Arnaud Poitou

Guillaume Racineux

A squeezing experiment showing binder migration in concentrated suspensions

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I. INTRODUCTION

Flow-induced migration of particles has been an active research field for many years. Acrivos' model ͓Leighton and Acrivos ͑1987͒; Phillips et al. ͑1992͔͒ states that a particle placed in a flow which presents a strain gradient receives from the other particles, on the average, more shocks on the side of the flow that has the higher shear rate and, therefore, tends to migrate in the opposite direction. Acrivos' model consists of representing this phenomenon with a diffusion equation. Thus, it is possible to describe a Couette device experiment or a Poiseuille flow. More recently, these results were extended to three dimension by numerical methods. However, since Biot's studies, other models have been developed which are closer to soil mechanics theories ͓Bowen et al. ͑1976͒; Lante ´ri et al. ͑1991͒; Lante ´ri et al. ͑1996a͒; Dutilly ͑1998͒; Louge ͑1996͒; Casella ͑1997͔͒. Nevertheless, regardless of the model, the main difficulty in these approaches is to establish an experimental methodology which allows one to identify the rheological parameters on industrial materials. The aim of this paper is to present a solution to this problem. First, we will show that in a squeezing experiment particle migration-or, more precisely, binder migration-occurs as long as the velocity at which the test is carried out does not exceed a critical value. Next, we will develop a mathematical model to describe the test as well as a complete methodology to identify the rheological parameters of this model. We will give an analytical expression of the critical velocity, which is related to the physical characteristics of the suspension. The exact identification of the parameters requires the use of a numerical program, which was developed specifically.

II. EXPERIMENTS

A. Materials

The concentrated suspension used in this study was a paste at a solid concentration of 60% per weight. The Procatalyse Company produces this saturated paste by extrusion in order to manufacture petrochemical catalysts. The T i O 2 powder is made of polydisperse grains with an average size of 4 m. The powder is obtained by sol atomization, which causes the grains to be constituted of agglomerates of finer particles. The sol-gel process results in calibrated particles which are, on the average, about 40 nm in diameter. These elementary spheres themselves are aggregates of anastase microcrystallites of about 65 Å ͓Jabbes ͑1995͔͒. Figure 1, is a typical scanning electron microscope photomicrograph which shows the morphology of the powder and the agglomerate size distribution. The different types of agglomerates can be seen: some have a diameter between 6 and 7 m while most of the others are around 1 m. Their shape is approximately spherical. The specific mass of the powder is 3.35 g/cm [START_REF] Casella | Etude du comportement d'un milieu pa ˆteux-Approche biphasique et identification[END_REF] and its specific area is 280 m 2 /g.

The binder is an aqueous ( f ϭ 10 Ϫ3 Pa s) solution of concentrated nitric acid which acts as a peptization agent. Because of capillary forces, the wet powder is initially agglomerated and aggregated. The acid induces repulsive interactions among the aggregates: the particles are electrostatically charged by protons through a phenomenon similar to the D.L.V.O. mechanism. This colloidal action combined with mechanical mixing allows the agglomerates to break up, which leads to a homogeneous paste.

The mixing device is a Z-blade internal mixer. These blades scrape the paste against the recipient walls with a complex trajectory. Thus, the paste is subjected to a history of high and almost homogeneous shear. Its behavior is very sensitive to the water content; therefore, it is very important to conduct the mixing process carefully in order to obtain reproducible results:

͑i͒
The mixer is cooled to avoid water losses by evaporation. ͑ii͒

The powder is introduced first in order to avoid fluid flow through interstices located at the point where the blades enter the mixing recipient. ͑iii͒ The dry powder is mixed a few minutes in order to break apart the largest agglomerates. ͑iv͒ The binder is introduced in one step. ͑v͒

The paste is then mixed for approximately 1 h to get a homogeneous product.

To control the agglomerate size after mixing, mercury porosimetry experiments were carried out on dry samples ͑i.e., without binder͒. The porosity of the skeleton is perfectly stable from one experiment to the next ͑void volume of 0.35 cm 3 /g and pore size between 10 and 100 nm͒.

