N Kacem

S Hentz

S Baguet

R Dufour

Non linear dynamics of Mathieu resonators for resonant gyroscope applications

to study the stability of resonant MEM gyroscopes.

Introduction:

The resonant sensing technique [START_REF] Roessig | IntegratedMEMStuning fork oscillators for sensor applications[END_REF] is highly sensitive, has the potential for large dynamic range, good linearity, low noise and potentially low power. However, when scaling sensors down to NEMS, nonlinearities occur sooner [START_REF] Gui | and hysteresis of resonant strain gauges[END_REF] restricting the benefits of the resonant sensors. Moreover, for resonant gyroscopes, the frequency of the coriolis forces becomes closer to the resonator frequency. The idea is to investigate the dynamic behaviour of nonlinear Mathieu resonators [START_REF] Demartini | Chaos for a Microelectromechanical Oscillator Governed by the Nonlinear Mathieu Equation[END_REF] in order to find the optimum physical conditions for gyroscope designers to maximise the sensors performances.

Device and equations:

The resonant output gyroscope [START_REF] Seshia | An integrated microelectromechanical resonant output gyroscope[END_REF] shown in Figure [START_REF] Roessig | IntegratedMEMStuning fork oscillators for sensor applications[END_REF], as its name implies, utilizes resonant sensing as the basis for Coriolis force detection. In its simplest form, the device consists of three resonating elements, a proof mass and two resonating sense elements. The dynamics of the device can be described by a series of coupled differential equations. The proof mass dynamics can be described for most part by a classical spring-mass-damper equation [START_REF] Roessig | IntegratedMEMStuning fork oscillators for sensor applications[END_REF]. The dynamics of the resonators subjected to an axial time-varying Coriolis force is described by a nonlinear partial differential equation ( 2) with boundary conditions (3) as follows:

( ) ( ) ( ) M F t X dt t dX Q dt t X d e = + + 2 2 2 δ δ ( ) [ ] ( ) ( ) ( ) ( ) ( ) [ ] 2 2 0 2 2 2 2 0 2 4 4 , 2 1 , , , , , t x w
g bV t t x w s t t x w c x t x w dx x t x w l ES t Cos F N x t x w EI R l c - = ∂ ∂ + ∂ ∂ + ∂ ∂               ∂ ∂ + + - ∂ ∂ ∫ ε ρ δ ( ) ( ) 0 ) , ( , ) , 0 ( , 0 = ∂
∂ = = ∂ ∂ = t l t w t l w t t w t w

Model:

The Galerkin discretization procedure was used in order to represent the solution of Equation 2 in terms of a linearly independent set of basis functions ( )

x n φ
where each basis function satisfies the boundary conditions described in Equation 3. This procedure permits transforming the nonlinear partial differential Equation 2 into a finite system of nonlinear Mathieu equations. Since we are interested in the response of the resonator at resonance when the first mode is dominant, we reduce the system of ordinary differential equations given by Galerkin discretization to a nonlinear Mathieu equation. A perturbation technique [START_REF] Nayfeh | Introduction to Perturbation Techniques[END_REF] was used in order to obtain two first order non-linear ordinary-differential equations which describe the amplitude and phase modulation of the response and permits the computation of its stability.

Results:

The effect of the Coriolis force frequency in the resonator frequency response at its primary resonance is shown in the Figure [START_REF] Gui | and hysteresis of resonant strain gauges[END_REF]. We observe the separation of the curve branches when the Coriolis frequency is closed to the resonator frequency. This particular case is treated in Figure (3) for different kinds of resonator behavior. It appears that the quality factor decreases when the angular rate of the microgyroscope increases. Figure [START_REF] Seshia | An integrated microelectromechanical resonant output gyroscope[END_REF] shows the effect of the parametric and the nonlinear terms in the periodicity of the resonator response and its amplitude.

Conclusions:

Another limitation for the full scale of a resonant microgyroscope is underlined: the frequency ratio between the proof mass actuation and the resonator sensing. It appears in Figure [START_REF] Nayfeh | Introduction to Perturbation Techniques[END_REF] that the symmetry can be broken between negative and positive Coriolis stress effect when the resonator lose the stability for high coriolis forces amplitude and frequency. The maximum of sensitivity is situated at 0.6 of frequency ratio.

Fig. 3 :

 3 Fig.3: Predicted forced frequency responses for different angular rates of the resonant microgyrosope at a frequency equal to the resonator frequency with different kinds of behaviour. Wmax is the normalized displacement with respect to the gap. .

Fig. 3 .

 3 Fig.3.b: Mixed behaviour .

Fig. 3 .Fig. 1 :

 31 Fig.3.c: Hardening behaviour .Resonator design: length =100µm, width = 5µm, thickness = 2µm.Resonator design: length =50µm, width = 0.5µm, thickness = 2µm.

Fig. 2 :Fig. 5

 25 Fig.2: Predicted forced frequency responses including Coriolis force at different frequencies. Microgyroscope angular rate = 150 °/s.

Fig. 4 .

 4 Fig.4.a: Phase plane .

Fig. 4 :

 4 Fig.4: Long time integration Resonator design: length =50µm, width = 0.5µm, thickness = 2µm Microgyroscope angular rate = 1000 °/s


Bastien Lange
email: bastien.lange@unilasalle.fr

Guillaume Delhaye

Sylvain Boisson

Nathalie Verbruggen

Pierre Meerts

Michel-Pierre Faucon

Variation in copper and cobalt tolerance and accumulation among six populations of the facultative metallophyte Anisopappus chinensis (Asteraceae)

Keywords: Hyperaccumulation, Intraspecific variation, Metallicolous, Non-metallicolous, Plant evolution

Population variation in Co tolerance and translocation is demonstrated

Introduction

Facultative metallophytes, i.e. metallophytes that are not restricted to soils strongly enriched in trace elements (TEs), represent powerful models for studying evolution of adaptation to metal toxicity [START_REF] Baker | Accumulators and excluders-strategies in the response of plants to heavy metals[END_REF][START_REF] Pollard | Facultative hyperaccumulation of heavy metals and metalloids[END_REF]. Studies of the past decades have mostly focused on species comprising populations adapted to Zn-, Cd-, Ni-and Cu-enriched soils [START_REF] Antonovics | Heavy metal tolerance in plants[END_REF][START_REF] Macnair | Heritability and distribution of variation in degree of copper tolerance in Mimulus guttatus at Copperopolis, California[END_REF][START_REF] Bert | Zinc tolerance and accumulation in metallicolous and nonmetallicolous populations of Arabidopsis halleri[END_REF][START_REF] Escarré | Zinc and cadmium hyperaccumulation by Thlaspi caerulescens from metalliferous and nonmetalliferous sites in the Mediterranean area: implications for phytoremediation[END_REF][START_REF] Dechamps | Reaction norms of life history traits in response to Zn in Thlaspi caerulescens from metalliferous and nonmetalliferous sites[END_REF][START_REF] Pauwels | A broad-scale analysis of population differentiation for Zn tolerance in an emerging model species for tolerance study: Arabidopsis halleri (Brassicaceae)[END_REF][START_REF] Meyer | Variability of zinc tolerance among and within populations of the pseudometallophyte species Arabidopsis halleri and possible role of directional selection[END_REF][START_REF] Faucon | Copper tolerance and accumulation in two cuprophytes of South Central Africa: Crepidorhopalon perennis and C. tenuis (Linderniaceae)[END_REF][START_REF] Schvartzman | Adaptation to high zinc depends on distinct mechanisms in metallicolous populations of Arabidopsis halleri[END_REF]. In rare facultative metallophytes, TEs tolerance is constitutive at the level of the species; such is the case for the Zn-and Cd-tolerant Arabidopsis halleri and Noccaea caerulescens [START_REF] Bert | Do Arabidopsis halleri from nonmetallicolous populations accumulate zinc and cadmium more effectively than those from metallicolous populations?[END_REF]. Most often, increased tolerance to one or several TEs has been found in metallicolous populations compared to their relatives originating from normal soils [START_REF] Gonnelli | Nickel and copper tolerance and toxicity in three Tuscan populations of Silene paradoxa[END_REF][START_REF] Assunção | Differential metal-specific tolerance and accumulation patterns among Thlaspi caerulescens populations originating from different soil types[END_REF]. However, the plant response when increasing soil TEs concentration can vary widely between species and populations, and [START_REF] Baker | Accumulators and excluders-strategies in the response of plants to heavy metals[END_REF] proposed three main conceptual strategies of tolerance, including so-called "excluder", "indicator" and true "hyperaccumulator" plants (i.e. accumulation occurring from either low or high available concentration in the growth medium).

