Igor Veselovskii

Qiaoyun Hu

Philippe Goloub

Thierry Podvin

Marie Choël

Nicolas Visez

Mikhail Korenskiy

Mie-Raman-Fluorescence lidar observations of aerosols during pollen season in the north of France

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Pollen grains represent a significant fraction of primary biological particles emitted from the biosphere into the atmosphere in certain seasons and locations [START_REF] Fröhlich-Nowoisky | Bioaerosols in the Earth system: Climate, health, and ecosystem interactions[END_REF].

There has been a growing interest in pollen study in recent years, because they can affect human health by causing allergy-related diseases and contribute to the cloud formation by acting as giant cloud condensation nuclei (CCN) [START_REF] Diehl | The ice nucleating ability of pollen: Part I: Laboratory studies in deposition and condensation freezing modes[END_REF][START_REF] Pope | Pollen grains are efficient cloud condensation nuclei[END_REF][START_REF] D'amato | Climate change and air pollution: Effects on pollen allergy and other allergic respiratory diseases[END_REF][START_REF] Steiner | Pollen as atmospheric cloud condensation nuclei[END_REF][START_REF] Lake | Climate Change and Future Pollen Allergy in Europe[END_REF][START_REF] Mack | Respiratory health effects of exposure to ambient particulate matter and bioaerosols[END_REF]. To investigate the processes of pollen transport and dispersion, the information about vertical distribution of pollen grains is needed, and this information can be obtained from lidar measurements. Pollen grains are large irregularly shaped particles of complicated morphology [START_REF] Frenguelli | Pollen structure and morphology[END_REF], causing strong depolarization of the backscattered laser radiation, which provides a basis for their identification. The first profiling of pollen with depolarization lidar was reported by [START_REF] Sassen | Boreal tree pollen sensed by polarization lidar: Depolarizing biogenic chaff[END_REF][START_REF] Sassen | Elliptical pollen corona from North American boreal paper birch trees (Betula papyrifera): strong fall orientations for near-spherical particles[END_REF]. His measurements over Alaska revealed that linear depolarization ratio of birch pollen plumes at 0.694 nm can exceed 30%. Further studies of pollen with elastic backscatter lidar at 532 nm were reported by Noh et al. (2013 a,b) and by Sicard et al. (2016). Their measurements confirmed high depolarization ratio of pollen grains (particle depolarization ratios as high as 43% were observed for aerosol mixture containing Platanus and Pinus pollen). Moreover, pollen grains backscattering demonstrated strong diurnal cycle, being highest near the noon. The use of multiwavelength observations increases capability of lidar technique for aerosol characterization. In recent studies of [START_REF] Bohlmann | Detection and characterization of birch pollen in the atmosphere using multiwavelength Raman lidar in Finland[END_REF] and [START_REF] Shang | CC BY 4.0 License. Raman polarization lidar in Finland: characterization of pure pollen types[END_REF] measurements performed with Polly XT lidar allowed to estimate mean values of the lidar ratios (about 45 sr and 55 sr at 355 and 532 nm respectively for birch pollen grains). The decrease of extinction and backscattering Angstrom exponents (EAE and BAE) during pollen episodes was also reported.

Atmospheric biological particles efficiently produce wideband fluorescence emission, when being exposed to UV radiation [START_REF] Pohlker | Autofluorescence of atmospheric bioaerosols-fluorescent biomolecules and potential interferences[END_REF][START_REF] Pan | Detection and characterization of biological and other organic-carbon aerosol particles in atmosphere using fluorescence[END_REF][START_REF] Miyakawa | Ground-based measurement of fluorescent aerosol particles in Tokyo in the spring of 2013: potential impacts of nonbiological materials on autofluorescence measurements of airborne particles[END_REF], which offers an opportunity for monitoring them with fluorescence lidars. Nowadays, lidar spectrometers based on multianode photomultipliers allow a simultaneous detection of fluorescence backscattering in 32 spectral bins [START_REF] Sugimoto | Fluorescence from atmospheric aerosols observed with a multichannel lidar spectrometer[END_REF][START_REF] Reichardt | Cloud and aerosol spectroscopy with Raman lidar[END_REF][START_REF] Reichardt | Fluorescing aerosols and clouds: investigations of coexistence[END_REF][START_REF] Saito | Remote detection of the fluorescence spectrum of natural pollens floating in the atmosphere using a laserinduced-fluorescence spectrum (LIFS) lidar[END_REF]. In particular, such lidar spectrometer was used in recent work of [START_REF] Saito | Remote detection of the fluorescence spectrum of natural pollens floating in the atmosphere using a laserinduced-fluorescence spectrum (LIFS) lidar[END_REF] for remote measurement of the fluorescence spectrum of atmospheric pollen grains. The results demonstrate that, for 355 nm stimulating wavelength, the fluorescence spectra of different pollen grains have maxima in the 400-600 nm range and the intensity peak at around 460 nm.

To achieve the highest sensitivity of fluorescence detection, in many tasks it is preferable to use a single channel monitoring, where a part of the fluorescence spectrum is selected with a wideband interference filter [START_REF] Immler | Fluorescence from atmospheric aerosol detected by a lidar indicates biogenic particles in the lower stratosphere[END_REF][START_REF] Rao | Preliminary measurements of fluorescent aerosol number concentrations using a laser-induced fluorescence lidar[END_REF][START_REF] Li | Fluorescent aerosol observation in the lower atmosphere with an integrated fluorescence-Mie lidar[END_REF]. In our recent 3 publication (Veselovskii et al., 2020) we reported the results obtained from a modified Mie-Raman lidar (LILAS lidar system in Laboratoire d'optique Atmosphérique) with one additional fluorescence channel at 466 nm. Such an approach has proved high sensitivity, allowing to detect fluorescence signals from weak aerosol layers and to calculate the fluorescence backscattering coefficient from the ratio of fluorescence and nitrogen Raman backscatters, thus making it potentially attractive for pollen monitoring.

In the present research we combine capability of multiwavelength Mie-Raman lidar for providing three backscattering, two aerosol extinction coefficients and linear depolarization ratio at three wavelengths with single channel fluorescence measurements for characterization of aerosol mixtures containing pollen grains. The measurements reported were performed during March-June 2020 period at the Lille Atmospheric Observation Platform (https://www-loa.univ-lille1.fr/observations/plateformes.html?p=apropos) hosted by Laboratoire d'Optique

Atmospherique, University of Lille, Hauts-de-France region.

Instrumentation

Mie-Raman-Fluorescence lidar

The measurements were performed using LILAS lidar systema multiwavelength Mie-Raman lidar, based on a tripled Nd:YAG laser with a 20 Hz repetition rate and pulse energy of 70 mJ at 355 nm. The backscattered light is collected by a 40 cm aperture Newtonian telescope. The full geometrical overlap of the laser beam and the telescope FOV is achieved at approximately 1000 m and to obtain the information about particles at lower altitudes, part of the measurements were performed at an angle of 30 degrees to the horizon. The system is designed for a simultaneous detection of elastic and Raman backscattering, allowing the so called 3β+2α+3 data configuration, including three particle backscattering (β355, β532, β1064), two extinction (α355, α532) coefficients along with three particle depolarization ratios (355, 532, 1064). The particle depolarization ratio , determined as a ratio of cross-and co-polarized components of the particle backscattering coefficient, was calculated and calibrated the same way as described in [START_REF] Freudenthaler | Depolarization ratio profiling at severalwavelengths in pure Saharan dust during SAMUM[END_REF].

The description of the system can be found in the recent publication of [START_REF] Hu | CC BY 4.0 License. transported Canadian smoke plumes in the lower stratosphere over northern France[END_REF].

To perform fluorescence lidar measurements, the water vapor Raman channel at 408 nm was replaced by a fluorescence channel, whose spectrum is captured by a wideband filter centered at 466 nm and of 44 nm width (Veselovskii et al., 2020). The fluorescence measurements were was derived by the Klett method [START_REF] Klett | Lidar inversion with variable backscatter/extinction ratios[END_REF]. The fluorescence backscattering coefficient F is calculated from the ratio of fluorescence and nitrogen Raman backscattering, as described in Veselovskii et al. (2020). This approach allows to evaluate the absolute values of F, if the relative sensitivity of the channels is calibrated and the nitrogen Raman scattering cross section is known.

Corresponding uncertainty we estimate to be below 50%. Parameters of detectors were not changed during the campaign, so uncertainty of relative variations of F was significantly lower and was determined by the statistical errors of fluorescence measurements.

To characterize the efficiency of the fluorescence in respect to elastic scattering, the fluorescence capacity 532

F F G    (1)
is also used. This parameter depends on the relative humidity (RH), so information about RH is important for data analysis. Radiosonde measurements are used to monitor water vapor, as the water vapor channel is replaced by the fluorescence channel in current lidar configuration. The closest available radiosonde data are from Herstmonceux (UK) and Beauvecchain (Belgium) stations, located 160 km and 80 km away from the observation site respectively. These radiosonde data are not collocated with the lidar measurements, so only qualitative analysis of humidification effects was possible.

Pollen in situ sampling

Airborne pollen grains and spores were collected by a Hirst-type volumetric sampler (VPPS 2000, Lanzoni s.r.l). The pollen sampler was located on the campus of the University of Lille (France) on the rooftop of a 20-m-high building where the lidar instrument was operated.

Ambient air was sampled at 10 L.min -1 flow rate, allowing the impaction of pollen and spores on an adhesive strip mounted on a rotating clockwork-driven drum. The impaction surface moves at 2 mm.h -1 behind the entrance slit, allowing a temporal resolution of 2 hours. The adhesive strip was substituted every 7 days after a full rotation of the drum, which is splitted into 7 parts, each corresponding to a day of monitoring. And then they are fixed on a microscope glass slide with gelatin and fuchsine dye. Pollen taxa were identified by light microscopy on the basis of their characteristic shape and size. Airborne pollen concentrations were expressed as a daily and dual hourly number of pollen grains per cubic meter of air.