B. Rheological measurement

Squeezing tests were carried out on an M.T.S. servocontrolled hydraulic tension test machine. The squeezing device consists of two coaxial circular plates ͑40 mm in diam-eter͒. The upper plate is fixed whereas the lower one moves vertically at a prescribed velocity V. The evolution of the squeezing force F is measured against the size h of the gap between the plates ͑see Fig. 2͒. The paste sample is placed between the plates and then squeezed at constant velocity. The initial gap h 0 between the two plates at the beginning of the tests is such that h 0 /D Ӷ 1, where D is the plate diameter.

Initial conditions

To ensure reproducible initial conditions, we prepared specimens with a constant initial thickness. Our procedure consisted of preforming them quickly in moulds with calibrated diameter D and thickness h 0 ͑Fig. 3͒. The preshaped sample was then placed on the lower plate, which was then moved upwards until the sample reached the upper plate ͑Fig. 4͒.

Boundary conditions

To avoid slipping of the paste at the wall, we used roughened plates, i.e., plates onto which we pasted abrasive papers in order to force sticking contact at the wall. We used self-adhesive papers intended for surface polishing because they present the following advantages:

͑i͒ Their grain size is controlled. ͑ii͒

Polishing papers resist shearing very well. ͑iii͒ Self-adhesive papers have good resistance to water, which means that they do not alter the composition of the paste during the test.

To confirm that the paste sticks on plates with abrasive papers, we tested plates with different surface roughness: polished plates made of rectified iron (Ra ϭ 0.4 m) and plates with abrasive papers of different grain sizes: P400 ͑35 m͒, P600 ͑26 m͒, and P800 ͑22 m͒. Figure 5 shows the results of the tests with an initial gap size h 0 ϭ 2.5 mm and piston velocity V ϭ 0.02 mm/s without abrasive paper and with P400, P600, and P800 abrasive papers. Two conclusions can be reached from these curves:

͑i͒
The tests carried out with different kinds of abrasive papers give the same results: the curves obtained with the different papers are superimposed on Fig. 5. ͑ii͒

The test carried out with polished plates ͑i.e., without paper͒ gives different results, indicating an interaction between the paste and the boundary. In this paper, we are interested only in tests with no slipping at the walls ͑i.e., with abrasive papers͒.

FIG. 4.

Samples are placed on the lower plate, which is moved upwards until it reaches the upper plate.

III. RESULTS AND DISCUSSION

A. Effect of velocity: dissociation

Figure 6 shows the squeezing force curve at different squeezing velocities and with abrasive papers ͑i.e., with sticking contact at the wall͒. Figure 7 shows the force as a function of velocity for a few fixed gap widths. These results show effects which appear contradictory: ͑i͒ For a squeezing velocity greater than a critical piston velocity V c Ӎ 1 mm/s, the force increases with the velocity, but less than linearly ͑i.e., the force is less than doubled if the velocity is doubled͒. ͑ii͒ On the contrary, for a squeezing velocity less than about 1 mm/s, the force increases when the velocity decreases. At the end of a test, the sample is very fine ͑0.2 mm͒ and it tears when one lifts the upper plate ͑Fig. 8͒. It was, therefore, impossible to take samples of paste in different places ͑at the center, on the edges...͒ with sufficient precision to measure the volume fraction of powder in the sample. Furthermore, in very fine samples, the binder evaporates very quickly. But after tests at low squeezing velocities, if we touched the sample we noted ͑at least qualitatively͒ that the paste remaining between the plates contains a higher volume fraction of powder at the end of the test than at the beginning and a higher volume fraction of powder in the center of plates than on the edges. This observation enables us to find an explanation for why the squeezing force at low velocities can be greater than at high velocities. A test carried out at low velocity is a drained test ͑the binder has had time to filter through the powder͒; therefore, at the end of the test, one measures the resistance of a quasicompact skeleton of powder. On the contrary, at higher velocity, the binder cannot flow through the powder and, thus, it behaves as a lubricant between the grains. The test results presented in Fig. 9 assumption which had already been made by many authors: Kendall ͓Kendall ͑1987͔͒, for example, observed this type of phenomenon on cement pastes; Lanos ͓Lanos ͑1993͔͒, on ash pastes, also observed a very strong migration of phases resulting in a solid nucleus at the center of the specimen. This reveals two interesting features of these materials:

͑i͒
It is, generally, impossible to test them in a homogeneous manner. ͑ii͒ However, at high velocities, the force increases with velocity, which is also an indication that a homogeneous test is possible in such a case.