Most often, facultative metallophytes from Cu and/or Co-enriched soils behaved as Cu and/or Co excluders or indicators in culture, with limited metal translocation from the roots to the leaves (Lange et al., 2017a). Foliar Cu and/or Co concentrations above 300 mg kg -1 dry weight (DW) may be due to passive transport at phytotoxic concentrations in the growth medium [START_REF] Clemens | A long way ahead: understanding and engineering plant metal accumulation[END_REF][START_REF] Verbruggen | Molecular mechanisms of metal hyperaccumulation in plants[END_REF]. Plant populations established on naturally Cu-and Co-enriched soils are almost restricted to southeastern Democratic Republic of Congo (D.R. Congo) [START_REF] Faucon | Implication of plant-soil relationships for conservation and restoration of copper-cobalt ecosystems[END_REF]copperflora.org). These plants can thrive on soils extremely enriched with Cu and/or Co (1 to 10 g kg -1 inorganic Cu-Co) deriving from the natural Cu-and Co-outcrops of the so-called "Katangan Copperbelt". Some of the facultative metallophytes from that region have been reported to accumulate Cu and Co above 300 mg kg - 1 DW in the leaves in situ (Lange et al., 2017a). Cobalt, with no physiological function established so far in higher plants, is known to be more highly phytotoxic compared to Cu [START_REF] Palit | Effects of cobalt on plants[END_REF][START_REF] Morrissey | The ferroportin metal efflux proteins function in iron and cobalt homeostasis in Arabidopsis[END_REF][START_REF] Pilon-Smits | Physiological functions of beneficial elements[END_REF], which contrasts the rather accepted Irving Williams theory on metal toxicity [START_REF] Stumm | Aquatic chemistry, chemical equilibria and rates in natural waters[END_REF]. It has been hypothesized that Co could be an important driver in the evolution of local adaptation in metallophytes [START_REF] Duvigneaud | Plantes cobaltophytes dans le Haut Katanga[END_REF][START_REF] Malaisse | The impact of copper and cobalt orebodies upon the evolution of some plant species from Upper Shaba, Zaïre[END_REF]. However, large variations in shoot biomass and foliar Cu and/or Co concentrations are observed between populations and plants [START_REF] Faucon | Soil influence on Cu and Co uptake and plant size in the cuprophytes Crepidorhopalon perennis and C. tenuis (Scrophulariaceae) in SC Africa[END_REF][START_REF] Lange | Prediction of the edaphic factors influence upon the copper and cobalt accumulation in two metallophytes using copper and cobalt speciation in soils[END_REF][START_REF] Delhaye | Community variation in plant traits along copper and cobalt gradients[END_REF]. These variations could be accounted for by variations in Cu and Co mobility in soil (varying widely among sites) [START_REF] Pourret | Assessment of soil metal distribution and environmental impact of mining in Katanga (Democratic Republic of Congo)[END_REF], but also by variation in the capacity to tolerate and accumulate Cu and/or Co [START_REF] Lange | Prediction of the edaphic factors influence upon the copper and cobalt accumulation in two metallophytes using copper and cobalt speciation in soils[END_REF]. However, only few works have investigated variation in Cu and Co tolerance and accumulation within the flora of Cu and Co-enriched soils. [START_REF] Baker | Studies on copper and cobalt tolerance in three closely-related taxa within the genus Silene L. (Caryophyllaceae) from Zaire[END_REF] observed a variation in Cu and Co tolerance between three Silene taxa. Variation in Cu-Co tolerance was found in the Silene burchelii complex, correlated to the Cu-Co status of native soils. More recently, increased tolerance to Cu was found in metallicolous populations of Crepidorhopalon tenuis [START_REF] Faucon | Copper tolerance and accumulation in two cuprophytes of South Central Africa: Crepidorhopalon perennis and C. tenuis (Linderniaceae)[END_REF] and Haumaniastrum katangense [START_REF] Peng | Differences in copper accumulation and copper stress between eight populations of Haumaniastrum katangense[END_REF]. Most often, increased tolerance is coupled to decreased Cu concentrations in shoots, as a result of specific tolerance mechanisms restricting Cu uptake and/or translocation [START_REF] Ouzounidou | Comparative responses of a copper-tolerant and a copper-sensitive population of Minuartia hirsuta to copper toxicity[END_REF][START_REF] Gonnelli | Nickel and copper tolerance and toxicity in three Tuscan populations of Silene paradoxa[END_REF]. The plant response to Cu treatment can vary widely depending on the concentration, the species and the growth medium including growth inhibition, no effect on growth to growth stimulation [START_REF] Jiang | Growth response and phytoextraction of copper at different levels in soils by Elsholtzia splendens[END_REF][START_REF] Lou | The copper tolerance mechanism of Elsholtzia haichowensis, a plant from copper-enriched soils[END_REF][START_REF] Lange | Copper and cobalt mobility in soil and accumulation in a metallophyte as influenced by experimental manipulation of soil chemical factors[END_REF]. Increased requirements of Cu were found in Crotalaria cobalticola [START_REF] Boisson | Copper tolerance of three Crotalaria species from southeastern D.R. Congo at the early development stage[END_REF] and Crepidorhopalon perennis [START_REF] Faucon | Copper tolerance and accumulation in two cuprophytes of South Central Africa: Crepidorhopalon perennis and C. tenuis (Linderniaceae)[END_REF] two strict endemic species of Cu-enriched soils. In contrast to Cu, studies on intraspecific variation in Co tolerance and accumulation are very scarce. [START_REF] Paliouris | Arsenic, cobalt and nickel tolerances in two populations of Silene vulgaris (Moench) Garcke from Ontario, Canada[END_REF] found increased tolerance to Co in a population of Silene vulgaris from mine tailings compared to a population from uncontaminated site. In another study, a genetic variation in Co tolerance was shown among clones of Agrostis gigantea [START_REF] Hogan | Tolerance and toxicity of cobalt, copper, nickel and zinc in clones of Agrostis gigantea[END_REF]. A more recent study has demonstrated that Co tolerance may be associated with the translocation of Co from the root to the shoot, and accumulation in Haumaniastrum katangense and H. robertii, two metallophytes from the Katangan Copperbelt [START_REF] Kabeya | Tolerance and accumulation of cobalt in three species of Haumaniastrum and the influence of copper[END_REF].