Fig. 1 shows the most abundant pollen taxa for the period from March to June 2020 in Lille.

These include: Betula (54.8% of total pollen taxa over the period), Fraxinus (8.2%), Quercus

Discrimination of pollen from other types of aerosol

Specific features of pollen containing aerosol mixture

In contrast to the observations performed over Alaska [START_REF] Sassen | Boreal tree pollen sensed by polarization lidar: Depolarizing biogenic chaff[END_REF][START_REF] Sassen | Elliptical pollen corona from North American boreal paper birch trees (Betula papyrifera): strong fall orientations for near-spherical particles[END_REF] or Finland (Bohlmann et al., 2011), where pollen concentration was high due to boreal forests surrounding, the pollen loading in the north of France is significantly lower. Long-term lidar and sun photometer observations performed at Lille University demonstrate that local aerosol is mainly of continental type, with predominance of the fine mode particles and low depolarization ratio. The emission of large pollen grains, should lead to strong spectral dependence of the depolarization ratio, because the backscattering at 1064 is less sensitive to the fine background particles than at shorter wavelengths, thus particle depolarization ratio at 1064 nm (1064) should be more sensitive to the presence of pollen grains, compared to 355 and 532. The particle depolarization ratio  of the mixture, containing background aerosol (b) and pollen (p), with corresponding depolarization ratios  b and  p , can be calculated as:

                              (2)
Here total backscattering b p    .

To estimate the dependence of depolarization of the aerosol mixture on the contribution of pollen to the backscattering coefficient 532 532 p   at 532 nm, a simplified simulation was performed.

Assuming that the depolarization ratios of pollen and background aerosol are spectrally independent and that  p =30% while  b =3%, the mixture depolarization ratios 355, 532, 1064 were calculated as a function of and for most of the samples 1064 exceeded 355. However, we should keep in mind that measurements in the chamber were performed at low RH, and depolarization ratios at higher RH may be different. Still, from Fig. 2 we conclude that the increase of the particle depolarization ratio with wavelength can be an indication of the presence of large, irregularly-shaped pollen grains.

We should recall that similar spectral dependence can be provided also by the dust particles in the aerosol mixture. However, as it will be shown later, pollen possesses significantly higher fluorescence capacity and this is how these particles can be discriminated from dust.

The presence of pollen should lead also to the decrease of the extinction and backscattering Angstrom exponents. EAE depends mainly on particle size, while BAE is sensitive also to the particle complex refractive index and shape, thus the measured profiles of EAE and BAE can present significant difference [START_REF] Veselovskii | Characterization of forest fire smoke event near Washington, D.C. in Summer 2013 with multi-wavelength lidar[END_REF]. In our study we analyze the EAE and BAE

Characteristics of background aerosol over observation site.

Long-term observations in Lille University demonstrate that aerosol over the observation site is mainly of continental type with predominance of the fine mode particles. Typical vertical profiles of the background aerosol parameters, observed on 3 June 2020, are given in Fig. 3, showing aerosol elastic and fluorescence backscattering coefficients, lidar ratios, Angstrom exponents and depolarization ratios at three wavelengths. The RH from Beauvecchain (Belgium) radiosonde observations, was below 50% in the height range considered. Particle depolarization ratios at all three wavelengths are below 7%, indicating that contribution of pollen to the total backscattering is insignificant. This agrees with the low values of pollen concentration provided by in situ measurements (Fig. 1a). The lidar ratios at both wavelengths (S355, S532) are close, varying in the 50-60 sr range, and the fluorescence capacity GF is below 0.35×10 -4 . The EAE and BAE

( 355/532 A  , 355/532
A  ) are in the 1.5-2.0 range. The presence of pollen should lead to a deviation of the particle intensive parameters, such as the fluorescence capacity, depolarization ratio, EAE and BAE, from the typical values of background aerosol.

Identification of the smoke particles

During the observation period the smoke elevated layers transported over Atlantic were frequently detected. Smoke particles, are characterized by high fluorescence cross section [START_REF] Reichardt | Fluorescing aerosols and clouds: investigations of coexistence[END_REF]Veselovskii et al., 2020) A  is another feature that will be used for aged smoke discrimination. Smoke fluorescence capacity is high, reaching up to GF=5*10 -4 for the period of observations, and this is one more feature, allowing to separate smoke from other types of aerosol.

Identification of the dust particles

Presence of dust particles and pollen in the fine background aerosol leads to some common characteristics in the lidar data, such as the decrease Angstrom exponents and increase of depolarization ratios. However, pollen and dust can be separated by the fluorescence capacity. The vertical profiles of particle parameters during dust episode on 27 May are shown in Fig. 6. The dust containing layer extends from 2000 m to 7000 m and the particle depolarization ratios 1064 and 532 in this layer are close to 20%. These values are lower than depolarization of pure dust. For example, [START_REF] Freudenthaler | Depolarization ratio profiling at severalwavelengths in pure Saharan dust during SAMUM[END_REF] for pure dust provide the values of 27% and 31% at 1064 nm and 532 nm wavelengths respectively, thus in our case transported dust particles may be mixed with local aerosols. The particle depolarization at 355 nm is not shown in the figure, because the scattering ratio in the dust layer was too low to compute 355 reliably. The fluorescence capacity of particles in the dust layer is about 0.1×10 -4 at 4000 m, which is factor 50 lower than GF of the smoke in Fig. 5. There is also a weak aerosol layer at 1600 m with 532 about 0.035 Mm -1 sr -1 . The fluorescence capacity in this layer is high (GF2.0*10 -4 ), suggesting that this layer may contain smoke or pollen particles.

Impact of particle hygroscopic growth

The vertical variation of observed aerosol properties may be a result of particle water uptake, which should be separated from the features related to pollen presence. Fig. 7 shows the profiles of the particle parameters for the episode on 15 June 2020, when the aerosol hygroscopic growth could take place. In the height range 900-1500 m the fluorescence backscattering F is stable, while the elastic backscattering 532 increases by a factor 3. Radiosonde measurements in Herstmonceux (UK) in this height range demonstrate an increase of RH from about 75% to 85%, while lidar measured extinction and backscattering Angstrom exponents decrease from 1.5 to 1.3, corroborating the presence the particle hygroscopic growth. The depolarization ratio 1064 at low altitudes exceeds 355 and 532, which can be an indication of pollen presence. This is supported by significant fluorescence capacity (GF=0.910 -4 at 750 m).

The number of fluorescent particles in the 900-1500 m range, does not present significant changes (F is stable), so observed vertical variations, i.e. the decrease of depolarization ratios at all three wavelengths and the increase of lidar ratios S355 and S532 from 50 sr to 65 sr, are probably the result of water uptake by the particles. Water uptake does not change the number of fluorescent molecules, however the fluorescence capacity decreases in the process of the hygroscopic growth, so GF can be a representative parameter of aerosol types only at the condition of low RH.

Results of lidar measurements in the presence of pollen

During March-June 2020, we had numerous measurement cases demonstrating the features in the profiles of the particle parameters, that can be attributed to pollen. The EAE for both nights is about 2.0 and does not show significant changes with height.

The BAE is lower (about 1.5 at 1000 m) and for both nights it shows some increase in 1250-2250 m range. The BAE, in contrast to EAE, depends strongly on the particle refractive index and shape, thus it may demonstrate higher sensitivity to the changes in aerosol mixture composition. Recall that backscattering and extinction Angstrom exponents are related as: A A    (Fig. 5).

If the depolarization ratios of pollen  p and background aerosol  b are known, the pollen backscattering coefficient p  can be calculated. Such approach is widely used for the separation of contributions of dust and smoke particles [START_REF] Sugimoto | Characteristics of dust aerosols inferred from lidar depolarization measurements at two wavelength[END_REF][START_REF] Tesche | Vertically resolved separation of dust and smoke over Cape Verde using multiwavelength Raman and polarization lidars during Saharan Mineral Dust Experiment 2008[END_REF] and the same technique was applied to separate pollen and background aerosol (Noh et al. 2013a;Sicard et al., 2016;[START_REF] Shang | CC BY 4.0 License. Raman polarization lidar in Finland: characterization of pure pollen types[END_REF]. For height independent depolarization ratios of pollen and background aerosol the pollen backscattering coefficient can be calculated as suggested by Tesche et al. ( 2009):

( ) (1 ) ( ) (1 ) b p p p b             , (4)
Here β and  are backscattering coefficient and particle depolarization ratio of the mixture. The

27 -28 March and 21 April 2020 observations.

According to the in-situ pollen sampling at rooftop level, the maximal pollen content was Agreements between results obtained from depolarization and fluorescence techniques in Fig. 8,10, corroborates the suggestion that the observed fluorescence is mainly due to the presence of pollen. However, in some episodes the particles with high fluorescence cross section, other than pollen, could interfere. In particular, such interference occurred in 20-23 April 2020 period. Fig. 11 shows the vertical profiles of particle parameters measured on 21 April. The depolarization ratio 1064=22% at 750 m was one of the highest during campaign. The RH is low, increasing from 30% to 45% in 800-1500 m range, according to Herstmonceux radio sounding. The back trajectory analysis demonstrates, below 1500 m the air masses are transported from Spain, while at 2000 m the transportation is from the Northern Europe.

Fluorescence backscattering is stable in 500-1500 m range and the fluorescence capacity at 1000 m is about 1.510 -4 , which is a typical value for the pollen. However, above 1250 m GF starts to rise, reaching the value of 2.510 -4 at 1750 m, meanwhile depolarization ratio decreases.

Such high GF is more typical for the smoke particles. The lidar ratios above 1250 m, as well as BAE, also increase. Such vertical variation of the intensive particle parameters was observed during 20-23 April period and it may indicate the presence of the biomass burning aerosol near the boundary layer top.

