker productions, bribes typically affected grammatical words (see examples 3 and 4). Whereas bribes were generally limited to two successive forms in the first year productions (example 3), in the fourth year productions, these were often longer (example 4):

3. les les parents ne l'aiment pas (Jo) 4. c'est ceux qui qui qui avaient le le droit de de chercher de l'asile (Vikram) Amorces or false starts were again found in both groups. However, whereas these were generally limited to monosyllabic word-initial amorces in the first year productions (example 5), there were longer, polymorphemic amorces in the fourth year productions (example 6):

5. i-il manque maintenant l'i-l'imagination (Stuart) 6. on on pourrait on pourrait le décrire comme euh_ -l'adversaire euh français euh le plus médiatique (Debbie)

Typically, in the first year productions, verticalisation through combinations of bribes and amorces was associated with linguistic "stumbling" and uncertain terminology (example 7). This has been identified as a communication strategy by learners [START_REF] Eva | Reference to Past and Verticalization in Advanced NNSs Dialogues[END_REF] and typically results in the examiner completing the paradigm before the learner can move on: 7. Rhys: il était très réalis-[ri@list] 7 -Examiner: réaliste Rhys: réali-réaliste <parce> <Examiner: hm> que euh -c'est -euh c'est pros-/c'est, c'é-/ c'était possible This appeared to be less the case in the fourth year productions, where greater linguistic mastery appeared to enable learners to dig their own way out of paradigms. And when the examiners did intervene, this generally involved more phatic productions and less scaffolding. Typically, the fourth years spent less time silent pausing, preferring to keep production going, albeit vertically (see example 8). Indeed our data showed a marked difference between first and fourth year productions in terms of silent pause length and frequency: silent pauses were found to be 3 times more frequent in the productions of first years. Also, whereas roughly half of all silent pauses in the first year productions were of medium or long duration (i.e. over a second), in the fourth year productions medium and long pauses accounted for only 30% of all silent pauses. 8. je je je crois que les raisons pour laquelle on veut pas donner maintenant peu-peut-être dans à l'avenir mais maintenant euh les mêmes droits c'est_ est est -est à cause du du fait que pr-euh actuellement les les immigrés ou l-ou bien les immigrants euh ils ils s-ils euh saignent euh le euh_ l'Etat euh de euh de des des bénéfices donc ils travaillent ils travaillent ici peut-être au marché noir -ou ils travaillent sans au-sans autorisation (Vikram)

Finally, we would stress there was considerable intra-group variation in terms of hesitation (both silent and filled). The observations here, then, are fairly general, and do not claim to account for the fine detail. Further analysis of these features is required.

(b) Left expansion

Left expansion involves returning along the syntagmatic axis in order to insert new elements into the existing structure and thereby expand the utterance (see example 2 above). This is sometimes referred to as anticipation insofar as the first version effectively contains elements that arrive "too soon" in relation to the final message (though not necessarily in an unsyntactic way). Thus in the following example by a fourth year learner, we see the speaker coming back and prising apart the determiner and the adjective to fit in another element:

9. son mandat pendant les der-euh les cinq dernières années (Lisa) son mandat pendant les der-euh les cinq dernières années

Whereas there was some evidence of this in the productions by first year learners (see example 10), 10. personnellement je je euh préfère euh --pour les pour les enfants je préfère euh -jouets --qui aident euh l'é-l'éducation (Rhys) this was found to be used fairly extensively by the fourth years:

(c) Right expansion

With right expansion, the speaker, rather than going back along the syntagmatic axis to insert elements, expands by going on to the right: we see a verticalisation within the existing paradigm (but no left-hand movement) before the syntagmatic development takes place (see grid for example 15). Whereas this procedure was found in small doses in the fourth year productions (examples 15 and 16), there was little or no evidence of this in the first year productions 8 : In native spoken French, right expansion is less productive than left expansion and is generally associated with the formal context [START_REF] Claire | Le français parlé: études grammaticales[END_REF]).

(d) Redefinition or "paradigm work"

Both this feature and the following one (normative corrections) are interesting insofar as they demonstrate active use of verticalisation (i.e. "paradigm work") as a means of showing sociolinguistic awareness: the speaker, aware that a particular term or expression does not fit either the requirements of the given production or indeed those of the situation, verticalises in order to reinstate a "better" version. Here, then, the speaker uses the vertical dimension of speech to redefine, in a more "precise" manner (not necessarily in real terms), what he/she is saying. This is in fact one of two processes which should perhaps be mentioned together: working towards precision through revision and working away from precision through revision. Whereas in the first case we see the speaker giving general information at first before going on to nuance this (example 17), in the second case, we see the speaker giving first a precise term which in turn becomes "un-precisioned" (example 18), whether through failure to find a better, more precise term or as part of a strategy to "dumb down" the original term, generally due to situational constraints: 17. elle était débordée par comment dire par ses par ses grossesses par ses allaitements 18. il y avait euh un chef de groupe euh un pionnier euh un truc

In the L2 productions, whereas we found little evidence of redefinition through paradigm work by first years, we did find this with the fourth years: 8 As with left expansion (and indeed other features), there are cases in the first year productions which could be considered. However, given the particular nature of these (frequently including very long pausing, interventions by the other speaker, agrammaticality) it is not easy to decide on any clear-cut cases.