Anisopappus chinensis (Asteraceae) is a widespread facultative metallophyte in tropical Africa. Compared to the few facultative metallophytes studied so far, A. chinensis appears to have a broader niche, being very widespread on non-contaminated soils in tropical Africa [START_REF] Lange | Tolérance et accumulation du cuivre et du cobalt chez les métallophytes facultatives d'Afrique tropicale[END_REF], and being particularly frequent on cobalt-rich soils in southeastern D.R. Congo [START_REF] Duvigneaud | Plantes cobaltophytes dans le Haut Katanga[END_REF]. The ecology and evolution of Cu and Co tolerance and accumulation in facultative metallophytes remains poorly studied, especially for Co and species that present a high occurrence on non-metalliferous soils. Our working hypothesis is that differentiation in Cu and Co tolerance and accumulation exists between populations of such species, and is related to variations in the level of Cu and Co in the native soil.

In this paper, we assessed Cu and Co tolerance and accumulation in four metallicolous and two non-metallicolous populations of the facultative metallophyte A. chinensis. We addressed the following questions: (i) Is A. chinensis constitutively tolerant to Cu and/or Co? (ii) Does intraspecific variation in Cu and Co tolerance, if any, related to Cu and Co concentrations in native soils? (iii) Does increased tolerance to Co, if any, rely on Co translocation and foliar accumulation?

Materials and methods

Plant material

Anisopappus chinensis L. Hook.f. & Arn. (Asteraceae, subfam. Asteroideae, tribe Anthemideae) is a perennial facultative metallophyte native of and widely distributed in tropical central, austral, western and eastern Africa (but also extended to China and southeastern Asia) [START_REF] Wild | A revision of the genus Anisopappus Hook. & Arn. (Compositae)[END_REF]. In south-eastern D.R. Congo, A. chinensis is widespread in grasslands of natural Cu-Co outcrops and the surrounding wooded savannas [START_REF] Séleck | Chemical soil factors influencing plant assemblages along copper-cobalt gradients: implications for conservation and restoration[END_REF]Lange et al., 2017b). Four metallicolous (M) and two non-metallicolous (NM) populations were sampled.

At each site, the seeds from 20 plants were harvested and pooled. The sites of harvest were: Etoile (Et, M), Fungurume V (F5, M), Goma II (G2, M), Niamumenda (Nm, M), Mikembo (Mi, NM) and Kiswishi (Ki, NM) (Table 1). In nature, M populations exhibit broad variations in leaf Cu and Co concentrations: 3-1335 and 4-2820 mg kg -1 DW, respectively [START_REF] Lange | Prediction of the edaphic factors influence upon the copper and cobalt accumulation in two metallophytes using copper and cobalt speciation in soils[END_REF].

Leucanthemum superbum D.H.Kent (Asteraceae, subfam. Asteroideae, tribe Anthemidae), an ornamental perennial herb which has not evolved mechanisms to tolerate Cu-Co, was chosen as control species (hereafter "Ctrl"). Seeds were commercially purchased (Vilmorin©). M = metallicolous, NM = non-metallicolous. Inorganic concentrations of Cu and Co in the rooting zone (ionic forms) were obtained by Cu and Co speciation modelling [START_REF] Lange | Prediction of the edaphic factors influence upon the copper and cobalt accumulation in two metallophytes using copper and cobalt speciation in soils[END_REF][START_REF] Pourret | Assessment of soil metal distribution and environmental impact of mining in Katanga (Democratic Republic of Congo)[END_REF]. Standard deviation is given in brackets.

Experimental design

A hydroponic experiment was conducted in a greenhouse from April to June 2015 (three months). Seeds were germinated in vermiculite watered with nutrient solution for three weeks before being transferred in hydroponic conditions. Three-week old seedlings were then grown superbum were grown for each treatment (one pot per treatment). Pots were replicated 10 times (n = 10) and randomized each week.

Measurements

After eight weeks in hydroponics, plants were harvested and separated into shoots and roots. Shoots were washed with Alconox ® 1% in deionized water, dried at 65 °C for 48 h and weighed [START_REF] Faucon | Revisiting copper and cobalt concentrations in supposed hyperaccumulators from SC Africa: influence of washing and metal concentrations in soil[END_REF]. Roots were washed with 0.1 mM EDTA for 5 min, rinsed three times for 5 min in deionized water before being dried at 65 °C for 48 h and weighed [START_REF] Chipeng | Copper tolerance in the cuprophyte Haumaniastrum katangense (S. Moore) P.A. Duvign. and Plancke[END_REF]. Leaf Cu-Co and root Cu-Co concentrations in A. chinensis and L. superbum were determined using Inductively Coupled Plasma-Mass Spectrometry (Thermo Scientific XSERIES2) after a dry ashing [START_REF] Faucon | Soil influence on Cu and Co uptake and plant size in the cuprophytes Crepidorhopalon perennis and C. tenuis (Scrophulariaceae) in SC Africa[END_REF]. Briefly, 0.2 g (accurately weighed) of leaf or root powder were ashed at 500 °C for 12 h in a muffle furnace. Then, ashes were dissolved in a 32.5% HNO3 solution (3 mL) at 80 °C for 5 min. Samples were then diluted to a 5% HNO3 solution (50 mL) and stored at 4 °C until being analyzed. The precision and accuracy of analysis were determined using the SRM 1547 (peach leaves, NIST); bias of ± 5% were obtained in this case. At each treatment, average tolerance indices (TI) of populations were calculated: average biomass on metal divided by average biomass on control.

Statistical analysis

Plant biomass, leaf Cu and leaf Co were analyzed by partially nested ANOVAs on logtransformed data. The "edaphic group" (i.e. M vs NM) and "treatment" factors were fixed factors. The "population" factor was a random factor nested in the "edaphic group" factor. The "block" factor was tested as a random factor and did not have a significant effect on any variable. Data for Cu and Co were tested independently. The population variation in leaf Cu, root Cu, leaf Co, root Co and leaf/root Cu-Co ratio (Cu-Co translocation factor) were investigated by one-way ANOVAs on log-transformed data. The control species was not considered for the population variation analysis. Pearson's correlations were used to test the relationship between shoot and root biomass and between leaf and root Cu-Co. The population variation in average TI was not tested statistically due to the lack of replication. Relationship between population TI and native soil concentration or translocation factor was tested for each treatment using Spearman's rank correlation (corrected by Monte Carlo permutation test based on 999 replications). The analyses were performed using R software 3.2.2 (www.Rproject.org).

Results

Response of shoot biomass and leaf copper and cobalt concentrations to Cu and Co in

the growth medium

Response to Cu

In all tested plants (including the Ctrl plants), except those from the M population F5, shoot biomass decreased dramatically with increasing Cu to +10 µM (Fig. 1). Growth was very poor at +50 µM Cu in all populations and plants from Mi (NM) did not survive at this concentration (Fig. 1). In contrast to other populations, leaf Cu of F5 plants did not increase significantly with an increase of Cu to +10 µM, whereas a steady increase was observed in all populations, up to 250-500 mg kg -1 DW in leaves, with increasing Cu from +10 to +50 µM.

Tolerance indices for Cu were consistently very low in NM populations at +10 and +50

µM Cu (TI ≤ 0.07), while they showed extensive variation among M populations (TI: 0.12-0.51 and 0.08-0.29 at +10 and +50 µM Cu, respectively). The ranking of population TI differed between +10 and +50 µM Cu and did not match the ranking in inorganic Cu concentration in the native soil (rank correlation not significant).

Shoot and root biomass were highly correlated at +10 and +50 µM Cu (r = 0.96 and r = 0.76; p<0.001, respectively). ANOVA on plant biomass did not show a significant effect of the edaphic group (i.e. M vs NM) (Table S1). Otherwise all factors and interactions were significant. Patterns of variation in plant biomass varied between +10 and +50 µM Cu (data not showed), in accordance with the highly significant interaction population within edaphic group*treatment.