Separation of pollen and smoke layers

During Fig. 13 shows the vertical profiles of the particle parameters for these two cases. On 13-14 April the fluorescence backscattering below 1000 m is stable, while 532 rises, which can be the result of the particle water uptake. Above 1000 m, the depolarization ratio 1064 increases up to 8%.

Results in Fig. 13a are averaged over 21:15-00:40 UTC temporal interval, but peak values of 1064 between 23:00 and 00:00 exceeded 12%. Fluorescence backscattering increases simultaneously with the depolarization. The aerosol backscattering coefficient of fluorescent layer is too low for a reliable calculation of 355 and 532, so only the profile of 1064 is provided.

On 26-27 May the backscattering coefficient of the fluorescent layer at 3400 m is lower than in Fig. 13a (5320.14 Mm -1 sr -1 ), so the depolarization ratio 1064 can be calculated only in the center of the layer and it is about 2%, which is significantly lower than that on 13-14 April.

However, the fluorescence capacity on 26-27 May is up to 3.510 -4 , which is typical for smoke.

Thus, we can conclude that the fluorescent layer on 26-27 May contains the smoke particles, due to high GF and low 1064. On 13-14 April, the fluorescence capacity is significantly lower (about 0.910 -4 ) and depolarization ratio 1064 exceeds 10%, which is more typical for pollen. Due to low backscattering coefficients of the fluorescent layers in Fig. 12, we are not able to provide a complete set of intensive parameters, such as Angstrom exponents and particle depolarization ratios at three wavelengths. However, based on the obtained fluorescence capacities and 1064

values, we conclude that the fluorescent layers probably contain pollen grain in Fig. 12a, and smoke particles in Fig. 12b.

Aerosol classification based on polarization and fluorescence measurements.

Table 1 summarizes the results in the campaign, showing the aerosol parameters, such as particle depolarization and lidar ratios, extinction Angstrom exponent, fluorescence backscattering and capacity for several days in March-June 2020 observations period, when the contribution of pollen to the total particle backscattering was significant. All available night observations were averaged and results are given for heights with the highest particle depolarization. Lidar ratios varied approximately in 40--70 sr range, wherein normally S355 is greater than S532. It must be emphasized that pollen lidar ratios may differ for different taxa and that the observed lidar ratios are not attributed to pure pollen, but to the aerosol-pollen mixture, so the values provided are influenced by the properties of background aerosol. Moreover, the shape of pollen grains depends on RH, [START_REF] Heidemarie | Illustrated Pollen Terminology. 2 e éd[END_REF], which may also lead to the variation of pollen lidar ratios. In most of the cases, the depolarization ratio presents strong spectral dependence and increases with wavelength. This spectral dependence is probably the result of mixing of strongly depolarizing pollen grains with fine background aerosol. The maximal value of observed fluorescence capacity of pollen-aerosol mixture is 1.610 -4 , which is significantly higher than that of background aerosol, but lower than fluorescence capacity of smoke. for smoke maximal GF is about factor 50 higher. The fluorescence capacity depends on the relative humidity, so strong scattering of measurement points can be partly also due to RH variations.

Maximal values of GF for pollen mixture were about 1.510 -4 , and the corresponding depolarization ratios 532 are about 18%. Thus, assuming that depolarization ratio of pure pollen is 30%, we can expect GF for pure pollen to be about 2.510 -4 , which is comparable with values for smoke.

Conclusion

We analyzed the measurements from a multiwavelength Mie-Raman-fluorescence lidar during March-June 2020 in the north of France, to reveal the features that can be attributed to pollen grains. Contrary to previous studies, where pollen was identified by the enhanced depolarization ratio at a single wavelength, our lidar system allowed to measure depolarization ratios at three wavelengths, simultaneously with the fluorescence backscattering at 466 nm. In numerous episodes during the campaign, high values of the particle depolarization ratio at 1064 nm, exceeding 15%, were observed. Moreover, depolarization ratio had strong spectral dependence, being the highest at 1064 nm and lowest at 355 nm, which is expectable for big particles of irregular shape mixed with fine, low depolarizing background aerosol. The increase of particle depolarization correlated with enhancement of the fluorescence backscattering corroborating that in these episodes we observed aerosol mixtures containing pollen.

The lidar ratios of aerosol-pollen mixtures observed during campaign varied in a wide range. At low altitudes, where particle presented strong depolarization and fluorescence, in many cases we observed lidar ratios below 40 sr at both wavelengths. However, we had also cases when the lidar ratios at both wavelengths were in 50-60 sr range. Thus, at the moment we are not capable to specify lidar ratios for pure pollen and additional measurement campaigns in the locations with high pollen content are strongly desirable.

Obtained results demonstrate, that simultaneous measurements of particle depolarization and fluorescence allows to separate dust, smoke particles and pollen grains. Moreover, the fluorescence measurements provide additional information that can be used in aerosol classification schemes. However, further studies are needed to make this technique applicable for the quantitative pollen characterization. In the data analysis it is important to account for the process of water uptake by the particles, because hygroscopic growth increases backscattering of background aerosol and influences the pollen grain shape. In the presented lidar configuration, the water vapor channel was absent and radiosonde RH data were not collocated with lidar, which prevented us from performing a quantitative analysis of the hygroscopic effects. Since December 2020, we recovered the water vapor channel in upgraded configuration of the lidar. Moreover, we added one more fluorescence channel centered at 549 nm, which will be used in the next pollen campaign in 2021. This additional channel should improve discrimination of the pollen from other aerosols. In coming campaign we will try to correlate our results with pollen concentration at different location in Europe by using the transport model, e.g. SILAM (System for Integrated modeLling of Atmospheric coMposition) (Sofiev et al, 2013[START_REF] Sofiev | Construction of the SILAM Eulerian atmospheric dispersion model based on the advection algorithm of Michael Galperin[END_REF]. The use of this model should help in identification of pollen type in our observations.

  https://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License. performed during night time only. The aerosol extinction and backscattering coefficients at 355 and 532 nm were calculated from Mie-Raman observations (Ansmann et al., 1992), while 1064

( 5 .

 5 8%), Urticaceae (4.6%), Salix (4.5%) and Cupressaceae (4.1%). The same figure shows also the fluorescence backscattering F measured by lidar. The results presented are obtained by averaging all available data during the night and the maximal values in 500 -1000 m range are shown. The highest fluorescence is observed in the end of March, when ash (Fraxinus) is the main pollinator. The period of intense birch (Betula) pollination (3-15 April 2020) correlates also with high βF. Strong fluorescence observed for 5-10 May and 28 May-2 Jun periods, can be due to grass (Poaceae) pollen contribution. By the end of June, βF decreases and becomes comparable with fluorescence backscattering of background aerosol. From Fig.1, we can conclude that there is no direct correlation between in situ and fluorescence lidar measurements, thus pollen observed in the boundary layer by the lidar are probably transported from other regions. However, comparing lidar and in situ observations, we should keep in mind, that maximum of pollen emission occurs near the noon, while lidar measurements were performed in the night.

  https://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License. for the wavelength pair 355/532 nm ( 355/532 A  and 355/532 A  ) only, because the extinction and backscattering coefficient involved are calculated from Mie-Raman observations. When analyzing Mie-Raman-fluorescence lidar measurements of pollen containing aerosol mixtures, the numerous factors should be taken into account. These factors include the fluorescence of background aerosol and other non-pollen aerosols that have strong fluorescence capacity, for example, smoke particles. Dust particles can contribute to the increase of depolarization ratio and, finally, the hygroscopic growth can modify the particle parameters. All these factors will be considered in following sections.

  and can interfere pollen fluorescence measurements. The temporal evolution of the range corrected lidar signal, volume depolarization ratio at 1064 nm and fluorescence backscattering for the smoke episode on the night 23 -24 June 2020 are shown in Fig.4. During the night the smoke layer with low depolarization and high https://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License. fluorescence is observed at approximately 5000 m height. Back trajectories (not shown) indicate that the layer is transported from Canada. Vertical profiles of the particle parameters for this episode are shown in Fig.5.

  https://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License.

  For representative cases we have chosen observations with high depolarization ratio and high fluorescence backscattering. The same time, we omitted the days with high relative humidity, to minimize the impact of the hygroscopic growth effects. Below we consider several measurement cases representing different scenarios, in particular, the episodes when pollen concentration decreases with height (30-31 May, 1-2 June) and the episodes when pollen grains are well mixed inside the boundary layer(27)(28).4.1. 30-31 May and 1-2 June 2020 observationshttps://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License.The results of lidar measurements during the campaign in many episodes can be interpreted as decrease of pollen concentration with height. Vertical profiles of the main particle parameters for two representative cases in the nights of 30-31 May and 1-2 June 2020, are shown in Fig.8.The HYSPLIT back trajectory analysis[START_REF] Stein | NOAA's HYSPLIT atmospheric transport and dispersion modeling system[END_REF] demonstrates that in 1000--2000 m height range the air masses were transported from the Northern Europe. At the ground level, the grass could be the main pollen contributor for this period, as shown in Fig 1.On 31 May (at 00:00 UTC) the RH measured by the radiosonde in Herstmonceux (UK) was about 40% at 500 m and it increased up to 70% at 2000 m. On 2 June the RH increased from approximately 40% to 60% in the same height range. For both nights the fluorescence backscattering decreases with height, indicating the decrease of the concentration of fluorescent particles (presumably pollen). This decrease of F correlates with decrement of the depolarization ratio at all three wavelengths. Particle depolarization 1064 is the highest (about 15% at 750 m), while 355 and 532 are significantly lower. Such spectral dependence of depolarization ratio agrees with model calculation in Fig.2. The lidar ratios are available above 1250 m and for both cases, S355 and S532 increase with height. It indicates that the lidar ratios of pollen in the two considered cases can be quite low: below 40 sr at 355 nm and below 30 sr at 532 nm, considering that pollen concentration decreases with height, which is inferred from the features of depolarization ratio and fluorescence backscattering

  https://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License.



  is estimated as 30%. From Fig.2 the expected ratio 1064/355 is about 2.3. It agrees with observed ratio, which is about 2.4. The profiles of βF and 532 p in Fig.8(b, e) behave similarly, decreasing with height. Above 2000 m the decrease of βF slows down due to the fluorescence of background aerosol. The profiles of the fluorescence capacity GF and relative contribution 532 532 p   also demonstrate a good correlation. Thus both depolarization and fluorescence techniques lead to the same conclusion: pollen concentration in the boundary layer for the considered episodes decreases with height.