22. ils sont persé-euh ils sont soumis à la persécution (Vikram)

les nouvelles routes autoroutes qui relient les grandes villes … (Mark) (e) Normative corrections

The first feature we looked at here was evidence of correction (typically through verticalisation) not so much to bring about increased precision as to show awareness of sociolinguistic markedness of certain forms (this is not to do with correction of grammatical erroneous forms). In other words, where it was felt by the speaker that the element he/she had used was not in keeping with the register he/she was trying to maintain, so an attempt was made to rectify this. Again, this was found to be quasi-exclusive9 to the fourth year productions: The second feature we looked at was correction accompanied by apologies: recognising a "mistake" was made, stating it, rectifying it openly: 27. on recense chaque année euh de moins en moins euh de plus en plus excusez-moi de plus en plus … (Debbie) 28. les raisons … de l'échec de euh de pardon de la réussite … (Lisa)

les journaux et la télévision … ont beaucoup influencé -euh beaucoup parlé médiatisé pardon les … actes d'agression (Lisa)

This feature was found to be exclusive to the fourth year productions. The only apologies as such (indirect) in the first year productions were ones accounting for breakdown in communication.

Syntactic units: organisation and typology

Situation and genre constraints

Research on oral productions by native French speakers has shown that syntactic patterning is typically affected by the type of speech situation, or how this situation is experienced by the speaker (see [START_REF] Claire | Le français parlé: études grammaticales[END_REF]. It has been noted, for example, that productions which are prepared (with the use of notes or writing) prior to delivery, or even those which are not prepared as such but rather delivered under pressure, appear to favour a certain syntactic homogeneity: the speaker, by reiterating the same unit pattern, typically produces a "stacking" effect (see examples 30 and 31). The following examples, one from a lawyer's defence speech (prepared in advance), the other produced by a school child in the presence of the teacher (under pressure), show this: 30. il s'apprête à rentrer chez lui il succombe à la tentation de boire un dernier verre il se laisse alors aller à prendre sur sa gauche il songe alors à rentrer chez lui (ex Bilger 1990) 31. on fait des sorties on va à la piscine on prend le sac tyrolien on s'en va dans la forêt (ex Bilger 1990)

The study (a) Syntactic patterns

The reiteration of a similar pattern (S-V-O with some variation, notably in terms of the addition of other functional sequences 10 ) was found in both groups. This was, though, more consistently used by the first years: In the fourth year productions, on the other hand, the unit types were found to be more varied (this does not mean there was no simple S-V-O reiteration), with a wider range of functional sequences before and after the governing verb, and typically serving different techniques of presentation and cohesion:

33. ben -vache folle organismes euh génétiquement modifiés et listeria il se passe euh -guère une semaine en France -sans qu'un de ces faits euh ne fasse la une /euh, au/ au journal de vingt heures -on parle -on parle en fait de plus en plus euh --des problèmes euh de contamination et de euh d'intoxication en en Francealors euh peut-on manger tranquilleest-ce qu'il y a euh contamination alimentaire ou euh intoxication médiatique en France --j'ai choisi euh j'ai choisi ce sujet euh -en fait -car j'ai passé euh environ quatre cinq mois en Normandiealors euh l-la première semaine j'ai vu euh une émission à la télévision sur euh le salon de l'agriculture (Debbie)

Also, features such as left dislocation (examples 34-38) and pseudo-clefting (examples 39 and 40) were found in the fourth year productions:

(b) Unit expansions

Unit expansions introduced by parce que were found in both groups. In the first year productions, these often surfaced in the question-answer part of the examination, coinciding with the intervention of the examiner, pressing for more information:

41.

Examiner: comment ils réagissent Jo: les les parents ne l'aiment pas parce que -euh -parce que toutes les choses qui symbolisent euh l'enfance -euh sont -sont jetées euh trop tôt

In the fourth year productions, unit expansions (to the left or the right of the governing verb -compare examples 45 and 46) were found to be introduced by a greater variety of forms. Thus, whilst parce que unsurprisingly topped the list (64 occurrences -proportionally slightly more than in the first year productions), there were also examples of car (5), afin de (10), lorsque (3) and bien que (1). None of these latter forms was found in the first year productions:

42. ils ont décidé de ne pas aller aux urnes parce qu'ils pensent que les hommes politiques … (Lisa) 43. j'ai choisi ce sujet euh -en fait -car j'ai passé environ quatre cinq mois … (Debbie) 44. il a essayé de de changer son image -euh afin de d'augmenter sa respectabilité (Lisa) 45. lorsque je fais mes courses -je euh je me pose régulièrement euh des questions (Debbie) 46. les prix sont trop élevés bien qu'ils aient diminué un peu (Mark)

In native-speaker productions, prepared interventions also typically include expansion through the use of relative clauses, as is exemplified by this passage from a town councillor's speech:

47. je voudrais signaler que ce qui n'a pas été dit c'est que l'expérience semble être prévue pour huit mois -ce qui est une durée -euh disons relativement courte -par rapport à la somme investie -qui elle est quand même relativement élevée This type of expansion, present in the productions of both groups, was found to be only slightly more frequent in the fourth year productions: roughly one in every 120 words as opposed to one in 140 for the first years. Also, in both groups, it generally occurred to the right of the governing verb, though some examples were found of left context (subject sequence) expansion:

48. les parents --des enfants qui v-qui veulent -Anim'Animaux euh -n'attendaient pas pour Noël (Anna) 49. d'autres facteurs déterminants qui ont empêché Jospin de réussir euh -étaient par exemple la fragmentation de vote … (Lisa)

(c) Other stylistic features

In the fourth year productions, we found subjects placed after the verb as well as subject sharing. Both of these features have been recognised as "stylistic" (i.e. not commonplace in everyday language) traits in first language French:

Discussion: style in second language

Stylistic variation in second language (but also to a large extent in first language) is typically evaluated according to the use of certain formal variable criteria identified for this purpose (e.g. "classics" such as negative ne deletion or on versus nous) and studied in contrasting situations. Results generally show that the lower level learners are globally more "rigid" in their productions, i.e. not deleting ne, not using on for "we". However, this type of result, though useful, tells us very little about learners as actual speakers, i.e. as conveyers of thoughts, notions, feelings and desires through language. So, in our data, whilst it was not surprising to see a certain amount of "rigidity" in the first year productions within the same situation (first years tended to use more linear, "simple" units, though they did have bribes and amorces and show evidence of unit expansion and variation), it was interesting to note that the fourth years, without giving the impression of sounding "casual" or disrespectful of the formal situational constraints, made more use of overt construction techniques (knowing "how to hesitate") and showed a wider stylistic palette within the same formal setting. These two aspects would appear to work together, with the former helping the cause of the latter: for example, verticalisation of the relative pronoun qui allows the speaker to plan better the following expansion whilst keeping the production "alive"; left expansion is used to insert an extra functional sequence to the left of the verb, etc.

We would suggest, then, that second language speaking style is not simply a matter of applying a general filter or being able to strike up an appropriate "level" of language (identified according to certain salient features) and stick to it, nor is it simply a matter of knowing the range of different possible structures. Rather, it is a question of being able to put all of this to use in an appropriate but also an effective manner in actual speech. This has been identified in first language French productions: for example, [START_REF] Bilger Mireille | L'oral ou la multiplication des styles[END_REF] identify "style multiplication" as the speaker assumes different roles during the interaction. This has also been observed elsewhere in advanced learners' productions (e.g. Tyne 2004 on the use of colloquial versus standard lexical items).

Conclusion

From the small data set explored here, albeit in rather general terms, differences have emerged between first and second year learners of French. Whilst it is true that individuals vary considerably from one to another (we did not concentrate on intra-group discrepancies here), and that each learner has/her own personality and his/her own history of learning, it would appear, nonetheless, that certain group tendencies can be observed. Still, in further studies, we should not be surprised to find "good" first years behaving like fourth years in terms of text construction and vice versa.

The combined study of the "end product" and the various stages in its forming is, we think, an instructive way of looking at learners as actual speakers of the second language. Whereas the first year speakers preferred typically to remain silent for long periods of time, as though waiting for the finished product to be conceptually "ready" before speaking, the fourth years were more willing to keep the production going (not letting communication break down) and, like native speakers whose linguistic competence is not put into question, work the message openly into its "final" state. It is as if the more advanced learners, through their greater linguistic ability, can allow themselves to dispense with the primary microsyntactic preoccupation of "getting the grammar right" (which is ideologically seen to include no "performance errors"!), and move on to the important communicational task of expressing things in the manner they feel is appropriate for the given situation or moment in the situation. We could perhaps deal with this as a part of "communicative competence". Or we could consider it in more general terms within the notion of second language fluency.

We believe that mastery of the overt nature of text construction is an essential part of speaking style since it enables the speaker to edit and refine the oral text as a part of the social interaction. Spontaneous text production typically requires overt text construction techniques, even in very formal situations, in order to be able to develop and to gain in complexity and richness (there will obviously be intra-group variation, with some speakers being more "Mozartian" than others). This is required if language is to serve in an effective manner as the expression of the thoughts, desires and sociolinguistic awareness of the speaker, but also if communicative principles are to be respected and cooperation between speakers to take place. As suggested by W. Chafe in his 1979 article "The Flow of Thought and the Flow of Language", we could say that the relation between style and language structure is apparent in the way the oral text is worked into an "appropriate" form through successive spates of formulation and reformulation (it does not have to be the whole of the utterance that is reformulated). The very fact that speech is socially situated means that the "final" version, whatever it turns out to be, is largely determined by the particular nature of the social situation.

  11. ils m'ont dit que les routes spéc-euh les chiffres pour les routes spécifiques étaient classés (Mark) 12. la raison pour laquelle la principale raison pour laquelle … (Vikram) 13. si on a des droits différents entre les les pays les quinze pays euh européens … (Vikram) 14. parce que oui il y a des avec toutes les crises il y a il y a des personnes (Debbie)

  15. je vais parler de euh de des facteurs des facteurs déterminants qui ont mené … (Lisa) savais pas je savais pas trop euh quel sujet choisir (Debbie)

  19. c'est quelqu'un euh c'est un personnage qui s'alimente … (Debbie) 20. je vais parler de je vais aborder le euh l'échec de … (Lisa) 21. parce que les gens euh les les Anglais auront … (Mark)

  24. parce que c'est ce sont toutes des constructions … (Mark) 25. euh ça veut dire euh ce-cela veut dire que … (Vikram) 26. pour avoir une politique commune euh pour -afin de résoudre quelques uns des problèmes … (Vikram)

  choisi le texte s'appelle Chat alors -euh il s'agit d'un nouvelle technologie -/au N-, en/ Noël --deux mille un une grand compagnie -a introduit une nouvelle jouet ce jouet [s] -s'appelle ch-Anim'Animaux ---(Anna)

34.

  Lionel Jospin il a fait une mauvaise campagne (Lisa) 35. Chevènement il a beaucoup euh mené à l'échec (Lisa) 36. moi je je je suis pas sûr (Mark) 37. les immigrants euh ils ils s-ils euh saignent euh le euh_ l'Etat (Vikram) 38. les Français ils n'ont pas besoin … (Debbie) 39. ce ce que je dirai(s) en conclusion c'est que au au début euh au début euh je croyais vraiment qu'il y avait un problème … (Debbie) 40. ce qui est important c'est c'est euh de garder euh de respecter euh le patrimoine français (Debbie)

  50. j'ai analysé … la relation qu'entretiennent les Français avec euh -avec euh_ leur alimentation (Debbie) 51. il a poli son image et aussi a poli son discours (Lisa) 52. les Britanniques achètent des maisons de vacances et aussi émigrent en France (Mark) 53. ses projets ont amélioré la ligne TGV et ont réduit le euh la durée du trajet (Mark) Interestingly, the one example of subject postposition we found in the first year productions was a word-for-word citation from the article presented in the oral examination (example 54). Similarly, of the few examples of subject sharing, half were citations (examples 55 & 56).