Response to Co

The response of shoot biomass to the increase of the concentration of Co in the nutrient solution varied widely among populations. The growth of M populations Et, F5 and G2 was not significantly affected, and leaf Co concentrations increased progressively up to >300 mg kg -1 DW (Fig. 2). For the M population Nm, increasing Co to +5 µM had no significant effect on shoot biomass, and leaf Co increased only slightly (+25 mg kg -1 ). However, increasing Co from +5 to +20 µM led to a significant decrease in shoot biomass and a steady leaf Co increase in these plants (from 25 to 250 mg kg -1 ). In both NM populations Mi and Ki and the Ctrl species, increasing Co in the growth medium had a strong negative impact on shoot biomass, coupled to dramatic increase in the leaf Co (>400 mg kg -1 ).

Tolerance indices were consistently lower in both NM populations and the Ctrl species at +5 and +20 µM Co compared to M populations. Large variations of TI were observed among populations at +20 µM Co (from 0.03 to 0.75). The ranking of populations for TI did not differ between +5 and +20 µM Co and matched remarkably the ranking of native soils for Co concentration: TI = 0.03 (Mi) < 0.08 (Ki) < 0.22 (Nm) < 0.66 (G2) < 0.74 (Et) < 0.75 (F5) (rs = 0.89; p=0.015 and rs = 0.94; p=0.005 at +5 and +20 µM Co, respectively).

Shoot and root biomass were highly correlated at +5 and +20 µM Co (r = 0.92 and r = 0.94; p<0.001, respectively). Similarly to Cu, ANOVA on plant biomass did not show a significant effect of the edaphic group, while all factor and interaction effects were significant (Table S2). The interaction population within edaphic group*treatment was slightly significant.

Population variation in leaf and root Cu and Co concentration

Leaf and root Cu concentration

Leaf Cu ranged from 16 to 338 mg kg -1 and from 76 to 761 mg kg -1 at +10 and +50 µM Cu, respectively. Root Cu ranged from 828 to 6813 mg kg -1 and from 4347 to 81 088 mg kg -1 at +10 and +50 µM Cu, respectively. Leaf Cu and root Cu were not significantly correlated.

ANOVAs on leaf and root concentrations showed a significant effect of all factors and interactions, except the interaction edaphic group*treatment interaction (Table S3). The few variations of leaf and root Cu observed at +10 µM Cu did not match TI (rank correlation not significant) (Fig. 3a). Leaf Cu did not vary significantly between populations at +50 µM Cu (Fig. 3b).

Leaf/root Cu ratio was very low in all populations and the Ctrl, and varied weakly at +10 µM (from 0.01 [Ctrl] to 0.05 [G2, M]) and at +50 µM (from 0.00 [Ctrl] to 0.04 [Ki]) (Fig. 3). Variations in leaf/root Cu ratio did not match TI (rank correlation not significant).

Leaf and root Co concentration

At +5 µM Co, leaf Co ranged from 19 to 149 mg kg -1 in M populations and from 49 to 116 mg kg -1 in NM populations; and root Co ranged from 204 to 1186 mg kg -1 in M populations and from 1729 to 6577 mg kg -1 in NM populations. At +20 µM Co, leaf Co ranged from 110 to 504 mg kg -1 in M populations and from 89 to 620 in NM populations; and root Co ranged from 485 to 2690 mg kg-1 in M populations and from 2995 to 7794 mg kg -1 in NM populations. Leaf

Co and root Co concentration were positively correlated (r = 0.51, p<0.001).

ANOVA on leaf and root Co showed a significant effect of all factors and interactions, except the population effect for leaf Co (Table S4). In contrast to Cu, leaf and root Co varied among populations at the intermediate and highest treatments (Fig. 4a,b). At +5 µM Co, root

Co was much lower in M populations (442 ± 120 mg kg -1 ) compared to NM populations ( 3967± 1318 mg kg -1 ) and the control species (2198 ± 575 mg kg -1 ). A similar pattern was observed at +20 µM Co. Variations in root Co among populations matched variations in TI at +20 µM Co, with higher root Co concentrations in populations having lower TI (Mi and Ki) (rs = -0.94; p<0.05).

Leaf/root Co ratio varied weakly at +5 µM Co (Fig. 4a). At +20 µM Co, leaf/root Co ratio was significantly higher in the M populations F5 (0.29) and G2 (0.24) compared to the NM populations Mi (0.06) and Ki (0.09) (Fig. 4b). A positive correlation was observed between leaf/root Co and TI at +20 µM Co (rs = 0.94; p<0.01).

Discussion

Low expression of Cu tolerance in metallicolous populations grown in hydroponics

Anisopappus chinensis is a facultative metallophyte occupying a broad niche including nonmetalliferous habitats and soils enriched in Cu and Co in southeastern D.R. Congo. The latter species presents a high occurrence on non-metalliferous soils, as opposed to species from the Copperbelt studied so far [START_REF] Lange | Tolérance et accumulation du cuivre et du cobalt chez les métallophytes facultatives d'Afrique tropicale[END_REF]. Results of growth under Cu stress for the control species and A. chinensis plants demonstrated that tolerance to Cu is not constitutive in the species. Surprisingly, a very poor expression of Cu tolerance was observed in hydroponics for metallicolous populations originating from soils extremely concentrated in Cu (Fig. 1, Table 1). We expected higher tolerance to Cu in metallicolous compared to non-metallicolous populations, with respect to the metallic status of their native soils. Despite somewhat higher tolerance indices in metallicolous populations, A. chinensis contrasts with other facultative metallophytes such as C. tenuis and H. katangense. The latter species consistently showed a strong increased tolerance to Cu in metallicolous compared to non-metallicolous populations [START_REF] Faucon | Copper tolerance and accumulation in two cuprophytes of South Central Africa: Crepidorhopalon perennis and C. tenuis (Linderniaceae)[END_REF][START_REF] Peng | Differences in copper accumulation and copper stress between eight populations of Haumaniastrum katangense[END_REF]. For C. tenuis, increasing Cu in the growth medium even showed a growth stimulation. Contrast in the realized ecological niche between A. chinensis and species that present a higher occurrence on metalliferous soils may account for contrast in the degree of tolerance and response to Cu of M populations.

Elevated tolerance to Cu was demonstrated in metallicolous plants of H. katangense and C.

tenuis growing in hydroponics or MS medium [START_REF] Chipeng | Copper tolerance in the cuprophyte Haumaniastrum katangense (S. Moore) P.A. Duvign. and Plancke[END_REF][START_REF] Faucon | Copper tolerance and accumulation in two cuprophytes of South Central Africa: Crepidorhopalon perennis and C. tenuis (Linderniaceae)[END_REF]. At an equivalent concentration, metallicolous plants of A. chinensis suffered from Cu toxicity.