  Fig.8, where fluorescence decreases with height, on 27 -28 March the fluorescent particles are

  the campaign we observed narrow layers with strong fluorescence. Two examples of such observations, in the nights 13-14 April and 16-27 May 2020, are shown in Fig.12. The https://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License. white arrows on this figure point to the fluorescent layers. On 13 April, a weak aerosol layer (5320.6 Mm -1 for 23:00 -00:00 UTC) is observed at the top of the PBL. This layer demonstrates volume depolarization ratios exceeding 10% and high fluorescence backscattering. On 26-27 May a weak layer with high fluorescence backscattering occurs between 3 km and 4 km. However, in contrast with the first case, it has low depolarization ratio, so the layers may have different nature.

  https://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License.

  The simultaneous observations of depolarization ratio and fluorescence capacity for different types of aerosol are summarized by Fig.14. On this plot, particle depolarization 532 is plotted versus GF. The diagram allows to separate four types of the particles: (i) dust particleshigh 532 and low GF; (ii) pollenhigh 532 and high GF; (iii) smokelow 532 and low GF; (iiii) background aerosol (continental type) -low 532 and low GF. Points corresponding to the pollen mixture provide extended pattern, because parameters depend on the concentration of pollen in the aerosol mixture. The dust measurements are also scattered, because dust over the instrumentation site is long transported and mixed with local aerosol. Minimum GF for dust is about 0.110 -4 while

  https://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License.

Mar 1 1 ..

 11 characterization: feasibility experiment, Atm.Meas. Tech., 13, 6691-6701, 2020. 583 doi.org/10.5194/amt-13-6691-2020. 584 585 Mar 11 Mar 21 Mar 31 Apr 10 Apr 20 Apr 30 May 10 May 20 May 30 Jun 9 Daily concentration of most abundant pollen taxa, for the period March-June 2020 in Lille 587 from in situ measurements on the rooftop. Open symbols show fluorescence backscattering F 588 measured by lidar. Lidar measurements are averaged over night and maximal value in 500-1000 589 m range is shown. 590 591 https://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License. Depolarization ratios of pollen ( p ) and background aerosol 596 ( b ) are assumed to be spectrally independent, with values of  p =0.3 and  b =0.03. The 597 backscattering coefficient of pollen is spectrally independent, while for background aerosol the /doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License. Measurements in the condition of background aerosol predominance. Vertical profiles of (a) 604 backscattering coefficients β355, β532, β1064 and lidar ratios S355, S532; (b) fluorescence 605 backscattering βF and fluorescence capacity GF= βF/β532; (c) particle linear depolarization ratios 606 355, 532, 1064 together with extinction and backscattering Angstrom exponents 355/532 A 2020 for 20:30 -23:00 UTC. Measurements were performed at 30 degree to the horizon. /doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License.

Fig. 4 .Fig. 5 .Fig. 6 .Fig. 8 .Fig. 10 .Fig. 13 .

 45681013 Fig.4. Range corrected lidar signal at 1064 nm, volume depolarization ratio at 1064 nm and fluorescence backscattering coefficient (in 10 -4 Mm -1 sr -1 ) on 23-24 June 2020.

  The lidar ratio is about 50 sr at 355 nm, while the lidar ratio at 532 nm increases within the smoke layer from 60 sr to 80 sr. This increase of S532 occurs simultaneously with decrease of 355/532A  from 1.5 to 0.75, indicating that the particle size inside the layer growths with height. Higher values of S532 in respect to S355 are typical characteristics for the aged smoke

	extinction Angstrom exponent	A 	in the center of layer is about 0.75, while	A 	is about
		355/532			355/532
	1.9 and shows no significant variation through the layer. High values of	A 	compared to
				355/532
	355/532

(e.g.

[START_REF] Müller | Raman lidar observations of aged Siberian and Canadian forest fire smoke in the free troposphere over Germany in 2003: Microphysical particle characterization[END_REF][START_REF] Nicolae | Characterization of fresh and aged biomass burning events using multi-wavelength Raman lidar and mass spectrometry[END_REF][START_REF] Hu | CC BY 4.0 License. transported Canadian smoke plumes in the lower stratosphere over northern France[END_REF]

. The depolarization ratio decreases with wavelength from 355=10% to 1064=1.5%. Strong spectral dependence of depolarization ratio and, in particular, low values of 1064, are the features, allowing to identify the smoke layers. The

Table 1 .

 1 Acknowledgement https://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License. Lidar measured aerosol parameters, such as particle depolarization ratios (355, 532, 1064), 455 lidar ratios (S355, S532), extinction Angstrom exponent ( 355/532 A  ), fluorescence backscattering 456 coefficient (F) and fluorescence capacity (GF) for several days during March -June 2020 period, 457 when contribution of pollen to the total particle backscattering could be significant. 458 ://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License.

	454
	Day	Height, m 355, %	532, %	1064, % S355, sr	S532, sr

https

https://doi.org/10.5194/amt-2021-80 Preprint. Discussion started: 24 March 2021 c Author(s) 2021. CC BY 4.0 License.

We acknowledge funding from the CaPPA project funded by the ANR through the PIA under contract ANR-11-LABX-0005-01, the "Hauts de France" Regional Council and the European Regional Development Fund (ERDF). The "Réseau National de Surveillance Aérobiologique" (RNSA) and the "Association pour la Prévention de la Pollution Atmosphérique" (APPA) are gratefully acknowledged for providing Hirst-collected pollen grains identification and for assistance with the pollen data handling.


A first French episode in the renewal

Keywords: Endogenous business cycle theory, Nonlinear dynamics, Non-Walrasian and Kaldorian macrodynamics 1

This paper focusses on some relatively neglected French contributions to the revival of nonlinear business theory around 1980. Drawing on the formal distinction between continuous and discrete-time modeling, we investigate the mathematical and analytical features of these contributions. The Benassy model exemplifies the use of the Poincaré-Bendixon theorem to prove the existence of endogenous cycles in a simple non-Walrasian framework. The discretization of Kaldor's model by Dana and Malgrange mobilizes recent advances in bifurcation theory and chaotic dynamics developed at the same time by French scholars in dynamical systems. It is shown that both contributions build on the aggregate macroeconomic framework, but differ substantially in their objectives and ambition. Benassy mobilizes the nonlinear approach to extend his non-Walrasian theory to short-term dynamics. On the contrary, Dana and Malgrange are more interested in the operational aspects of nonlinear modeling. For the these reasons, this second line of research will contribute to further important developments in nonlinear dynamics in France, albeit in a different perspective.

1. Especially by scholars trained in the Grandes Ecoles, like ENS or Polytechnique. For a systematic investigation on the history of nonlinear dynamics in France during the period 1880-1940, see Ginoux 2017. After war let mention the International Conference on Economic Dynamics held in 1955 in Paris, the contributions of Allais to endogenous business cycle in the 1950s or the 1962 special issue of the Cahiers du Séminaire d'Econométrie dedicated to business cycles with a detailed mathematical analysis of Goodwin's nonlinear models. On these points see [START_REF] Raybaut | Toward a nonlinear theory of economic fluctuations : Allais's contribution to endogenous business cycle theory in the 1950s[END_REF].

2. [START_REF] Lorenz | Nonlinear dynamical economics and chaotic motion[END_REF], p.5.

3. See e.g. Medio 2000, p.1 4. For a methodological appraisal of the different tools for proving the existence of endogenous cycles in two dimensional continuous systems, see [START_REF] Ragupathy | Existence proofs in nonlinear endogenous theory of the business cycle on the plane[END_REF]. Notice that the Hopf bifurcation theorem proper for two dimensional flows was not yet used in the economic literature under consideration. It was also the case for 3 dimensional flows and strange attractors.

5. The Cepremap, at that time "Centre pour la Recherche en Economie Mathématique

On the one hand, Benassy develops a simple nonlinear Keynesian framework to study the emergence of endogenous business cycles as a succession of short run non-Walrasian equilibria. In this contribution, the technique mobilized is not totally innovative. The existence of closed solutions is classically proved with the Poincaré-Bendixon theorem for two-dimensional continuous-time systems. But, it is applied in an elegant way to a specific dynamic model in the non-Walrasian perspective largely initiated by the author.

On the other hand, Dana and Malgrange starting with the well-known Kaldor's 1940 model, produce an original analysis in discrete dynamics. In particular, they build on emerging mathematical advances in bifurcation theory and chaotic dynamics of mappings developed mainly by French scholars and not yet diffused in economic analysis.

These advances in nonlinear dynamics build on the aggregate macroeconomic framework. Indeed, Benassy and Malgrange have used this modeling framework in several previous works, albeit from a different perspective. Benassy has developed a number of macroeconomic models applying the non-Walrasian methodology. In the early 1980s he aimed notably at bridging the gap between traditional Keynesian macroeconomics and the non-Walrasian theory with the help of synthetic models 6 . Malgrange has contributed to a wide range of issues in macroeconomic modeling. He was in particular involved in the development of a simplified version of the 'core' Keynesian macroeconomic model of Cepremap 7 . Hence, he made insightful exercises in the use of small scale macroeconometric models. Analyzing the long and short run properties of these models, he showed notably how the linearization procedures and the use of numerical methods may be highly misleading with respect to stability 8 .