  54. en mille --neuf cent -quatre-vingt quinze -euh -naît le Tamagotchi (Anna) 55. c-c'est une chat qui miaule ronronne frotte sa joue remue la queue -siffle et s'endort ---(Anna) 56. Rhys: c'était possible -pour le /j-, 0/ -le jouet de -euh --de euh ---SOUPIR ---euh de [Ro~Ro~] -et s'endort --et s-s'est en-s'endort -et -aussi de miaule---miaule----<euh> <Examiner: miauler> miauler oui

We are grateful to Ros Mitchell for her comments on a draft of this paper.

Groupe Aixois de Recherches en Syntaxe.

First years: Anna, Jo, Rhys, Stuart; Fourth years: Debbie, Lisa, Mark, Vikram. No attempt is made here to look at gender differences.

DEscription Linguistique Informatisée sur Corpus.

Although GARS is known for its work on spoken French, the analytical framework within which it works (Pronominal Approach -see[START_REF] Claire | Pronom et syntaxe: l'approche pronominale et son application au français[END_REF] is by no means specific to the study of spoken language. Indeed, one of the main aims of GARS's studies has been to show that spoken and written French are not two separate languages (see Blanche-Benveniste & Jeanjean 1987), as some would claim.

Examples which appear without specific reference are taken from[START_REF] Claire | Le français parlé: études grammaticales[END_REF]. All others are referenced, with details of either the published source or, in the case of our corpus, the learner's first name.

All phonetic transcriptions use SAMPA.

One example (après ça après cela) was found in the first year productions (Rhys).


Francois Mauger

Cristel Chandre

Turgay Uzer

Simulated annealing algorithm for

Keywords: Periodic orbits, Simulated annealing, Hamiltonian systems

published or not. The documents may come

Introduction

Periodic orbits play a central role in the description and analysis of dynamical systems as they represent the skeleton of the dynamics [START_REF] Cvitanović | Chaos: Classical and Quantum[END_REF]. It means that some important dynamical properties can be deduced from these orbits. By continuity of the flow a typical trajectory approaching a periodic orbit will mimic its dynamics. The time delay during which the trajectory is caught by the periodic orbit depends on the stability of the orbit: the more stable the periodic orbit, the longer the trajectory will stay in its neighborhood. As a result, the knowledge of the periodic structures and their stability properties enables one to predict the dynamical organization of the flow in their neighborhood. A fair amount of information is provided by the linear stability properties of these invariant structures. As an example, cycle expansions according to the length, stability or action of these orbits, are carried out to describe long time behavior such as averages of observables [START_REF] Cvitanović | Chaos: Classical and Quantum[END_REF][START_REF] Lan | Cycle expansions: From maps to turbulence[END_REF]. For chaotic systems, the symbolic dynamics describes a hierarchy between periodic orbits, and the resulting expansion converges exponentially or superexponentially with the cycle length. For autonomous Hamiltonian systems, the eigenvalues of the Jacobian matrix from which the linear stability properties are determined come in quadruplet (λ, 1/λ, λ * , 1/λ * ). In addition, there are at least two marginal eigenvalues corresponding to the time translation invariance (along the periodic orbit) and the energy conservation. These eigenvalues allow the classification of periodic orbits according to their linear stability properties. For instance, for Hamiltonian systems with two degrees of freedom, the periodic orbits can be sorted in three categories depending on their linear stability property: they are either elliptic (in general, linearly stable), hyperbolic (linearly unstable) or parabolic (linearly neutral). Elliptic periodic orbits are generally surrounded by an elliptic island inside which trajectories are trapped (and stay on invariant tori). In this region, the dynamics is mainly ruled by the central periodic orbit and the size of the island is determined by nonlinear stability properties. For hyperbolic periodic orbits, neighboring trajectories (linearly or locally) exponentially diverge in time. In general, these orbits are surrounded by a chaotic dynamics of stretching and folding since their stable and unstable manifolds intersect an infinite number of times to create a chaotic tangle.

Of course the influence of various kinds of periodic orbits strongly depends on the problem at hand. For atomic and molecular systems, the typical duration of a process is short compared to the other physical processes. This is particularly true for systems driven by short and intense laser pulses. Periodic orbits longer than this typical duration (or even of the same order) will not influence drastically the dynamics. On the contrary, periodic orbits much shorter than this duration will have a chance to trap the trajectory in its neighborhood and hence significantly affect the dynamical properties.