B. Effect of initial thickness h 0

Figure 9 shows results obtained at constant squeezing velocity V ϭ 0.02 mm/s with polished plates 20 mm in diameter and for different initial gap sizes h 0 . These results show that for fixed gap size h and velocity V, the measured force depends on the initial gap size h 0 and, therefore, on the strain history of the paste. If we assume that the strain can dissociate a specimen, then forming the specimen with an initial thickness h 0 has to be undertaken with great care. In particular, it is not possible to presqueeze it slowly. This is the reason why we preshaped the samples.

IV. MATHEMATICAL MODELING

A. Physical concept: existence of a critical velocity

In our material, we can distinguish three different causes of dissipation: ͑i͒ The intrinsic viscous dissipation of the fluid ͑binder͒ on the microscopic scale:

D f ϭ ͵ ⍀ f ␥ ˙f 2 dv,

͑1͒

where ␥ ˙f is the fluid shear rate, whose dimension is v f /h, where h is the gap between plates.

͑ii͒ The intrinsic viscous dissipation caused by friction between particles:

D s ϭ ͵ ⍀ K␥ ˙s mϩ1 dv with ␥ ˙s ϭ v s h , ͑2͒
where ␥ ˙s is the solid shear rate and m the coefficient of shear rate sensitivity. ͑iii͒ An interaction dissipation due to the viscous drag through the powder ͑Darcy's law͒:

D i ϭ ͵ ⍀ k͑v f Ϫv s ͒ 2 dv. ͑3͒
We use a highly concentrated suspension ͑at a solid concentration of 60% per weight͒ and an aqueous binder whose viscosity is very low ( f ϭ 10 Ϫ3 Pa s). To a first approximation, it seems reasonable to neglect the intrinsic dissipation of the binder. Then, from a physical point of view, particle migration in our material results from a balance between two contradictory effects:

͑i͒
On the one hand, particles tend to migrate toward zones where the intrinsic viscous dissipation is low, i.e., where the solid shear rate is low. ͑ii͒

On the other hand, particle migration implies dissipation because of the viscous drag of the flow through the powder skeleton.

The result of this conflict is that migration occurs if the reduction in viscous dissipation when particles migrate towards low shear strain areas is greater than the dissipation required for filtration. It turns out that, for most pastes, the coefficient of rate sensitivity is strictly less than 1. Moreover, the viscous dissipation has the dimension ␥ ˙s mϩ1 and the filtration dissipation has the dimension (v s Ϫv f ) 2 : therefore, the ratio between filtration and skeleton deformation is negligible at low squeezing velocities and large at high velocities. This indicates that at low velocities it is ''worthwhile'' for particles to migrate. The situation is reversed at large squeezing velocities for which filtration ''costs'' too much to enable migration. The limit between the two regimes occurs when filtration is no longer negligible. We evaluate an order of magnitude of the critical velocity:

v s Ϫv f Ӎ v s 10 ␥ ˙s Ӎ v s h v s Ӎ V ͑4͒
at least near the center

V c ϭ 100 ͩ K kh mϩ1 ͪ 1/͑lϪm ͒ . ͑5͒

B. Mechanical three-dimensional analysis

To extend this physical concept to three dimensions, we can model the heterogeneous behavior of a concentrated paste by describing the global behavior of the paste as the superposition of the behavior of each of its constituents. This approach is used in soil mechanics to model the behavior of drained porous media. Thus, the powder is assumed to consist of a quasirigid skeleton in which the binder can circulate ͓Coussy ͑1987͔͒. More recently, this approach has been used to model the behavior of concentrated paste in injection ͓Lante ´ri et al. ͑1991͒; Lante ´ri et al. ͑1996a͒; Dutilly ͑1998͔͒ and extrusion ͓Casella ͑1997͒; Louge ͑1996͔͒ processes. Two main approaches have been proposed in the literature to obtain macroscopic equations for multiphase media:

͑i͒ The first is the mixture theory, which is a generalization of continuum mechanics ͓Bowen et al. ͑1976͒; Poitou ͑1988͒; Lante ´ri et al. ͑1996a͔͒. Conservation laws are written directly for each phase on the macroscopic scale, which leads directly to the model equations.