Besides the formal aspect mentioned above, the background of the protagonists in question helps to understand some substantial differences in their

Appliquée à la Planification", created in 1967 was a particular example of these institutions at the interface between academic research and public administrations, notably the Plan, INSEE and CNRS, with specific seminars and a working papers series, ("emphCouverture Orange"). On the specific role of these institutions see Fourcade (2009). 6. It is beyond the scope of this paper to detail these advances. See e.g. [START_REF] Benassy | The economics of market disequilibrium[END_REF] 7. On this point see Deleau, Malgrange and Muet (1981), "Une maquette représentative des modèles macroéconomiques", Annales de l'INSEE n˚42 and "A Study of Short Run and Long Run Properties of Macroeconometric Models by Means of an Aggregative Core Model" in Malgrange P. and Muet P.A. Eds (1984), Contemporary Macroeconomic Modelling, Blackwell, Oxford. The book contains the contributions and comments presented at an international seminar organized jointly by the French Commissariat du Plan and the Cepremap in 1982.

8. See e.g. [START_REF] Malgrange | Note sur le calcul des valeurs propres d'un modèle macroéconométrique[END_REF]. The existence of complex dynamics in the nonlinear difference equation used by Malgrange in this work was established by [START_REF] Uebe | A Nonlinear Difference Equation of Malgrange[END_REF].

contributions in nonlinear dynamics. For Benassy, the adoption of a nonlinear approach partakes of his non-Walrasian research agenda. Accordingly, he mobilizes this method to extend the scope of the non-Walrasian framework to short-term dynamics, having in mind the nonlinear Keynesian models of the business cycle 9 . On the contrary, Dana and Malgrange are more interested in the operational aspects of nonlinear modeling. In this perspective, their goal was mainly to set out a rigorous example showing the properties and the limits of a discrete-time analysis in macrodynamics 10 . As we will see, But it is precisely for these reasons that this second line of research will influence future developments in nonlinear dynamics in France.

The paper is organized as follows. The first section illustrates the use of the Poincaré-Bendixon theorem by Benassy. The second section is dedicated to the new advances in discrete dynamics made by Dana and Malgrange. The last section makes some concluding remarks on further developments in French nonlinear dynamics.

The Poincaré-Bendixon theorem in a Non-Walrasian framework : Benassy's model This section is dedicated to the non-Walrasian dynamic model of the business cycle developed by Benassy.

The model was prepared for the Conference on the dynamics of decentralized market economies held near Stockholm in August 1983, on the invitation of Richard Day 11 12 . The aim of the conference was to provide a forum confronting different views on the dynamics of market economies, focusing on crucial 9. In this perspective, he mentions the main attempts developed since the 1950s 10. Some connections also exist between the protagonists. Benassy thanks Malgrange for stimulating discussions on his model. In the early seventies, he had collaborated directly with Malgrange on production functions and technical progress. See [START_REF] Benassy | Substitution et progrès technique dans les modèles générations de capital[END_REF], [START_REF] Benassy | Estimation d'une fonction de production à générations de capital pour l'industrie française[END_REF]. Rose-Anne Dana was at the time Associated Professor of mathematics at University Pierre and Marie Curie (Paris VI). In 1970-1973 she was Teaching assistant of mathematics and research assistant of economics at University of California in Berkeley, when Benassy was writing his Ph.D. She also made a Ph.D. in Berkeley in 1974 on optimal growth under the supervision of Roy Radner.

11. Day made several visiting periods in Paris between 1976 and 1982. Day mentions four effective periods of study facilitated by Jean pierre Aubin during Springs 1976Springs , 1978Springs , 1979Springs and 1982 Benassy's interest for dynamic issues emerges naturally in his contributions on the microeconomic theory of non-Walrasian equilibria and their macroeconomic applications. But the different non-Walrasian concepts he developed in the 1970s give only some insights into how the market economies work dynamically 14 . Indeed, the explicit derivation of dynamic findings in a disequilibrium perspective was difficult because of the added structure required which even presses downward the analytical benefit-cost ratio, especially in the microeconomic framework. This may partly explain why Benassy did not really produce a dynamic analysis until the early eighties 15 . The macroeconomic framework adopted by Benassy in this contribution disposes of these difficulties 16 . In this way, as Day put it, Benassy was able "with a few deft Keynesian strokes to describe a disequilibrium macro model and derive its salient dynamic properties." 17 Several growth models were developed in France during the 1980's in the so-called disequilibrium dynamic approach. But this contribution is a rare exception in short-run dynamics 18 . The objective is to explore this dyna-13. 1983 marked the hundredth anniversary of Joseph Schumpeter's birthday. Day has always been convinced that the dynamics of market processes works in perpetual disequilibrium. He has developed a number of adaptive and nonlinear models of growth and business cycle.( See e.g. [START_REF] Day | Complex Economic Dynamics, An Introduction to Dynamical Systems and Market Mechanisms[END_REF]. His own contribution "Disequilibrium Economic Dynamics : A post-Schumpeterian Contribution" is a description of the research program he has carried out in this perspective. Let also mention among the other leading theoretical contributions, a critique of neoclassical macroeconomics by Simon, an analysis of non-tatonnement dynamics by Clower and the presentation of the Micro-Macro model by Gunnar Eliasson. The rest of the papers discussed the role of institutions, markets, technological progress, innovations and entrepreneurial activity from an historical and empirical view points and the new evolutionary perspective by Winter. The main contributions, including Benassy's paper, with supplementary pieces developed from the discussions were reprinted in the book The Dynamics of Market Economies edited by Richard [START_REF] Day | Complex Economic Dynamics, An Introduction to Dynamical Systems and Market Mechanisms[END_REF]Gunnar Eliasson in 1986. 14. Béraud (2017) 16. The model largely builds on [START_REF] Benassy | The economics of market disequilibrium[END_REF]17. Richard Day (1983), comments on Benassy in [START_REF] Day | The dynamics of market economies[END_REF], p.148. Acknowledging Benassy for this contribution he adds that "it is thus especially noteworthy that a mathematical economist of the general equilibrium school should venture into this macroeconomic realm where much serious analytical effort needs to be allocated." Ibid.

18. In the non-Walrasian perspective, the contribution of Somonovits ( 1982) is ano-mics in a very simple non-Walrasian IS-LM framework, showing that it can exhibit endogenous cycles. The business cycles are then analyzed as a succession of non-Walrasian equilibria. According to Benassy, the comparative advantage of the non-Walrasian approach is its capacity to give a consistent determination of prices and quantities, contrary to traditional Keynesian business cycle models 19 . A second striking property of the model is the use of topological qualitative tools to prove the existence of endogenous cycles without resorting to special functional forms or ad-hoc assumptions, contrary to previous non linear Keynesian models of the business cycle.

The model

The starting point is a simple IS-LM framework with fixed money supply, no public spending, flexible price and fixed wage.

The Demand Side is thus described by the equations y = C(y, p) + I(x, r) and L(y, r, p) = M , where y is current output, p current price, r the interest rate and M the fixed quantity of money. As mentioned by Benassy, here, the sole novelty is the explicit introduction of expected demand x in the investment function, usually captured by some function of past or current income 20

Next, consider the Supply-side of the model. Labor supply is inelastic and equal to l 0 . The firm operates on a competitive market with a decreasing returns technology F (l). She his not quantity constrained, thus for a given stock of capital, the demand for labor is given by F [F ′-1 ( w p )]. The wage w evolves according to a Phillips curve with the traditional decreasing shape :

ẇ = H(u) (1)
where u = l 0 -l is the level of unemployment. Using the fact that l = F -1 (y), this relation can be rewritten as an increasing function of the level of output

y : ẇ = G(y) (2)
To complete the model, Benassy considers that expected demand x adjusts ther attempt in short-run nonlinear dynamics, albeit in the line of Honkapohja and Ito's approach.

19. Benassy mentions in particular the models based on discrepancies between investment and savings in the short run, "which could be made sense of only by invoking inventory adjustments not formalized in the models." [START_REF] Benassy | A Non-Walrasian Model of the Business Cycle[END_REF] 20. These two functions are C 1 and their first derivatives verify the standard assumptions 0 < Cy < 1,Cp < 0,Ir < 0, Iy > 0,Ix > 0, Ly > 0,Lr < 0 and Lp > 0. In addition, Benassy assumes some type of real balance effect with strictly positive consumption for any finite price level, even with zero income.

to actual demand y with a speed of adjustment µ > 0 :

ẋ = µ(y -x) (3)
The level of income y is a temporary equilibrium solution between aggregate demand and supply. For any level of expected demand x, this solution can be written as y = Z(w, x), where Z w < 0 and Z w > 0. 21 Consequently, the dynamics of the economy is fully described by the planar dynamical system in w and x.

Dynamics and existence of limit cycles

Before investigating the dynamics of this nonlinear system, Benassy recalls that the assumptions made above on the IS-LM structure, the Philips curve and the adaptive revision of expected demand, ensure the existence of a unique stationary solution w = w * and x = x * = y, where y = F (l 0u) and u is the non inflationary rate of unemployment. Then, the he considers classically the linearisation of the system about this stationary solution.

The determinant of the Jacobian matrix -µG ′ .Z x is always strictly positive, which discards saddle path dynamics. Thus, the long run equilibrium (w * , y) is locally stable or unstable according to the sign of the trace of this Jacobian. Thus, the economy always converges to its long run equilibrium position when Z x > Z, where Z = 1 -G ′ Zw µ . On the contrary, the singular point is locally unstable when Z x > Z > 1. 22 From an economic point of view, local instability will arise when the accelerator coefficient is large enough. In such case, it is shown that there exists at least a limit cycle surrounding the long run equilibrium. 23 Next, Benassy proceeds to show rigourously how this instability condition eventually results in an endogenous cyclical dynamics.