Metallicolous populations of A. chinensis may have evolved contrasting tolerance to Cu, which could be the result of specific soil-plant interactions not expressed in hydroponics. For instance, specific rhizosphere processes involving the microbial communities might account for the ability to grow under high Cu concentration on metalliferous soils. Among them, metal-tolerant arbuscular mycorrhizal fungi (AMF) and endophytic bacteria can have a beneficial effect on Cu tolerance by plants [START_REF] Cornejo | Glomalin-related soil protein in a Mediterranean ecosystem affected by a copper smelter and its contribution to Cu and Zn sequestration[END_REF][START_REF] Sun | Genetic diversity and characterization of heavy metal-resistant-endophytic bacteria from two coppertolerant plant species on copper mine wasteland[END_REF][START_REF] Ma | Plant growth promoting rhizobacteria and endophytes accelerate phytoremediation of metalliferous soils[END_REF][START_REF] Lenoir | Arbuscular mycorrhizal fungal responses to abiotic stresses: A review[END_REF]. Future work should test the presence of AMF and endophytic bacteria in M populations of A. chinensis and, if any, explore possible relationships with the ability to grow under high soil Cu levels. Co (>300 mg kg -1 DW). In our experimental conditions, the foliar concentration increase up to >300 mg kg -1 DW in all population with increasing Cu to +50 µM might be explained by passive absorption [START_REF] Baker | Heavy metal tolerance in plants: evolutionary aspects[END_REF]Verkleij and Schat, 1989). This study is the first demonstration of a high degree of tolerance to Co associated with foliar Co accumulation up to >300 mg kg and interaction between is still poorly understood and particular adaptation mechanisms are expected.

Increased tolerance to Co in populations from

Conclusion

Tolerance to Cu and Co are not constitutive in Anisopappus chinensis. The contrast between Cu toxicity in our study and field observations on Cu-enriched soils needs to be clarified. For instance, rhizospheric processes and/or biotic interactions which can have a beneficial effect on metal tolerance by plants but cannot be expressed in hydroponics, such as associations with mycorrhizal fungi and/or endophytic bacteria, should be investigated. Such processes might also account for the high concentrations of Cu in field growing plants. In contrast, populations form Co-enriched soils did express increased tolerance to Co. Variation in the degree of tolerance among these populations was related to the concentration of Co in the native soil, strongly suggesting that Co tolerance has evolved in response to directional selection. This study is the first to report population differentiation in Co tolerance in a facultative metallophyte depending on Co soil status. Interestingly, Co tolerance was positively correlated to Co translocation to the leaves up to >300 mg kg -1 dry weight at phytotoxic concentration in the nutritive solution (+20 µM Co). This suggests that A. chinensis from soils highly enriched in available Co would behave as Co indicator sensu [START_REF] Baker | Accumulators and excluders-strategies in the response of plants to heavy metals[END_REF] and that increased capacity to translocate Co may have evolved in the most tolerant populations.

Acknowledgments

The Belgian Fund for Scientific Research (FRS-FNRS) is acknowledged for financial support to Bastien Lange as a research fellow of the Fonds pour la Recherche dans l'Industrie Finally, thank to Philippe Jacolot, from UniLaSalle (Beauvais, France), for his precious help in ICP-MS analyses.

Author contributions

BL, GD, SB, NV, MPF and PM designed the experiment. BL conducted the experiment with the participation of GD. BL did the mineral and statistical analyses. BL wrote the manuscript with the participation of NV, MPF and PM. All authors critically revised the manuscript and gave their final approval.

  for eight weeks in 5 L polyethylene pots filled with a nutrient solution composed of: 20 µM Fe-EDDHA (iron salt of ethylenediamine-di-o-hydroxyphenylactic acid), 0.88 mM K2SO4, 0.25 mM KH2PO4, 10 µM NaCl, 1000 µM Ca(NO3)2, 1000 µM MgSO4, 10 µM H3BO3, 1 µM ZnSO4, 1 µM MnSO4, 0.5 µM CuSO4, 0.01 µM (NH4)6Mo7O24 (adapted from[START_REF] Meyer | Variability of zinc tolerance among and within populations of the pseudometallophyte species Arabidopsis halleri and possible role of directional selection[END_REF]. The nutrient solution was adjusted to pH 5.8 with KOH (0.01 M) and changed each week. Five treatments were performed using Cu (II) sulfate (CuSO4•5H2O) or Co (II) sulfate (CoSO4•7H2O): Control (with no addition of Cu and Co), +10 µM Cu, +50 µM Cu, +5 µM Co, +20 µM Co. Plants of A. chinensis (one plant from each population) and one plant of L.

Figure 1

 1 Figure 1 Shoot mass and leaf Cu as a function of increasing Cu in the nutrient solution in four metallicolous and two non-metallicolous populations of Anisopappus chinensis. Leucanthemum superbum was used as a control. Three Cu treatments: Control, +10 μM Cu, +50 μM Cu. Mean ± standard deviation (n = 10). Mean values with the same superscript are not significantly different (one-way ANOVAs on log-transformed data followed by post-hoc multiple comparison, Tukey HSD test).

Figure 2

 2 Figure 2 Shoot mass and leaf Co as a function of increasing Co in the nutrient solution in four metallicolous and two non-metallicolous populations of Anisopappus chinensis. Leucanthemum superbum was used as a control. Three Co treatments: Control, +5 μM Co, +20 μM Co. Mean ± standard deviation (n = 10). Mean values with the same superscript are not significantly different (one-way ANOVAs on log-transformed data followed by post-hoc multiple comparison, Tukey HSD test).

Figure 3

 3 Figure 3 Leaf and root Cu concentrations in four metallicolous and two non-metallicolous populations of Anisopappus chinensis at two concentrations of Cu in the nutrient solution a. +10 μM Cu in the nutrient solution b. +50 μM Cu in the nutrient solution. Et (Etoile), F5 (Fungurume V), G2 (Goma II) and Nm (Niamumenda) are metallicolous (M) populations. Mi (Mikembo) and Ki (Kiswishi) are non-metallicolous (NM) populations. Ctrl is the control species: Leucanthemum superbum. Shoot mass and root mass, as well as leaf Cu and root Cu were tested independently, without the Ctrl. Translocation factors (leaf/root Cu ratios), including the Ctrl, were also calculated. Mean ± standard deviation (n = 10) with the same superscript are not significantly different (results of one-way ANOVAs on logtransformed data followed by post-hoc multiple comparison, Tukey HSD test).

Figure 4

 4 Figure 4 Leaf and root Co concentrations in four metallicolous and two non-metallicolous populations of Anisopappus chinensis at two concentrations of Co in the nutrient solution a. +5 μM Co in the nutrient solution b. +20 μM Co in the nutrient solution. Et (Etoile), F5 (Fungurume V), G2 (Goma II) and Nm (Niamumenda) are metallicolous (M) populations. Mi (Mikembo) and Ki (Kiswishi) are non-metallicolous (NM) populations. Ctrl is the control species: Leucanthemum superbum. Shoot mass and root mass, as well as leaf Co and root Co were tested independently, without the Ctrl. Translocation factors (leaf/root Co ratios), including the Ctrl, were also calculated. Mean ± standard deviation (n = 10) with the same superscript are not significantly different (results of one-way ANOVAs on logtransformed data followed by post-hoc multiple comparison, Tukey HSD test).