͑ii͒ The second is the averaging method, which consists of writing conservation laws for each constituent on the microscopic scale ͓Gilbert ͑1987͒; Ni and Beckerman ͑1991͔͒. These equations are then integrated with convolution products to obtain conservation laws on the macroscopic scale for each phase. On this scale, each constituent is considered to be a continuous medium.

A distinction has to be made between constituents ͑the binder and the granular mate-rial͒ on the microscopic scale and phases ͑solids and fluid͒ on the macroscopic scale. In particular, we do not determine the behavior of the granular material alone, but of a solid phase consisting of the granular material lubricated by the binder.

Representative volume

In order to be able to write macroscopic conservation laws, it is necessary to define a representative elementary volume or macroscopic point. This volume has to be large compared to the size of the solid particle, but small compared to the gap size.

Conservation laws

Under the hypothesis of the existence of a representative elementary volume, our paste can be described as the superposition of two continuous media. At each macroscopic point, both phases c coexist proportionately to their volume fraction c . We designate the fluid and the solid phases, respectively, by the indexes f and s. Then, each phase is described by: ͑i͒ its apparent density c ϭ c (c) , where (c) is the density of constituent ͑c͒. ͑ii͒ its macroscopic velocity v គ c , which can differ from the velocity of another phase. ͑iii͒ its macroscopic stress field c and strain rate D c ϭ 1 2 ͓grad(v គ c )ϩgrad(v គ c ) t ͔; and ͑iv͒ its own boundary conditions.

Mass balance:

In the case of a nonreactive medium, we can deduce from the global mass balance of phase c the local form

ץ c ͑c͒ ץt ϩdiv͑ c ͑c͒ v គ f ͒ ϭ 0. ͑6͒
If the constituents ͑c͒ on the microscopic scale are individually incompressible, the mass balance for each phase is expressed by

ץ f ץt ϩdiv͑ f v គ f ͒ ϭ 0 ץ s ץt ϩdiv͑ s v គ s ͒ ϭ 0. ͑7͒
These equations indicate an apparent compressibility of each macroscopic phase which enables dissociation without any diffusion mechanism. Moreover, if the mixture ͑as is the case of our paste͒ is saturated ͑no air or gas inside͒, then the volume fraction verifies:

f ϩ s ϭ 1. ͑8͒
Momentum balance: Our paste is at a concentration of 60% per weight; its global viscosity is about 300 Pa s and the powder density is 3.35 g/cm [START_REF] Casella | Etude du comportement d'un milieu pa ˆteux-Approche biphasique et identification[END_REF] . Mass and inertia forces can be neglected with respect to viscous effects. The momentum balance for each phase can thus be written as ͓Lante ´ri et al. ͑1991͔͒:

div͑ f ͒ϩ គ f ϭ 0 គ div͑ s ͒ϩ គ s ϭ 0 គ . ͑9͒
Since the flow considered is isothermal, it is not necessary to write the balance of energy and entropy.

Constitutive equations and boundary conditions

Momentum supply: Assuming that the filtration is isotropic, it is generally agreed that the interaction force is the sum of a Darcy term and a reaction caused by the gradient of volume fraction:

គ s ϭ Ϫ គ f ϭ k͑v គ f Ϫv គ s ͒ϩp grad s , ͑10͒
with s ϭ 1Ϫ f , and k being the interaction coefficient between the solid and the fluid, i.e., the ratio between the viscosity f of the binder and the permeability B of the skeleton ͓Costet and Sanglerat ͑1975͔͒. B depends on many parameters, such as the size of the particles, their shape, their morphology, their chemical properties, etc. In order to minimize the number of adjustable parameters, we assume here that the permeability B follows the Kozeny expression

B ϭ b f 3 ͑1Ϫ f ͒ 2 with b ϭ 1 2CT 2 S 2 Ӎ D 2 100 , ͑11͒
where T is the tortuosity of the pores, S the specific surface of the powder, C the circularity coefficient which defines the shape of the pores, and D the diameter of the particles.