21. In this non-Walrasian framework with inelastic labor supply, the level of output is normally given by the relation

y = min[F [F ′-1 ( w p )], l0] (4)
The type of short-run equilibrium, with full employment when y = y0 = F (0) or with underemployment when y = F [F ′-1 ( w p )], depends on which part of the supply curve the system is. However, the assumption made by Benassy for the shape of the Phillips curve implies that the full employment equilibrium will never be reached, as wage increases become infinite when y approaches y0. As a consequence, y = F [F ′-1 ( w p )], with y ′ p > 0 and y ′ w < 0. 22. It is easy to check that the isokine ẋ = 0 is strictly above the horizontal axis with a positive slope if Zx > 1 and a negative slope if Zx < 1, which is a sufficient condition for local stability. In addition, the locus ẇ = 0 is always increasing.

23. Indeed, Zx the sensitivity of production to expected demand is positively related to a kind of accelerator coefficient Ix, by the relation Zx = IxΩ, where Ω = ypLr ypLy Ir +(1-Cy )ypIr +LpIr -CpIr > 0.

The proof of the existence of at least one limit cycle resorts to the Poincaré-Bendixon theorem. From a formal point of view, this theorem theorem ensures that a non empty compact limit set of a continuously differentiable dynamical system in R 2 which contains no fixed point, is a closed orbit (Hirsch and Smale 1974, p.252). As Medio put it, « the basic condition required by the theorem is the existence of a 'trapping region' which contains no fixed point, such that the orbits of the system can enter but cannot leave it. » (Medio 2000, p. 120). Without entering into the details of the proof, the argument is here to show that when the unique fixed point is locally unstable, the assumptions made above for the presentation of the model are sufficient to ensure that the the trajectories remain in a bounded the set.

As shown on the figure above, this set is given by the compact formed by the bold line ABCDE which contains at least one limit cycle. 24 From an economic point of view, the emergence of these bounded trajectories is explained by the interplay between a destabilizing tendency on the quantity side and a stabilizing dynamics due to prices. Indeed, the destabilizing effect on the quantity side is the consequence of the adaptive revision of expected demand, the level of investment being determined by an unstable accelerator mechanism. But this instability is contained by the damping direct effect exercised by prices on aggregate demand and the adjustment of wages. The latter is captured by a traditional Phillips curve which tend to 24. The sole ambiguity concerns the segment AB. In order to be sure that the dynamics points inwards on AB, that is ẏ < 0, Benassy needs to assume that the location of point C defined by ŷ satisfies the condition G( y) > -µy 0 Zx Zw . This sufficient condition is not restricting, but arbitrary. bring back the system to its long run equilibrium position.

As noticed by Day in his comments, Benassy introduces price adjustment explicitly, "the absence of which is perhaps the most important restriction in the standard IS-LM framework. The results illustrate the important point that the mere existence of price adjustments can not guarantee the emergence of equilibrium, a mistaken presumption at the root of much discussion of the market system." 25 . More generally, this type of nonlinear models with low order equations "permits the derivation of sharp and interesting results that go well beyond the question of existence and comparative statics". But this payoff "necessitates difficult global analysis of dynamic behavior" 26 .

In this perspective, Benassy recalls that the Poincaré-Bendixon theorem has already been applied to various Keynesian nonlinear models. However, the implementation of this technique in a dynamic non-Walrasian framework is totally original. In addition, the elegance of the qualitative proof should be emphasized. Contrary to the other contributions, no specific ad-hoc assumptions on the functional forms of the model need to be assumed 27 . But this type of qualitative and global analysis of economic dynamic presents several difficulties.

First, as mentioned by Benassy himself in his concluding remarks, this simple IS-LM framework could be judiciously enriched by additional ingredients, like expectations in an augmented Philips curve or capital accumulation and growth. But the use of the Poincaré-Bendixon theorem limits the analysis to two dimensional systems. Second, the theorem establishes the existence of at least one limit cycle and it remains in general impossible to elicit the exact number of closed trajectories. Finally, this framework deals with the qualitative and global properties of the dynamics. Thus, it is not possible to say much on the effective trajectories of the economy in the short run as in traditional business cycle modeling.

These difficulties, reinforced by the rapid decline of the aggregate perspective in mainstream macroeconomic analysis, may explain the limited influence of the model. Day (1983), comments on Benassy, in [START_REF] Day | The dynamics of market economies[END_REF], p.148 26. ibid. 27. As it was the case for example, on a nonlinear Phillips in the pioneering contribution of Rose (1969) or the well-known sigmoid Kaldorian investment function in Chang and Smyth (1971) and other reformulations. Notice that simultaneously to Benassy's model, the contribution of [START_REF] Schinasi | Fluctuations in a Dynamic Intermediate-Run IS-LM Model : Applications of the Poincare-Bendixon Theorem[END_REF] is also based on the qualitative dynamics of an IS-LM model, but in a classic Walrasian framework augmented of a budget constraint and a nonlinear investment function.

Richard

The model was naturally discussed in the specific Keynesian literature on nonlinear dynamics 28 . A larger audience could be expected with the publication in 1986 of Benassy's book Macroeconomics An Introduction to the Non-Walrasian Approach 29 . The book gives an overview of the non-Walrasian approach and discussed its applications to traditional macroeconomic issues, emphasizing in particular their policy implications. Dealing with the problem of business cycles Benassy resorts to his non-Walrasian business cycle model to discuss how the dynamic evolution of a short-run non-Walrasian equilibrium of the IS-LM type may generate cycles 30 . But in the meantime, Benassy has rapidly abandoned this IS-LM framework, moving to more sophisticated optimizing models with non-Walrasian features, notably imperfect competition. However, these contributions still remind us of the advantages of relatively simple benchmarks with analytical solutions, privileging the basic properties of the mechanisms at work in the dynamics 31 .

New advances in a well-known model : the pioneering analysis in discrete dynamics by Dana and Malgrange The second type of contribution adopts a different perspective, building on Kaldor's 1940 classic business cycle model 32 .

The reformulation into a nonlinear dynamical framework of Kaldor's initial insights is not new in the early 1980s. Let us simply recall among others, the 28. See notably Semmler (1986). In the early 1990s, the model was still mentioned in the main syntheses on nonlinear economic dynamics. For example [START_REF] Gabisch | Business cycle theory[END_REF] discuss lengthly Benassy's "complicated compact set". The model is also referred in [START_REF] Lorenz | Nonlinear dynamical economics and chaotic motion[END_REF] or [START_REF] Dohtani | Existence and stability of business cycles[END_REF] for a mathematical perspective. For a specific presentation and economic discussion of the model see [START_REF] Dore | The macrodynamics of business cycles : a comparative evaluation[END_REF] and the Appendix 1 in [START_REF] Chiarella | The Dynamics of Keynesian Monetary Growth : Macro Foundations[END_REF] 29. Accordingly, the Web of Science data bases mention only 11 citing references (with a sum of times cited equal to 71) for the 1986 JEBO article. As to Macroeconomics An Introduction to the Non-Walrasian Approach, the data bases mention 43 citing references (with a sum of times cited equal to 251.

30. See Benassy (1986), Chapter 11, pp.173-85. 31. In this perspective, see for example Benassy (1996), "Analytical solutions to an RBC model with imperfect competition, increasing returns and underemployment", Recherches Économiques de Louvain Vol. 62, No. 3/4. 32. Several Kaldorian studies where developed inCepremap in the 1980s, dedicated to the empirical and theoretical analysis of the relation between technical progress, productivity and growth. On this issue, see the different contributions of Boyer and Petit, notably the references in [START_REF] Boyer | Progrès technique, croissance et emploi : un modèle d'inspiration kaldorienne our six industries européennes[END_REF]. In addition, let us recall that the French translation of some selected contributions of Kaldor, Economie et Instabilité, including his 1940's business cycle model, was set about under notably the direction of Robert Boyer and Pascal Petit.

Japanese attempts in the 1950s with the contribution of [START_REF] Ichimura | Towards a general nonlinear macrodynamic theory of economic fluctuations[END_REF] 33 , the application of catastrophe theory by [START_REF] Varian | Catastrophe theory and the business cycle[END_REF] and the seminal reformulation into a nonlinear planar differential system resorting to the Poincaré Bendixon theorem by Chang and Smyth (1971). Dana and Malgrange emphasize that most contributions deal with autonomous planar systems in continuous time. This class of nonlinear systems can only give birth to stable or unstable fixed points, saddle path dynamics or periodic orbits (limit cycles). Their aim is to develop a discrete version of the model. Building on the new mathematical literature on bifurcation and discrete dynamics, they show that more complex patterns, like deterministic chaos, can also emerge in this framework. Thus, the contribution gives an insightful comparison of the dynamics of discrete and continuous time models. Indeed, an important methodological issue in macroeconomic modeling is whether to work in continuous or discrete time. Starting from a theoretical model in continuous time, many contributions opt directly for discrete time dynamics. Dana and Malgrange show that the transformation from continuous to discrete time dynamics is far from neutral, particularly in a nonlinear framework 34 .

The discrete reformulation

This discrete version is based on a slightly modified version of Chang and Smyth (1971). Indeed, one difficulty with this reformulation is still the absence of long term growth. Cycles take place around stationary equilibrium values of the capital stock and output. Dana and Malgrange generalize the model to the case of cyclical fluctuations around an exogenous growth trend. This trend is classically explained by public expenditures G 0 growing at a constant rate g. 35 . 36 Then, the dynamics of the model in reduced variables (y, k) is described by a system of two first order difference equations :

y t+1 = 1 1 + g [y t + α(I(y t , k t ) -S(y t , k t ) + G 0 )] (5)
k t+1 = 1 1 + g [(1 -δ)k t + I(y t , k t )] ( 6
)
33. On the different Japanese attempts see Velupillai 2008 34. This choice is often implicitly due to econometric considerations. Indeed, the contribution was presented at a workshop held at Erasmus University in 1982, on how the structure of complex theoretical and econometric models might be tricked out. Thus, a version of the paper was published in the book edited by J.P. Ancot (1984)Analysing the Structure of Econometric Models that contains a collection of the conference papers.