  Co-enriched soils as a result of local adaptationResults showed that tolerance to Co is not constitutive in A. chinensis. Nonmetallicolous populations of A. chinensis have not evolved mechanisms to tolerate concentrations of Co that are toxic for nontolerant plants such as L. superbum. In contrast to Cu, large intraspecific variation in the response to increasing Co in the growth medium was observed in A. chinensis. This study is only the second one to report intraspecific variation in tolerance to Co in a metallophyte.[START_REF] Paliouris | Arsenic, cobalt and nickel tolerances in two populations of Silene vulgaris (Moench) Garcke from Ontario, Canada[END_REF] demonstrated an increased tolerance to Co in Silene vulgaris from a Co-contaminated soil compared to its relative on uncontaminated soil. Interestingly, these authors did not find any relationship between tolerance indices and Co concentrations in the native soils (water extract). Our work is the first to report population differentiation in Co tolerance related to the level of Co in the native soil. Tolerance indices at each Co treatment matched remarkably well the ranking of native soils for Co concentration. Selective pressure stemming from Co toxicity in highly contaminated soils may have induced local adaptation and enhanced tolerance to Co in the populations Et, F5 and G2. Therefore, the broad ecological niche of A. chinensis in southeastern D.R. Congo might be ascribed to within species genetic variation in Co tolerance. A similar conclusion was drawn by Faucon et al. (2012), for the facultative metallophyte Crepidorhopalon tenuis for Cu. The existence of such Co-tolerant ecotypes suggests that Co tolerance can explain the broad ecological niche of facultative metallophytes growing on Cu-enriched soils. 4.3. High accumulation of Co and increased translocation in populations adapted to extreme Co-levels Relatively high foliar concentrations of Cu and Co have been reported from field growing plants of A. chinensis among M populations. Anisopappus chinensis from M populations was classified by Lange et al. (2014) as a Cu-Co hyperaccumulator based on operationally defined threshold values proposed by van der Ent et al. (2013) for foliar Cu and

  et l'Agriculture(FRIA, Belgium). The authors are grateful to Tenke Fungurume Mining S.A.R.L., which allowed seed collection of Anisopappus chinensis. Pacifique Kizila and Héritier Bahati from the Université de Lubumbashi (UNILU, D. R. Congo) are thank for their help in collecting seeds. Kristel Wart, from the Université Libre de Bruxelles (Brussels, Belgium) is acknowledged for her essential assistance in the experimental measurements.

Table 1 Populations of Anisopappus chinensis: location and habitat description

 1

	Sites	Status	Coordinates	Habitat	Soil inorganic	Soil inorganic
			(GCSWGS84 DD)		Cu	Co
					(mg kg -1 )	(mg kg -1 )
	Etoile (E)	M	11.38'S 27.35'E	Cu-Co hill disturbed by	953 (2836)	1311 (2541)
				mining. Secondary
				vegetation on mine
				waste.
	Fungurume 5 (F5)	M	10.62'S 26.29'E	Natural Cu-Co hill not	4774 (9070)	2558 (2685)
				disturbed by mining.
				Grassland.

  -1 DW in a facultative metallophyte largely distributed among non-metalliferous soils. Interestingly, Co tolerance was positively correlated to Co translocation to the leaves at elevated concentration, implying that the strategy of Co tolerance in A. chinensis does not rely on Co exclusion. The lower root Co concentration in the most tolerant compared to non-tolerant populations seems to confirm this hypothesis. Our results suggest that A. chinensis from soils highly enriched in available Co would behave as Co indicator rather than true hyperaccumulator Intraspecific variation in leaf and root Co accumulation was shown for the first time in a facultative metallophyte. Interesting work could be to test the possible use of the most tolerant populations of A. chinensis, which translocate and accumulate Co above 300 mg kg -1 DW in leaves, in phytomining applications, for the production and valorization of a Co bio-ore[START_REF] Faucon | Element case studies: copper and cobalt[END_REF]. Interestingly, the latter populations that also present the lowest root Co concentrations in our study, originate from soils with the highest level of inorganic Co.Selective pressure stemming from Co toxicity in the soil of origin may have induced a particular ability in (i) limiting Co storage in the root system and (ii) translocating Co to the leaves in plants originating from Co-enriched soils. The ecophysiology of Co tolerance and accumulation

	sensu Baker (i.e. accumulation occurring from either low or high available concentration in the
	growth medium). The latter hypothesis should be verified by looking at the response of
	metallicolous plants grown on a natural metalliferous soil of reference with different levels of
	Co.


Najib Kacem
email: najib.kacem@cea.fr

Sébastien Hentz
email: sebastien.hentz@cea.fr

Sébastien Baguet
email: sebastien.baguet@insa-lyon.fr

Régis Dufour
email: regis.dufour@insa-lyon.fr

Nonlinear dynamics of Mathieu resonators for resonant gyroscope applications Dynamique non linéaire des résonateurs de Mathieu pour les applications de gyromètres résonants

Keywords:

Operation at resonance is not only advantageous for micromechanical actuators but also for sensing.

The sensor is comprised of elements that operate at their characteristic resonance and the effect of the measurand is detected as a change in the resonant characteristics of these elements. Resonant sensing often involves the detection of an input measurand by means of a resonant frequency shift in the sensing device.

 restricting the benefits of the resonant sensors. Moreover, for resonant gyroscopes, the frequency of the Coriolis forces becomes closer to the resonator frequency. The idea is to investigate the dynamic behaviour of nonlinear Mathieu resonator [3] in order to find the optimal physical conditions for gyroscope designers to maximise the sensors performances.

RESUME

Travailler à la résonance n'est pas seulement avantageux en actuation micromécanique mais aussi en détection. Un capteur à détection fréquentielle comprend un élément sensible résonant qui oscille à sa fréquence propre. L'effet de la mesurande modifie cette fréquence de résonance, souvent avec une loi quasi linéaire qui s'étend sur une gamme de mesure identifiable. Le paramètre physique à mesurer est détecté sous forme d'un décalage fréquentiel.

Le mode de détection fréquentielle [START_REF] Roessig | IntegratedMEMStuning fork oscillators for sensor applications[END_REF] a pour avantages d'être extrêmement sensible, a le potentiel d'avoir une large gamme dynamique, peu sensible au bruit, une large gamme de linéarité et une faible consommation électrique. Cependant, en réduisant la taille des détecteurs vers les NEMS, les non-linéarités apparaissent rapidement [START_REF] Gui | Nonlinearity and hysteresis of resonant strain gauges[END_REF] limitant ainsi les avantages des capteurs fréquentiels. De plus, pour les gyromètres résonnants, la fréquence des forces de Coriolis devient comparable à la fréquence du résonateur. L'idée est d'examiner le comportement dynamique des résonateurs non linéaire de Mathieu [START_REF] Demartini | Chaos for a Microelectromechanical Oscillator Governed by the Nonlinear Mathieu Equation[END_REF] afin de trouver les conditions physiques optimales pour les concepteurs de gyroscope dans le but de maximiser les performances de ce type de capteur.

M ots clés: MEMS \ gyromètre \ résonateur de Mathieu \ dynamique non linéaire \ gamme dynamique

Introduction

The permanent quest for cost cuts has led to the use of potential "In-IC" compatible thin SOI-based technologies, which imposes drastic size reduction of the sensors. Combined with the need for in-plane actuation for fabrication and design simplicity, this implies a large reduction in detectability. Moreover nonlinearities [START_REF] Kacem | From MEMS to NEMS : Modelling and characterization of the non linear dynamics of resonators, a way to enhance the dynamic range[END_REF] occur sooner for small structures which reduce their dynamic range.

In the context of a "small" z-axis resonant gyroscope (Figure 1) designed with high actuation frequency, the dynamic of the sensing part governed by a nonlinear Mathieu equation has been modelled.

The device consists of a proof mass suspended by flexures attached to a rigid frame. The proof mass is driven relative to the outer frame using embedded lateral comb drive actuators. If an external rotation θ is applied to the chip about the z-axis, the Coriolis force F c = 2M θV acting on the proof mass is transmitted to the outer frame and communicated axially onto two resonators.

The periodic compression and tension of the resonators by the Coriolis force at the proof mass drive frequency modulates the resonant frequency of these force sensors. Each resonator embedded in the feedback loop of an oscillator circuit. Thus, by demodulating the oscillation frequency, the rotation rate applied to the device can be estimated.