Constitutive equation for the fluid phase: The binder used in our paste is a mixture of water and nitric acid. Assuming that the binder is Newtonian and incompressible on the microscopic scale and that the particles are rigid and not in direct contact with one another, along with Gilbert ͑1987͒ we write:

f ϭ Ϫ f pIϩ2 f D f , where f ϭ cst,

͑12͒

where f is the viscosity of the fluid and v គ f its macroscopic velocity.

Constitutive equation of the solid phase:

The macroscopic constituent referred to as the solid phase is made up of solid particles surrounded by a fine layer of adsorbed fluid. This constituent cannot be tested separately and it is impossible to measure its behavior. Therefore, we can reasonably assume, like Lante ´ri et al. ͓Lant e ´ri et al. ͑1996b͔͒, that the solid phase has purely viscous behavior which depends only on the velocity of the solid phase:

s ϭ Ϫ s pIϩ2 s D s with s ϭ Kͱ2tr͑D s 2 ͒ mϪ1 , ͑13͒
where K is the consistency and m the coefficient of rate sensitivity.

Many experimental and theoretical approaches ͓Krieger and Dougerthy ͑1959͒; Quemada ͑1977͔͒ lead us to assume that the consistency K diverges as the volume fraction approaches the maximum particle packing m . We extend these results to our power law behavior:

K ϭ K 0 ͩ 1Ϫ s m ͪ ␣ with ␣ Ͼ 0,

͑14͒

where ␣ is chosen in order to be consistent with Einstein's model for the viscosity of dilute sphere suspensions:

K ϭ K 0 ͑1ϩ2.5 s ͒. ͑15͒
Then, one obtains

␣ ϭ 2.5 m . ͑16͒
To a first approximation, we assumed that the maximum particle packing density m is equal to the compact packing density of disorderly rigid spheres: 64%. Boundary conditions: We assume that the fluid and solid phases stick to the wall:

v គ f ϭ v គ s ϭ v គ wall . ͑17͒
At the outflow of the plates, the boundary condition on both phases expresses local equilibrium with the outside medium:

f n គ ϭ Ϫ f p a n គ s n គ ϭ Ϫ s p a n គ , ͑18͒
where n គ is the outward normal vector and p a the ambient pressure.

C. Numerical resolution

Since the problem is unsteady, we adopted an incremental approach in three steps ͓Racineux ͑1999͒; Racineux et al. ͑1999͔͒. ͑i͒ In the first step, we assume that the volume fraction is known and we solve the flow problem, which is of the same level of complexity as the Stokes' problem. We use the finite element method with a velocity-pressure formulation and a Newton-Raphson algorithm to solve the nonlinearity.

͑ii͒ Then, the volume fraction is actualized using an implicit algorithm and a Taylor-Galerkin formulation by solving the equation of mass transport balance. The difficulty of the calculation results mainly from the hyperbolic nature of this equation.

͑iii͒ At the end of a time increment the mesh is convected in order to keep the size of the elements constant.

V. IDENTIFICATION

A. Behavior of the solid phase

The behavior of the solid phase ͑m, K͒ is identified at high velocity, i.e., in flow conditions which make the paste remain homogeneous. In this particular case, the velocity of the fluid phase is equal to the velocity of the solid phase and, therefore, the tensor of homogeneous equivalent stresses can be expressed as:

ϭ ϪpIϩ2Kͱ2tr͑D 2 ͒ ͑ mϪ1 ͒ D s .
͑19͒

Using the Hele-Shaw approximations ͑since the tests were carried out for h/R Ӷ 1͒, one shows that the squeezing force is

F ϭ 2 ͩ 2mϩ1 m ͪ m 1 ͑mϩ3͒ K 2 ͑mϪ1/2͒ V m h ͑ 2mϩ1 ͒ R ͑ mϩ3 ͒ . ͑20͒
We identified m and K on the force curve obtained at a compression velocity of 10 mm/s ͑Fig. 10͒. Using a least-square method, we perform a best-fit approximation of the power law to experimental data, which leads to:

m ϭ 0.01 K ϭ 22 600.
The coefficient of strain rate sensitivity is very close to zero, which means that the homogeneous paste behaves like a rigid-plastic material ͑Bingham fluid͒.