35. The authors assume moreover that the investment I(Yt, Kt) and saving function S(Yt, Kt) are homogenous of degree 1 in global income Yt and capital Kt 36. In the initial Cepremap working paper and the published version, Dana and Malgrange, start by considering the version in continuous time, give an existence theorem of at least a limit cycle using the Poincaré Bendixon theorem, and find a numerical example calibrated on French data 1960-74.

where, δ is the rate of capital depreciation and α is the speed of adjustment between global supply and demand.

On this basis, the contribution builds on several mathematical advances in discrete dynamics, not yet diffused in economic dynamics analysis at that time 37

The first point is to consider that if the state of the system in period t is described by (y t , k t ), then in period t + 1, this state will be given by (y t+1 , k t+1 ) = T α (y t , k t ), where, T α is a mapping from R 2 + into R 2 +, with

T α (y t , k t ) = 1 1 + g [y t + α(I(y t , k t ) -S(y t , k t ) + G 0 ), (1 -δ)k t + I(y t , k t )] (7)
Notice that this mapping is defined for a given value of α, where this speed of adjustment is a bifurcation parameter. Quite interestingly, this mapping is continuous and derivable, but not always invertible for large values of α, thus very complex trajectories (deterministic chaos) may exist.

From this standpoint, this model gives a clear illustration of [START_REF] Ruelle | Les attracteurs étranges[END_REF] conjecture on the role that strange attractors and chaotic dynamics may play in economic dynamics, « where periodic processes (economic cycles) are well-known » 38

The author begin their investigations with numerical simulations of an example calibrated on French quarterly data for the period 1960-74. Different configurations are then directly observed when the parameter α is increased between 0.5 to 20. Convergence to the steady state is obtained for small α.

For intermediary values a "periodic regime" is observed, then quasi periodic solutions and eventually transition to turbulence for high α are obtained.

Bifurcation theory and dynamics for mappings : the "Nice School"

Next, in order to investigate more closely these findings, the authors refer to recent developments in the theory of bifurcation and dynamics of nonlinear mappings. Contrary to continuous system, in mathematics and physics 37. Let us recall that the fully-fledged working paper is recorded March 1981, but the study began in the end of the 1970s, notably 1978 for the example. In the same perspective, the sole economic exception are the yet unpublished papers of Day and Benhabib and Day, mentioned by the authors. 38. Ruelle (1980), quoted by [START_REF] Dana | On a discrete version of Kaldors's 1940 cycle model[END_REF] p.10. Ruelle supposes that the dynamics of the macroeconomic system may depend on a parameter µ. Then, like in physics, « we would guess that for small µ the economy is a steady state and that as µ increases periodic or quasi periodic cycles may develop. For high µ chaotic behavior with sensitive dependence on initial condition would be present » (ibid.).

the systematic investigation in the theory of discrete dynamics develop only in the 1970s. In this perspective, the contribution of [START_REF] Dana | On a discrete version of Kaldors's 1940 cycle model[END_REF] provides one of the first synthetic presentation in an economic paper of some main findings 39 .

It is obviously outside the scope of this paper to survey this growing and not yet completely coordinated technical literature 40 We simply focuss on the major advances made by different scholars in Nice at that time.

Indeed, a local culture of nonlinearity and chaos existed in Nice since the early 1970s. The French mathematician and astronomer Michel Hénon a leading figure in the field of stellar dynamics moved from the Institut d'Astrophysique de Paris to the Nice Observatory in 1968 where he spent the rest of his career. Hénon inspired a very active group of scholars in nonlinear dynamics in Nice. His name is in particular linked to the discover in 1976 of the so-called Hénon map which is one of the most studied chaotic systems 41 . But he also contributed with other astrophysics of the Nice Observatory, notably Claude Froeschlé, to the implementation of performing numerical techniques using Monte Carlo methods 42 . In the meantime, two French mathematicians Gérard Iooss and Alain Chenciner, have joined the University of Nice, making breakthroughs in the analysis of nonlinear dynamics and bifurcation theory 43 In 1977 an important international meeting between physicists and mathematicians about non-linear problems and their applications supported by the French Physical Society was held in Nice 44 In 39. Simultaneously with Day's papers, dedicated more specifically to chaotic dynamics, notably on Li and Yorke (1975) and [START_REF] May | Simple mathematical models with complicated dynamics[END_REF] results, and not to bifurcation theory for mappings 40. For surveys of the period, see e.g. Marsden and Mac-Cracken (1976), Ruelle and Takens (1971), Ruelle (1977) and the seminal synthesis of [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields[END_REF].

41. [START_REF] Coullet | History of chaos from a French perspective[END_REF] explain that Hénon found out this mapping during lunch time after a seminar at the Nice Observatory where Yves Pommeau and Jean Louis Ibanez have presented some new findings on Lorenz's attractor. 44. Or Société Française de Physique (SFP) the main professional society of French physicists founded in 1873. The proceedings, with notably contributions of Michel Hénon, Gérard Iooss, Alain Chenciner, Pierre Coullet and Charles Tresser, were published in Journal de Physique C5, Vol.39, 1978 August. For an assessment of this meeting see Petitgirard (2004). Petitgirard describes how this conference shaped the research in these fields with an increasing implication of the French National Center for Scientific Research this context, Pierre Coullet and Charles Tresser discovered at the same time as Feigenbaum in the US the scenario of transition to turbulence by accumulation of period doubling by iteration of simple maps 45 .

In addition, [START_REF] Iooss | Bifurcation of Maps and Applications[END_REF] gives one of the first systematic discussion and new results on Hopf bifurcation theory for maps, directly mobilized in Dana and Malgrange 46 . Indeed, Iooss' (1979) theorem is illustrated with a detailed application to the discrete Kaldor model 47 . The existence of an invariant orbit is proved theoretically. The period and the amplitude of the cycled are also investigated.

As shown in [START_REF] Iooss | Bifurcation of Maps and Applications[END_REF], this analysis needs to consider the normal form in polar coordinates and is rather intricate, mobilizing the concept of rotation number of an associated homeomorphism f α which captures the dynamics of the initial map T α restricted to the invariant curve Γ α 48 . The authors refer to an algorithm recently developed by [START_REF] Arneodo | Simple computations of bifurcating invariant circles for mapping[END_REF], showing that the amplitude can be approximated by a square root function depending on the parameter α and coefficients of the normal form of the model. But computations were not so easy at today, and they conclude that it is impossible to derive economic practical estimations on theses theoretical advances 49 . However, a main limitation with these findings is the local property of the (CNRS) during the 1980s and the creation in 1991 of the Nonlinear Institute of Nice-Sophia Antipolis (INLN) by Iooss and Coullet. 45. Coullet was at that time a young CNRS research fellow and Tresser had no position at the university. Coullet mentions that they discussed the results with Hénon who had just received a preprint from Feigenbaum describing related findings. See [START_REF] Coullet | History of chaos from a French perspective[END_REF] 46. As we may recall, this type of bifurcation was initially investigated by Neimark in 1959 and Sacker in 1965. See [START_REF] Medio | Nonlinear dynamics : a Primer[END_REF].

47. Iooss ' (1979) theorem states in a nutshell the following : Let consider a mapping Tα of class C k with k 6, from an open set of R 2 into R 2 . Let assume that for each value of the bifurcation parameter α, the mapping admits a fixed point x(α). Assume moreover, that : i) There exists a critical value of the bifurcation parameter α0 such that the jacobian of Tα admits for this value two conjugate roots λ(α) of modulus 1. ii) d dα |λ(α)|α=α 0 > 0. Then, there exists a neighborhood of α0 such that at α0, there is an exchange of stability of the fixed point and an invariant circle for Γα.

48. The period of the cycle is defined as the inverse rotation number. See [START_REF] Medio | Nonlinear dynamics : a Primer[END_REF] p.158-59 for a synthetic discussion on the rotation number in local bifurcations for maps.

49. In the working paper, for α = 0.9, the period is estimated to 107 units of time. The published version indicates α0 = 0.73 with no longer estimation of the period. Notice that in both cases the authors do not give results on the stability of the cycle (du to their complexity) and assume that the invariant circle is locally attractive. discrete dynamics. The initial conditions should belong to a neighborhood of the stationary state and the bifurcation parameter closed to its critical value. This limitation contrasts with the continuous systems where global results can be obtained, notably on the existence of limit cycles with the Poincaré-Bendixon theorem.

Dynamics on the interval : The Kaldor's map Finally, Dana and Malgrange consider Kaldor's conjecture that firms adjust production much faster than the capital stock in the very short run. It is then possible to analyze the adjustment of global income for a given level of capital. Consequently, the dynamics of y can be represented by a one dimensional mapping, T k (y), the « Kaldor map ». We have

y t+1 = T k (y t ) = 1 1 + g [y t + α{I(y t , k 0 ) -S(y t , k 0 ) + G 0 }] (8)
For a given value of the parameter α, it can be shown that a typical « Kaldor map » admits an invariant interval [y 0 , y 1 ] ∈ R+, such that T k (y) is unimodal.