The Mathieu equation has been widely studied in the context of parametric resonance, but our study will be restricted to the primary resonance on the way of design optimisation for high drive frequencies in order to enhance the dynamic range of resonant micro-gyroscopes.

Device and equations

The resonant output gyroscope [START_REF] Seshia | An integrated microelectromechanical resonant output gyroscope[END_REF] shown in figure [START_REF] Roessig | IntegratedMEMStuning fork oscillators for sensor applications[END_REF], as its name implies, utilizes resonant sensing as the basis for Coriolis force detection. In its simplest form, the device consists of three resonating elements, a proof mass and two resonating sense elements.

Equation of motion

The dynamics of the device can be described by series of coupled differential equations. The proof mass dynamics can be described for most part by a classical spring-mass-damper equation. The dynamics of the resonators subjected to an axial time-varying Coriolis force is described by a nonlinear partial differential equation. The respective equations can be written as :

d 2 X( t) d t2 + δ Q dX( t) d t + δ2 X( t) = F e M cos δt (1)
EI ∂ 4 w(x, t) ∂ x4 + ρbh ∂ 2 w(x, t) ∂ t2 + c ∂ w(x, t) ∂ t = Ñ + Fc cos δt + Ebh 2l l 0 ∂ w(x, t) ∂ x 2 dx ∂ 2 w(x, t) ∂ x2 + 1 2 ε 0 bC n V dc + V ac cos( Ωt ) 2 (g -w(x, t)) 2 (2)
w(0, t) = w(l, t) = ∂ w ∂ x (0, t) = ∂ w ∂ x (l, t) = 0 (3)
where X is the proof mass displacement along the drive axis, Q is the drive quality factor, δ and F e are the drive angular frequency and the electrostatic drive force applied to the proof mass M . x is the position along the microbeam length, E and I are the Young's modulus and moment of inertia of the cross section. Ñ is the applied tensile axial force due to the residual stress on the silicon or the effect of the measurand, t is the time, ρ is the material density, h is the microbeam thickness, g is the capacitor gap width, and ε 0 is the dielectric constant of the gap medium. The last term in equation ( 2) represents an approximation of the electric force assuming a complete overlap of the area of the microbeam and the stationary electrode including the edge effects by the coefficient C n .

Normalization

For convenience and equations simplicity , we introduce the nondimensional variables :

w = w g , x = x l , t = t τ (4)
Where τ = 2l 2 h 3ρ E . Substituting equation ( 4) into equations ( 2) and (3), we obtain :

∂ 4 w ∂x 4 + ∂ 2 w ∂t 2 + c ∂w ∂t -α 2 [V dc + V ac cos(Ωt)] 2 (1 -w)) 2 = N + F c cos δt + α 1 1 0 ∂w ∂x 2 dx ∂ 2 w ∂x 2 w(0, t) = w(1, t) = ∂w ∂x (0, t) = ∂w ∂x (1, t) = 0 (5)
The parameters appearing in equations ( 5) are :

c = cl 4 EIτ , N = Ñ l 2 EI , F c = Fc l 2 EI , δ = δτ α 1 = 6 g h 2 , α 2 = 6 ε 0 l 4 Eh 3 g 3 , Ω = Ωτ (6)

Resolution

A reduced-order model is generated by modal decomposition transforming equations ( 5) into a finitedegree-of-freedom system consisting of ordinary differential equations in time. We use the undamped linear mode shapes of the straight microbeam as basis functions in the Galerkin procedure. To this end, we express the deflection as :

w(x, t) = n k=1 a k (t)φ k (x) (7)
Where a k (t) is the k th generalized coordinate and φ k (x) is the k th linear undamped mode shape of the straight microbeam, normalized such that 1 0 φ k φ j = 0 for k = p and governed by :

d 4 φ k (x) dx 4 = λ 2 k φ k (x) (8) φ k (0) = φ k (1) = φ k (0) = φ k (1) (9)
Here, λ k is the k th natural frequency of the microbeam. We multiply equations ( 5) by φ k (x)(1 -w) 2 , substitute equations ( 7) into the resulting equation, use equations (8) to eliminate d 4 φ k (x) dx 4 , integrate the outcome from x = 0 to 1, and obtain :

äk + c k ȧk + λ 2 k a k -2 n j=1 λ 2 j a j 2 + c j a j ȧj + a j äj 1 0 φ k φ j 2 dx + n j=1 λ 2 j a j 3 + c j a j 2 ȧj + a j 2 äj 1 0 φ k φ j 3 dx - n j=1 (N + F c cos δt) a j + α 1 a j 3 1 0 φ j 2 dx 1 0 φ k φ j dx +2 n j=1 α 1 a j 4 1 0 φ j 2 dx + (N + F c cos δt) a j 2 1 0 φ k φ j φ j dx - n j=1 α 1 a j 5 1 0 φ j 2 dx + (N + F c cos δt) a j 3 1 0 φ k φ j 2 φ j dx = α 2 [V dc + V ac cos(Ωt)] 2 1 0 φ k dx (10)
Noting that the first mode should be the dominant mode of the system and the other modes are neglected, so it is sufficient to consider the case n = 1. Equation (10) becomes :

ä1 + (500.564 + 12.3 (N + F c cos δt))a 1 + (1330.9 + 38.

3 (N + F c cos δt))a 1 2 + (927 + 28 (N + F c cos δt) + 151α 1 ) a 1 3 + 471α 1 a 1 4 + 347α 1 a 1 5 +c 1 ȧ1 + 2.66c 1 a 1 ȧ1 + 1.85c 1 a 1 2 ȧ1 + 2.66a 1 ä1 +1.85a 1 2 ä1 = - 8 3π α 2 [V dc + V ac cos(Ωt)] 2 (11)
To analyse the equation of motion (11), it is convenient to invoke perturbation techniques [START_REF] Seshia | An integrated microelectromechanical resonant output gyroscope[END_REF] which work well with the assumptions of "small" excitation and damping, typically valid in MEMS resonators.

In order to facilitate the perturbation approach using the method of averaging, a standard constrained coordinate transformation is introduced, as given by :

                       a 1 = A(t) cos [Ωt + β(t)] ȧ1 = -A(t)Ω sin [Ωt + β(t)] ä1 = -A(t)Ω 2 cos [Ωt + β(t)] (12)
In addition, since near-resonant behavior is the principal operating regime of the proposed system, a detuning parameter, σ is introduced, as given by :

Ω = ω n + εσ (13)
Separating the resulting equations and averaging them over the period 2π Ω in the t-domain results in the system's averaged equations, in terms of amplitude and phase, which are given by :