Various algorithms have been developed for finding periodic orbits. Among them, some are commonly used in atomic systems with a few number of electrons : Newton-Raphson algorithm or modified version of it such as the damped Newton-Raphson algorithm offers high speed convergence provided one has a sufficiently accurate initial guess for the periodic orbit. The precision of this initial guess is a thorny problem since the basin of attraction shrinks exponentially with the instability and the length of the orbit [START_REF] Davidchack | Efficient algorithm for detecting unstable periodic orbits in chaotic systems[END_REF][START_REF] Koh | Finding periodic orbits of higher-dimensional flows by including tangential components of trajectory motion[END_REF]. A solution to overcome this difficulty is to consider a parallelized version of the algorithm through a multishooting strategy [START_REF] Cvitanović | Chaos: Classical and Quantum[END_REF]. Independently of the chosen version, a crucial point in Newton-Raphson methods is the evaluation of the derivative of the trajectory which can be a delicate point from a numerical point of view. For a flow with dimension N , the tangent flow is given by the evolution of a N × N matrix which increases significantly the dimensionality of the problem at hand (even if in some cases, the structure of the problem may enable one to reduce the actual number of components). Some methods were set up to overcome this issue, and are generally based on relaxation algorithms [START_REF] Davidchack | Efficient algorithm for detecting unstable periodic orbits in chaotic systems[END_REF][START_REF] Biham | Characterization of unstable periodic orbits in chaotic attractors and repellers[END_REF][START_REF] Schmelcher | Detecting unstable periodic orbits of chaotic dynamical systems[END_REF][START_REF] Diakonos | Systematic computation of the least unstable periodic orbits in chaotic attractors[END_REF][START_REF] Davidchack | Towards complete detection of unstable periodic orbits in chaotic systems[END_REF]. Overall, the goal is to set up a deterministic dissipative dynamical system which has the periodic orbit as its attractor (with hopefully a wide basin of attraction). Using the same philosophy, variational methods have been designed to determine periodic orbits. For instance, a variational principle combined with a Newton descent [START_REF] Lan | Variational method for finding periodic orbits in a general flow[END_REF] consists in setting up a (dissipative) fictitious-time dynamics in a space of loops such that it drives a loop to a true periodic orbit of the considered dynamical system. Similarly some other algorithms are designed based on minimizing a function (or a functional). For instance one can consider a Newton-Gauss method by looking at minimization of the distance between the starting and final points of an orbit on a particular surface intersecting the periodic orbit: Global vanishing minima correspond to periodic orbits.

Independently of the chosen algorithm, the chance to see the algorithm converging strongly depends on the basin of attraction of the periodic orbit of the deterministic dynamical system. Often, there is a trade-off between the size of the basin of attraction with the speed of convergence of the algorithm. This is in particular the case for the Newton-Raphson and its damped version. In this article we propose a method which extends this basin of attraction by use of systematized trial and error converging procedure. In order to do that, we combine these deterministic algorithms with a Simulated Annealing (SA) algorithm to approximate the periodic orbits. In other words, the goal of this stochastic method is to enable the determination of initial guesses with sufficient accuracy to lay them into the basin of attraction of a fast converging algorithm like the Newton-Raphson algorithm. As a consequence of the underlying stochastic nature of the algorithm, it enables one to determine several different periodic orbits for the considered dynamical system by launching the algorithm several times. One of the advantages of the SA algorithm is that it does require neither the computation of the tangent flow nor a high accuracy in the integration of the trajectory. We show below that it makes this method a tool of choice for systematic search for periodic orbits in phase space.

To illustrate the feasibility of the SA algorithm for finding periodic orbits, we consider the classical helium atom in its ground state. The dynamics is modeled by the following Hamiltonian with soft Coulomb potentials [START_REF] Javanainen | Numerical simulations of multiphoton ionization and above-threshold electron spectra[END_REF]:

H (x 1 , x 2 , p 1 , p 2 ) = |p 1 | 2 2 + |p 2 | 2 2 - 2 |x 1 | 2 + a 2 - 2 |x 2 | 2 + a 2 + 1 |x 1 -x 2 | 2 + b 2 . ( 1
)
where x i is the position of the i-th electron (the nucleus being set at the origin), and p i is its canonically conjugate momentum. The norm |•| is the Euclidian norm. Depending on the dimensionality of the model, the vectors x i and p i can be considered in R, R 2 or R 3 . Tuning the softening parameters a and b enables to reproduce various atomic configurations. The parameter a is chosen as to reproduce the ground state energy (defined as the sum of the first and second ionization potentials) while preventing any self ionization of the atom. The parameter b is chosen as to allow significant energy exchange between the electrons when they are close to each other. For helium, one usually considers a = b = 1 and the ground state energy is E g = -2.24 a.u. [START_REF] Haan | Numerical study of autoionizing states in completely correlated two-electron systems[END_REF].

The dynamical organization corresponding to Hamiltonian (1) is crucial for understanding the (multiple) ionization processes of these atoms driven by an intense and short laser pulse. In Ref. [START_REF] Mauger | Strong field double ionization : The phase space perspective[END_REF] it was argued from a one dimensional model that the dynamics of Hamiltonian ( 1) is mainly organized by a reduced set of periodic orbits which naturally places one electron close to the nucleus and one further away. This distinction between the two electrons is crucial when understanding the action of an intense linearly polarized laser field when it is applied to the system : The electron far away from the nucleus is picked up by the field and hence quickly ionizes while the other electron remains bound to the nucleus. This distinction between the two electrons is at the origin of the classical interpretation of the recollision or the "'three-step"' model which is the keystone of strong-field physics [START_REF] Corkum | Plasma perspective on strong field multiphoton ionization[END_REF][START_REF] Schafer | Above threshold ionization beyond the high harmonic cutoff[END_REF][START_REF] Becker | Many-electron strong-field physics[END_REF]. The question we are addressing here is whether this emerging picture of an inner (close to the nucleus) and an outer electron (further away) still exists in two or three dimensions, or if it is just an artifact of the one dimensional model which is very peculiar given the presence of the nucleus. We show below that this dynamical picture still persists and and it does so because of the organization of short periodic orbits in phase space.

In Sec. 2, we describe the algorithm we use to determine periodic orbits. In Sec. 3, we analyze the lay-out of these periodic orbits in phase space and their implication for the dynamical organization of the helium atom in the ground state.