The dynamics of this class of maps on the interval was quite well understood in the early 1980's. Thus, Dana and Malgrange apply directly these advances on iterated maps into the interval to their model. 50 In particular, the concept of negative Scharwtzien derivative in unimodal functions on the interval is mobilized. 51

The interesting point is to focuss on mappings with high values of the parameter α for which complex dynamical patterns emerge 52 . In this perspective, the authors consider the iterations of the critical points of their mapping T k (y), for different values of the capital level and draw the bifurcation diagram, showing the transition to turbulence 53 . We reproduced below 50. They refer notably to the contributions of [START_REF] May | Simple mathematical models with complicated dynamics[END_REF], [START_REF] Collet | Iterated map on the interval as dynamical systems[END_REF] 51. The Scharwtzien derivative of a map T is defined by

S(T ) = T ′′′ /T ′ -3/2(T ′′ /T ′ ) 2 (9)
. For unimodal maps on the interval with negative Scharwtzien derivative, several important and generic results on the existence of periodic or non periodic orbits have been established (see e.g Medio 2000 p. 215 for details). These properties will be extensively mobilized in the economic literature, notably by [START_REF] Grandmont | On endogenous competitive business cycles[END_REF] in OLG models. See notably [START_REF] Barnett | Economic complexity : Chaos, sunspots, bubbles and nonlinearity[END_REF] and Michel and Wignolle (1996). 52. The authors refer here to the work of Arneodo, Coullet and Tresser (1980) The diagram clearly shows the successive branch splitting and the period doubling scenario typical of the transition to turbulence within the black areas. Thus, the discrete reformulation of Kaldor's model may exhibit a richer dynamics than the mere limit cycles obtained in the continuous time frameworks.

This pioneering contribution exemplify the benefits of a discrete approach in nonlinear economic dynamics 55 . But Dana and Malgrange also perceive the methodological difficulties raised by this approach. They mention the potential bias in calculation induced by complex numerical computations.

In addition, the time paths generated by such dynamical systems were extremely sensitive to initial conditions and changes in the parameters. That would make it hard to formulate robust macroeconometric estimates and reliable representations of reality. In this perspective, it became also clear that formulating reliable forecasts would become extremely difficult in this type of nonlinear framework 56 .

54. Dana and Malgrange 1983, f.n. 8 p. 129. These elements are added in the published version, with the new results on the dynamics on the interval. On the contrary, the developments of the working paper inspired by [START_REF] Varian | Catastrophe theory and the business cycle[END_REF] in terms of catastrophe theory disappear.

55. The initial 1981 Cepremap working paper and its published version in the Cahiers du Séminaire d'Econométrie (1983) is most likely the first economic application of these advances in France, with the comment of [START_REF] Uebe | A Nonlinear Difference Equation of Malgrange[END_REF] on [START_REF] Malgrange | Note sur le calcul des valeurs propres d'un modèle macroéconométrique[END_REF]. The international economic literature on chaotic dynamics emerges at the same time with Day and Day and Benhabib, but the bifurcation diagram was introduced latter. See notably [START_REF] Day | Complex Economic Dynamics, An Introduction to Dynamical Systems and Market Mechanisms[END_REF].

56. Theses remarks anticipate further discussions notably by Baumol and Quandt (1985) in "Models and their implications for forecasting.",Eastern Economic Journal, Jan-Mar, 11, pp.3-15. The point was particularly sensitive for Malgrange involved in applied macroeconomic modeling.

Concluding remarks

This paper focussed on some relatively neglected French contributions to the revival of nonlinear business theory. Drawing on the formal distinction between continuous and discrete-time modeling, we investigated the mathematical and analytical features of these contributions. The Benassy model exemplifies the use of the Poincaré-Bendixon theorem to prove the existence of endogenous cycles in a simple non-Walrasian framework. The discretization of Kaldor's model by Dana and Malgrange mobilizes some recent advances in bifurcation theory and chaotic dynamics.

Beyond the formal aspects discussed in detail in the text, these two contributions differ substantially in their objectives and ambition.

Benassy's main goal was the extension of the non-Walrasian macroeconomic model to a short-run dynamic setting. Facing the revival of business cycle theories, the non-Walrasian approach had to take part in the debates. From this standpoint, the implementation of nonlinear tools was a necessary detour for proving rigourously the existence of endogenous Keynesian cycles. In the meantime, the focuss shifted to the New-Keynesian research programme and the intertemporal optimizing framework. In addition, the main critique to Lucas and the equilibrium business cycle approach had to be formulated by Grandmont in his model of endogenous cycles 57 . This may partially explain why Benassy did not contribute latter to nonlinear dynamics, privileging the introduction of non-Walrasian features into the stochastic general equilibrium framework. On the contrary, for Dana and Malgrange, the Kaldorian setting plays basically the role of a pedagogical framework in nonlinear discrete dynamics. Accordingly, their discrete version contributes to highlight the relevance as well as the methodological difficulties of this new approach in macroeconomic modeling. In this perspective, the contribution also emphasizes that the discretization of a continuous time macroeconomic model may be critical in a nonlinear context.

For the these reasons, this second line of research will contribute alone to further important developments in nonlinear economic dynamics, albeit in a different perspective. Pierre Malgrange will co-sign with Grandmont in 1986 the introduction of the special issue of the Journal of Economic Theory dedicated to nonlinear dynamics. In addition, as we may recall, [START_REF] Grandmont | On endogenous competitive business cycles[END_REF] thanks Dana and Malgrange for his introduction to the mathematics on nonlinear discrete dynamics. In a recent contribution based on several interviews of Jean- Michel Grandmont, Linnemer and Visser (2018) clarify the crucial role played by Dana and Malgrange. They explain that before a 57. On the specific case of Grandmont on Keynesianism, see [START_REF] Grandmont | Keynesian issues and economic theory[END_REF].

session of the Roy seminar in 1983, Grandmont was working on Lucas' model in the seminar room, "drafting ideas about how behavior could converge to several cyclical equilibria in the same type of models used by the New Classical economists. By chance, Pierre Malgrange saw the graphs Grandmont was drawing and asked about them. (...) Jean-Michel explained that he was lost, to which Pierre retorts 'You are lost because this is chaos theory.'" 58 . Pierre Malgrange was precisely working at the time with Rose-Anne Dana on their discrete version of Kaldor's model. His familiarity with the generic properties of nonlinear mappings was certainly decisive for bringing to Grandmont's attention the explanatory power of bifurcation theory and chaotic dynamics. On this basis, Grandmont was able to worked out completely his answer to Lucas on endogenous business cycles and stabilization policy 59 .

Consequently, the first French episode in the renewal of nonlinear business cycle theory ends with the publication of Grandmont's article 60 . Indeed, these different contributions make a breakaway in the nonlinear perspective. The 'old-type' aggregated macroeconomic framework is abandoned for the overlapping generations model or the inter-temporal framework with infinite horizon agents. In this perspective, the 1990s see a growing diffusion of the mathematics on nonlinear difference equations, bifurcations and chaos within French theoretically oriented macro-economists 61 . But this is beyond the scope of this article.

58. [START_REF] Linnemer | Jean-Michel Grandmont : not playing second fiddle[END_REF]Visser, (2018), p.19 59. Firstly in his 1983 Cepremap working paper "On Endogenous Competitive Business Cycles", followed in 1984 by the related Standford technical report and the lecture at the Meetings of the Econometric Society, which form the basis of his 1985 Econometrica article. We do not enter here into the content of this well-known contribution. A synthetic presentation is given in [START_REF] Linnemer | Jean-Michel Grandmont : not playing second fiddle[END_REF]. They also mention that it is now Grandmont's most cited publication with more than 1,200 Google Scholar citations.

60. Among the related contributions, let mention different Cepremap working papers on endogenous cycles by Grandmont, Grandmont and Laroque and Laroque and the 1986 special issue of the Journal of Economic Theory. We did not consider the literature on sun spots. See JET (1986) and [START_REF] Barnett | Economic complexity : Chaos, sunspots, bubbles and nonlinearity[END_REF] 61. These advances were mainly developed by several young French scholars directly supervised or advised by Grandmont. Let mention Stefano Bosi, Jean-Pierre Drugeon, Patrick Pintus, Thomas Seegmuller or Alain Venditti. For a comprehensive list see [START_REF] Linnemer | Jean-Michel Grandmont : not playing second fiddle[END_REF]. Phillipe Michel was another main French proponent of discrete dynamics in the overlapping generations growth model. The survey of [START_REF] Michel | Une présentation simple des dynamiques complexes[END_REF] is a good illustration of these French developments on complex discrete dynamics during the 1990s. For the investigations on bifurcation, cycles and indeterminacy in the optimal growth model, see notably [START_REF] Venditti | Croissance optimale et fluctuations endogènes : un arbitrage entre préférences, technologie et impatience[END_REF]

  at the University of Paris Dauphine, Centre de Recherche des Mathématiques et de la Décision. 12. The conference was organized jointly by Gunnar Eliasson for the Industrial Institute of Economic and Social Research in Stockholm and Richard Day for the Journal of Economic Behavior and Organization, founded by Day and Sidney Winter in 1980. The model was immediately published in 1983 in the JEBO and reprinted in a book in 1986 (see below) issues as disequilibrium, structural change and micro-macro relations in the spirit of Joseph Schumpeter 13 .

  42. See Chazottes and Monticelli (2015) 43. Both have studied at the École Polytechnique during the same period, Chenciner from 1963 to 1965 and Iooss from 1964 to 1966. Iooss joined in 1974 the University of Nice where he retired in 2007 ; Chenciner arrived in 1975 moved back to the University of Paris VII in 1978. Dana and Malgrange in their initial Cepremap working paper acknowledge Alain Chenciner for his help during the preparation of their paper. Notice that Jean-Michel Grandmont mentions that he also benefited from the expertise of Alain Chenciner, in particular from a course he gave on the Hopf bifurcation in Marseille in 1986.

  and the advances made in Nice on the doubling period scenario. See above 53. A point is said to be a critical point if T ′ k (y) = 0 this bifurcation diagram for α = 20 and 120 k 220, computed by Tresser from the research team in nonlinear dynamics of Nice 54 .