Ȧ ε = - Ac 1 2 -0.23A 3 c 1 - 7A 3 cos[4β]F c 8π (δ -4ω n ) + 7A 3 cos 4β -2πδ ωn F c 8π (δ -4ω n ) - 0.76A 2 cos[3β]F c δ -3ω n + 0.76A 2 cos 3β -2πδ ωn F c δ -3ω n - 0.5Acos[2β]F c δ -2ω n + 7A 3 sin 2β -πδ ωn sin πδ ωn F c 2π (δ -2ω n ) + 0.5Acos 2β -2πδ ωn F c δ -2ω n - 0.76A 2 cos[β]F c δ -ω n + 0.76A 2 cos β + π 2 -2δ ωn F c δ -ω n + 8VacVdcsin[β]α 2 3πω n + 0.76A 2 cos[β]F c δ + ω n - 0.76A 2 cos β + 2π(δ+ωn) ωn F c δ + ω n + 0.5Acos[2β]F c δ + 2ω n + 7A 3 cos[2β]F c 4π (δ + 2ω n ) - 0.5Acos 2 β + π 2 + δ ωn F c δ + 2ω n - 7A 3 cos 2 β + π 2 + δ ωn F c 4π (δ + 2ω n ) + 0.76A 2 cos[3β]F c δ + 3ω n + 7A 3 cos[4β]F c 8π (δ + 4ω n ) - 7A 3 cos 4β + 2π 4 + δ ωn F c 8π (δ + 4ω n ) - 0.76A 2 cos 3β + 2π 3 + δ ωn F c δ + 3ω n (14) β ε = - 0.98sin 2πδ ωn F c δ + 0.7A 2 ω n + Aσ - 7A 2 sin[4β]F c 8π (δ -4ω n ) + 7A 2 sin 4β + π 8 -2δ ωn F c 8π (δ -4ω n ) + 0.76Asin 3β + π 6 -2δ ωn F c δ -3ω n - 9A 2 cos[β]sin[β]F c ω n δ 2 -4ω 2 n - 4.6Asin[3β]F c ω n δ 2 -9ω 2 n + 2.3Asin β + π 2 -2δ ωn F c δ -ω n - 347.6A 2 ω n - 10.5A 2 N ω n - 56.6A 2 α 1 ω n - 108.4A 4 α 1 ω n - 0.85VacVdccos[β]α 2 Aω n - 2.3Asin β + 2π(δ+ωn) ωn F c δ + ω n - 4.6Asin[β]F c ω n δ 2 -ω 2 n - 7A 2 sin 2 β + π 2 + δ ωn F c 2π (δ + 2ω n ) - 0.76Asin 3β + 2π 3 + δ ωn F c δ + 3ω n + 7A 2 sin[4β]F c 8π (δ + 4ω n ) - 7A 2 sin 4β + 2π 4 + δ ωn F c 8π (δ + 4ω n ) - 21A 2 sin 2πδ ωn F c 4πδ + 7A 2 sin 2β + π 4 -2δ ωn F c 2π (δ -2ω n ) - 4cos[β]sin[β]F c ω n δ 2 -4ω 2 n - 0.5sin 2 β + π 2 + δ ωn F c δ + 2ω n + 0.5sin 2β + π 4 -2δ ωn F c δ -2ω n (15)
The steady-state motions occur when Ȧ = β = 0, which corresponds to the singular points of the equations system ( 14) and (15). Thus, the frequency-response equation can be writen in its parametric form as :

           σ = f [β, δ] A = f [β, δ] (16)
3 Results

The effect of the Coriolis force frequency in the resonator frequency response at its primary resonance is shown in Figure [START_REF] Gui | Nonlinearity and hysteresis of resonant strain gauges[END_REF]. The ratio between the drive and the resonator frequency varies from 0.25 to 1 and the external rotation applied to the device about the z-axis is about 150˚/s. We observe the separation of the curve branches when the Coriolis frequency equals the resonator frequency. It appears also that the quality factor decreases when the external rotation applied to the microgyroscope increases. For detection stability, embedding the resonators in the feedback loop of an oscillator circuit [START_REF] Seshia | An integrated microelectromechanical resonant output gyroscope[END_REF] permits the conservation of a uniform sensitivity through a defined full scale limited by a critical amplitude.

0.012 0.014 0.016 w max q = 250 ±ês q = 100 ±ês q = -100 ±ês q = -250 ±ês Resonator design: length =100µm, width = 5µm, thickness = 2µm.

Resonator design: length =50µm, width = 0.5µm, thickness = 2µm. Figure [START_REF] Kacem | From MEMS to NEMS : Modelling and characterization of the non linear dynamics of resonators, a way to enhance the dynamic range[END_REF] shows the effect of the parametric and the nonlinear terms in the periodicity of the resonator response and its amplitude(length =50µm, width = 0.5µm, thickness = 2µm). Moreover, the transition to the quasi-periodicity is shown in the phase plane when the drive frequency equals the resonator frequency and for an external rotation θ = 1000˚/s. This phenomenon is due to the resonator frequency modulation by the parametric terms but it can be also a consequence of the energy transfer between the parametric terms in the Mathieu equation and the nonlinear Vander pool damping term [START_REF] Nayfeh | Introduction to Perturbation Techniques[END_REF]. The quasi-periodicity of the resonator response may lead to a chaotic [START_REF] Demartini | Chaos for a Microelectromechanical Oscillator Governed by the Nonlinear Mathieu Equation[END_REF] device. w max q = 250 ±ês q = 100 ±ês q = -100 ±ês q = -250 ±ês ifferent angular rates of the resonant microgyrosope at a frequency equal to the ior. Wmax is the normalized displacement with respect to the gap. width = 0.5µm, thickness = 2µm. In this paper, it proves that increasing the drive frequency of the proof mass for resonant microgyroscopes can be advantageous in order to increase the detectability of the device (the maximum is situated at 0.6 of frequency ratio between the drive part and the sense part). This improvement has a direct effect on the full scale of the microgyroscope which will be limited by the instability of the Mathieu resonator.

Fig. 1 -

 1 Fig. 1 -Schema of the resonant microgyroscope

Fig. 3 .

 3 Fig.3.b: Mixed behavior .

Fig. 3 .Fig. 2 :

 32 Fig.3.c: Hardening behavior ., width = 5µm, thickness = 2µm.Resonator design: length =50µm, width = 0.5µm, thickness = 2µm.

Fig. 4 :

 4 Fig.4: Long time integration Resonator design: length =50µm,

Fig. 2 -

 2 Fig. 2 -Predicted forced frequency responses.

Fig. 3 :

 3 Fig.3: Predicted forced frequency responses for different angular rates of the resonant microgyrosope at a frequency equal to the resonator frequency with different kinds of behavior. Wmax is the normalized displacement with respect to the gap. .

  Fig.3.a: Softening behavior .

Fig. 3 .

 3 Fig.3.b: Mixed behavior .

Fig. 3 .

 3 Fig.3.c: Hardening behavior .

Fig. 1 :

 1 Fig.1: Schema of the resonant microgyroscope..

Fig. 2 :Fig. 3 -

 23 Fig.2: Predicted forced frequency responses including Coriolis force at different frequencies. Microgyroscope angular rate = 150 °/s.

FcFig. 3 :

 3 Fig.3: Predicted forced frequency responses for different angular rates of the resonant microgyrosope at a frequency equal to the resonator frequency with different kinds of behavior. Wmax is the normalized displacement with respect to the gap.

Fig. 3 .

 3 Fig.3.b: Mixed behavior .

Fig. 3 .Fig. 1 :

 31 Fig.3.c: Hardening behavior .Resonator design: length =100μm, width = 5μm, thickness = 2μm.Resonator design: length =50μm, width = 0.5μm, thickness = 2μm.

Fig. 2 :Fig. 4 Fig. 5 Fig. 4 -

 2454 Fig.2: Predicted forced frequency responses including Coriolis force at different frequencies. Microgyroscope angular rate = 150 °/s.

4 ConclusionsA

 4 complete analytical model describing the nonlinear dynamics of Mathieu resonators applied to resonant MEM gyroscopes has been presented. The model is based on the Galerkin discretization coupled with a perturbation technique (the averaging method) enabling the study of the dynamic behaviour of the sensing parts of these devices. In addition, this model constitutes a quick and powerful tool on the way of understanding the different dynamics phenomena and the description of the dynamics transition from periodicity to quasi-periodicity.

Fig. 5 Fig. 4 :Fig. 5 -

 545 Fig.5 Variation of Wmax with the frequency ratio between the coriolis force and the sensing resonance. θ is the angular rate of the microgyroscope.