Simulated annealing algorithm to determine periodic orbits

The Simulated Annealing (SA) algorithm [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF] is a metaheuristic algorithm designed for optimization under constraints which generalizes the ideas developed from the Metropolis algorithm [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF]. The main idea of the algorithm is to automate a trial and error search for a better solution within a controlled neighborhood whose diameter is successively reduced. Given a dynamical system Ẋ = F (X), we first consider a Poincaré section Σ and the associated Poincaré map Φ :

Σ → Σ. If a trajectory cuts Σ in a finite set of points (X 1 , X 2 , . . . , X n ) such that Φ (X n ) = X 1 and Φ (X i ) = X i+1 , 1 ≤ i < n it
corresponds to a periodic orbit with period n on the section (in other words Φ n (X 1 ) = X 1 ). The periodic orbit search is obtained by minimizing the distance between the starting point and the n-th return on Σ using the SA algorithm. From a random initial guess X (0) ∈ Σ we build up a sequence X (k) k∈N ∈ Σ N which aims at minimizing the Euclidean distance between X (k) and Φ n X (k) . If the SA algorithm converges, the limit X (∞) of the sequence exists and fulfills the condition Φ n X (∞) = X (∞) , therefore corresponding to a periodic orbit (with period n on Σ). For the problem at hand, finding periodic orbits of Hamiltonian (1) in the ground state, an additional constraint to take care of during the SA algorithm is the energy of the system (which is constant for autonomous Hamiltonian systems). There are several ways of doing it : The first one is to ensure that the points X (k) remain on the energy surface after each step of the SA algorithm by an additional procedure which adjusts one coordinate. The second way is a specific prescription on the function to minimize which only ensures that the limit X (∞) .

For a given function f : Γ → R, where Γ is a subset of Σ, called cost function, the goal is to approximate the global minimum of f over Γ. First a random initial condition X (0) ∈ Γ is chosen to initiate the SA algorithm and we define two parameters T 0 , the initial temperature, and κ ∈ R + which serves as error tolerance.

The SA algorithm proceeds in successive stages of annealing and cooling steps.

• During the annealing process, a random initial guess X (1) is chosen in the neighborhood of X (0) in Γ (the size of the neighborhood is controlled by T 0 ; for simplicity one usually chooses a ball with diameter T 0 ). It corresponds to a melting of the system with a controlled temperature T 0 . If f X (1) < f X (0) then X (1) replaces X (0) in the following step. If not, X (1) replaces X (0) with a probability exp(-(f (X (1) )f (X (0) ))/κT 0 ). Otherwise, the same X (0) is kept in the next step. The process of melting for a fixed temperature is iterated until the system reaches a steady state.

• where user-defined constants are labeled by # and % starts comments. The function ReturnDist plays the role of the function f as defined by Eq. ( 2). Finally, the function rand is a random generator uniformly distributed over [0, 1]. The melting process ( * ) is draffed in its simplest way above and it has to be followed by a projection on Γ, which has to be chosen accordingly to the problem at hand. In our case, after the melting ( * ), the new initial condition is neither on Σ nor on the ground state energy. The projection on Σ is simply done by setting x 1 = 0. The projection on the ground state is done by changing p x,1 accordingly to the energy constraint and we select the positive solution for p x,1 in order to be sure to remain on the Poincaré section. We also define a threshold distance under which the algorithm is considered as converged.

The projection of the perturbed position on the ground state energy surface ensures that the series of guesses stay on the energy surface at each iteration. However, one can also see the ground state energy as an additional constraint to minimize together with the n-return distance. In this case, one adds the energy error to the cost function

f : X → |X -Φ n (X)| + γ |H (X) -E g | , (3)
where γ > 0 is a constant. Then, after each perturbation, we only project the new guess on the Poincaré surface Σ. We have successfully implemented both methods for n ∈ {1, 2, 3}. All numerical results displayed in this paper and the values given for the parameters correspond to the projection (on both the Poincaré and ground state energy surfaces) method. Many improvements of the algorithm can be implemented depending on the problem at hand. For instance, there is no requirement for choosing the initial condition randomly. If one has an approximate guess for the location of a periodic orbit, or any subset of phase space where it is included (e.g., given the symmetries of the problem), it is more suitable to start the SA algorithm with this particular point. In such a case, it can be interesting to set the initial temperature cooler than for a random initial guess.

For improving the convergence, it is more efficient to combine the SA algorithm with a second method based on a Newton-Raphson method for instance. Here we use the function fsolve of Matlab, which is a Newton-Gauss minimizing scheme.

Application to two electron atoms with soft Coulomb potentials

For two spatial dimensions the complexity of the system is increased compared to one dimensional models since the number of degrees of freedom is increased from 2 to 4. In addition, for two spatial dimensions, the system has a continuous symmetry which corresponds to a global invariance by rotation of the atom in phase space. The associated conserved quantity is the total angular momentum x 1 × p 1 + x 2 × p 2 , where × is the cross product in R 2 : x × p = xp yyp x . It means that any periodic orbit generates an infinite family of periodic orbits deduced from this single one by a rotation of the atom in phase space. In addition, the system also has some discrete symmetries (the exchange of the role of the two electrons for instance). To simplify the analysis, in what follows, we identify as a single one all periodic orbits which can be deduced by one of those symmetries from another periodic orbit. We notice that these periodic orbits have the same period and the same linear stability properties. In the numerical implementation, we consider that a periodic orbit has converged when the distance in phase space between the starting and final points on the Poincaré section is smaller than 10 -10 . We consider a periodic orbit to be a new one when the difference in period or in the spectrum of the monodromy matrix is larger than 10 -2 from any already known periodic orbit in the list.

For actual use of the SA algorithm we set the initial temperature around 5. A tolerance parameter κ = 0 + (a perturbation is accepted if an only if it reduces the cost function) is enough for our problem when we perform the projection on the ground state energy surface. It should be noted that if we add the energy error to the cost function [see Eq. ( 3)], κ ≈ 5 × 10 -2 is a better choice. For the cooling we set the temperature ratio to α = 0.75. Finally, we iterate the melting step a fixed number of times (3000 melting steps during the annealing process for each temperature). We stop the cooling when the temperature has reached a threshold value (here the threshold temperature is 10 -5 ). We also define a threshold d crit for which the algorithm is stopped as soon as the current guess X satisfies f (X) < d crit , and here d crit = 10 -3 .

For actual use of the SA, there are two main strategies when choosing the temperature ratio and the number of melting steps. One strategy is to choose a fast cooling law (i.e. α small) and a large number of melting steps for each annealing. Another strategy is to consider a slow cooling rate (i.e. α ≈ 0.95) and a much reduced number of melting per annealing iteration. For the problem at hand, we have used the first strategy when considering a projection on the ground state energy surface, and we have used the second strategy when adding the energy error to the cost function.

The total angular momentum is a conserved quantity of the dynamics and to model a configuration closer to the quantum ground state, one may wish to impose it to be zero. That becomes a new constraint for our system and it decreases the actual dimensionality of the problem at hand, but still it is worth looking for periodic orbits which may organize this particular subclass of the dynamics. To do so, there are two straightforward ways to adapt the algorithm presented here. The first one consists in adding the total angular momentum to the cost function (in the same way as for the energy constraint)

f : X → |X -Φ n (X)| + γ |x 1 × p 1 + x 2 × p 2 | ,
where γ is constant. The total angular momentum is then considered as an additional constraint and the SA will try to minimize it together with the n-th return distance function. Another solution is to project, after each melting iteration, the guess on the intersection of the ground state energy and zero total angular momentum surfaces. We have successfully adapted our algorithm to find one dimensional, three dimensional and zero total angular momentum two dimensional (considering the projection alternative) periodic orbits.

Since we are interested in finding as many periodic orbits as possible up to a finite period due to the finite duration of laser pulses in experiments, we launch a large number of times the algorithm, to explore the whole accessible phase space. We do it for periodic orbits of Hamiltonian (1) which have up to three intersections on the Poincaré section. Because of the rotational invariance of the system, the monodromy matrix associated with a periodic orbit has an additional pair of eigenvalues 1 (in addition to the pair associated with time-translation along the orbit). As a result, for two spatial dimensions, four eigenvalues are equal to one in the monodromy matrix and the four others determine the linear stability properties. Consequently the system can exhibit nine kinds of linear stability, (elliptic, parabolic or hyperbolic)-(elliptic, parabolic or hyperbolic). However, for numerical computations, is it difficult to distinguish precisely the parabolic features, e.g., between a parabolic and an elliptic or a weakly hyperbolic periodic orbit. When checking the linear stability of the periodic orbits we have found, we arbitrarily group together parabolic and elliptic features (it is worth noting that we have found no fully parabolic periodic orbit).

For the case of two spatial dimensions for Hamiltonian (1), we have found a total of 155 periodic orbits with period smaller than 65 a.u. whose repartition is the following one : 43 periodic orbits with period one, 74 periodic orbits with period two (which cannot be reduced to period one, that is to say, prime periodic orbits) and 38 periodic orbits with period three. Regarding their stability properties, 22 are elliptic-elliptic or parabolic-elliptic, 71 are elliptic-hyperbolic or parabolic-hyperbolic and 62 are hyperbolic-hyperbolic. In Fig. 1, we display the period T as a function of the modulus of the largest eigenvalue λ max of the monodromy matrix for each periodic orbit determined by the algorithm. By no means are these periodic orbits the only ones in phase space. However, these are the ones obtained by launching a large number of times the SA algorithm (with the restrictions mentioned above). Period (a.u.)

Figure 1: Period T versus the norm of the largest eigenvalue λmax of the monodromy matrix for the detected periodic for Hamiltonian [START_REF] Cvitanović | Chaos: Classical and Quantum[END_REF] with two spatial dimensions. Elliptic-elliptic and parabolic-elliptic periodic orbits are labeled with red crosses, elliptic-hyperbolic and parabolic-hyperbolic ones with cyan squares and hyperbolic-hyperbolic ones with blue circles. Hyperbolic periodic orbits into the dashed black rectangle are depicted in Fig. 2.

There are five periodic orbits which are weakly hyperbolic and with relatively low period, and which are likely to organize the dynamics of Hamiltonian (1) (see dashed black rectangle in Fig. 1). This is so because all of those periodic orbits have a small period (around 15 a.u.) which is consistent with the short duration of laser pulses (typically of a few hundreds of a.u. which correspond to few tens of fs) and are weakly hyperbolic. These five hyperbolic periodic orbits are well separated, in the (λ max , T ) diagram, from the other periodic orbits which are significantly longer and more hyperbolic. We display these five periodic orbits in the configuration space (x, y) in Fig. 2. We notice that all of them are composed of one electron close to the nucleus and another further away. A closer inspection at Fig. 1 also shows a group of four weakly hyperbolic periodic orbits with a larger period (around 30 a.u.). Looking at their shape in phase space (not shown here) they are also composed of one electron close to the nucleus and the other one further away. It means that for a typical trajectory of Hamiltonian (1) under the influence of these periodic orbits it is possible to identify, at any time, an inner (close to the nucleus) and an outer (further away) electron with possible exchanges of the roles of the two electrons as the trajectory visits different areas of influence. The observation that all small period and weakly hyperbolic periodic orbits are clearly composed of an inner and an outer electrons, supports what we have shown for the model with one spatial dimension where we identified four dominant periodic orbits in the organization of phase space (which actually reduce to only one orbit by symmetry) [START_REF] Mauger | Strong field double ionization : The phase space perspective[END_REF][START_REF] Mauger | Strong field double ionization : what is under the 'knee' ?[END_REF]. However, for models with two spatial dimensions, the variety of periodic orbits and the invariance by rotation make it difficult to identify a reduced number of dominant orbits in the skeleton of the dynamics. In order to show that a typical trajectory has an inner and an outer electron, we consider two distances in phase space as functions of time : one from the nucleus and the other one from the boundary of the admissible phase shape for a typical trajectory of Hamiltonian (1) as shown in Fig. 3. At any time an inner and an outer electron are identified with quick and frequent exchanges of their roles : The inner electron is close to the nucleus and thus have a short distance from it while the outer one, further away will be closer to the edge of the accessible phase space. It is worth noticing that due to the exchanges of the roles of the two electrons, the invariance by rotation and the dimensionality of the problem, a projection on the configuration plane does not clearly show any significant organization of the dynamics, contrary to the case with one spatial dimension which clearly shows the specific role played by mainly one periodic orbit (and its symmetric orbits).

The same study can be carried out in three spatial dimensions. The numerical results suggest that in most of the periodic orbits, the two electrons are co-planar or nearly co-planar (but not all). One of them is depicted in Fig. 4. These periodic orbits display a similar organization with one electron close to the nucleus and one further away. As a consequence, from a systematic inspection of the periodic orbits in phase space, we are able to draw a picture of the dynamics consisting of one electron close to the nucleus and another one further away (with fast exchanges between the two), confirming a result obtained with a reduced system with only one spatial dimension. [START_REF] Cvitanović | Chaos: Classical and Quantum[END_REF] with two spatial dimensions for a = b = 1, for a typical trajectory over the admissible phase space. On both panels, each electron is labeled by a constant color (red or blue). Left panel: Distance, in phase space, of each electron from the nucleus (continuous curves) and from the edge of the accessible phase space (dashed curves). Right panel: Projection of this trajectory in configuration space (x, y).

  The cooling consists in decreasing the temperature to a new temperature T 1 < T 0 , realized at the end of the annealing process. Then a new iteration of annealing with the updated temperature T 1 , is

Figure 2 :

 2 Figure2: Projection in the two dimensional configuration plane (x, y) of the five hyperbolic periodic orbits inside the dashed black rectangle in Fig.1. Periodic orbits are sorted in increasing norm of the largest eigenvalue of the monodromy matrix, from left to right and up to down. Each electron is labeled with a different color (blue or red).

Figure 3 :

 3 Figure 3: Typical trajectory for Hamiltonian[START_REF] Cvitanović | Chaos: Classical and Quantum[END_REF] with two spatial dimensions for a = b = 1, for a typical trajectory over the admissible phase space. On both panels, each electron is labeled by a constant color (red or blue). Left panel: Distance, in phase space, of each electron from the nucleus (continuous curves) and from the edge of the accessible phase space (dashed curves). Right panel: Projection of this trajectory in configuration space (x, y).

Figure 4 :

 4 Figure 4: Projections on the (x, y) plane (left panel) and (x, z) plane (right panel) of a periodic orbit of Hamiltonian (1) with three spatial dimensions.

Acknowledgments C.C. and F.M. acknowledge financial support from the PICS program of the CNRS. This work is partially funded by NSF.

performed.

The successions of annealing and cooling steps are iterated until the system reaches a global steady state which is the resulting approximation of the global minimum. Since the size of the neighborhood where the perturbation is performed is controlled by the temperature, it has to shrink with iterations of cooling. At the limit for zero temperature, the neighborhood of control is reduced to a single point.

The constant κ is used to avoid the algorithm to be trapped in local minima. It has to be chosen carefully as two critical situations may arise:

• If κ is too large, the algorithm will not converge since too many "bad" perturbations are allowed.

• If κ is too small, one runs the risk to be trapped in local minima.

The cooling law also has to be chosen carefully to allow the algorithm to sufficiently explore the region around the initial guess. If the cooling is too fast, the system is frozen in its current position and the algorithm may not converge. If the cooling is too slow, the algorithm spends a lot of time in useless explorations which slows down the process. Usually, an exponential decrease is considered for the cooling by multiplying the current temperature with a constant, i.e. T n+1 = αT n where 0 < α < 1.

Recall that our goal is to find periodic orbits for Hamiltonian [START_REF] Cvitanović | Chaos: Classical and Quantum[END_REF] with the constraint to be on the ground state, and to do that, we are looking for n periodic points on a Poincaré surface. For instance, we choose the Poincaré surface Σ of equation x 1 = 0 with ẋ1 > 0, where x 1 is the first component of x in the canonical basis (other surfaces which intersect the ground state energy surface might also be suitable). For convenience, we denote X = (x 1 , x 2 , p 1 , p 2 ) the vector position in phase space. Our problem is to find a periodic point X (∞) on the Poincaré (and the ground state energy) surface such that X (∞) = Φ n X (∞) . We define the cost function as the n-th return distance function

(

such that periodic orbits with period n correspond to positions where the global minimum of the function f is reached and thus may be investigated using the SA algorithm. This function is defined on Γ which is the intersection of the ground state energy surface {X s.t. H (X) = E g } with the Poincaré section Σ. Schematically, the algorithm is written as