B. Interaction coefficient

The interaction coefficient k is the ratio between the binder viscosity f and the permeability of the skeleton B:

k ϭ f B . ͑21͒
The permeability is identified at low velocity, i.e., in a test in which dissociation actually occurs ͑drained test at 0.1 mm/s͒. In this case, the flow equations have no analytical solution. Therefore, we use the finite element code to simulate the squeezing test. Since we must identify only one parameter, we proceed by dichotomy and obtain the interaction coefficient: C. Discussion

k ϭ k 0 s 2 ͑1Ϫ s ͒ 3

Simulation results

The rheological parameters were identified with two experimental curves. Figure 11 shows the superposition of experimental results and numerical simulations for the squeezing test at every compression velocity. Figure 12 shows the evolution of the volume fraction calculated with our model at a point located at the center, under the upper plate, for different compression velocities. Three remarks must be made about these curves: ͑i͒ From Fig. 11, our model verifies the existence of paste dissociation. In particular, for velocities greater than 1 mm/s, the calculated force increases with the velocity. On the contrary, for velocities less than 1 mm/s, the force increases as the velocity decreases. ͑ii͒

For high velocities ͑10 and 1 mm/s͒, the calculated dissociation is very low. This result is consistent with our strategy of identifying K and m at large velocities, assuming that the sample then remains homogeneous. ͑iii͒ For low velocities, the calculated volume fractions indicate migration which is directly related to the increase of force as the velocity decreases.

Thus, there is good agreement between experimental results and simulations, at least until h/2 reaches about 0.3 mm.

Order of magnitude of the interaction coefficient

The order of magnitude of k can be evaluated as a function of the viscosity f of the binder and the permeability B of the skeleton:

k ϭ f B . ͑23͒
From relations ͑11͒, ͑22͒, and ͑23͒ one gets an estimate of k 0 as a function of the particle diameter D and the fluid viscosity f :

k 0 ϭ 100 f D 2 . ͑24͒
With the experimental value k 0 ϭ 5.10 12 Pa s m 2 , this leads to a particle diameter:

D Ӎ 0.1 m. ͑25͒
This value is approximately five times larger than the particle size suggested by the mercury porosimetry measurement performed on dry samples ͑specific area 280 m 2 /g or particle size 20 nm͒. This result, however, is in good agreement with our model if one notes that Eq. ͑11͒ holds for monodisperse granulometry only and that for polydisperse powders the larger pores have a predominant influence on permeability.

Critical dissociation velocity

From the above expression of the critical velocity ͓Eq. ͑5͔͒ and with our experimental value for the interaction coefficient, we calculate a theoretical critical velocity V c ϭ 2 mm/s which is also consistent with the experiments ͑see Fig. 6͒. More generally, Eq. ͑5͒ combined with Eqs. ͑11͒, ͑22͒, and ͑23͒ yields

V c ϭ 100 ͩ K k h mϩ1 ͪ 1/͑1Ϫm ͒ ϭ 100 ͩ K D 2 100 f h mϩ1 ͑ 1Ϫ s ͒ 3 s 2 ͪ 1/͑1Ϫm ͒ . ͑26͒
In this expression, if the volume fraction is fixed, m and D are the two main materials parameters that govern the transition between the two regimes. The larger m͓0,1͔, the lower the critical velocity ͑Fig. 13͒. In the same way, the larger D, the lower V c ͑Fig. 14͒. From a geometrical point of view, the lower h, the larger V c ͑dissociation occurs more easily at the end of the test than at the beginning͒.

Limitations

At the end of the simulation, the results diverge greatly, particularly at low squeezing velocities. There are several possible explanations for this: the packing fraction introduced in the model could have been underestimated, which would lead to a sharp force increase when dissociation becomes important. Another explanation of the discrepancy between simulations and experiments at the end of the tests could be some numerical errors which tend to increase because of the hyperbolic nature of the mass balance equation. But, more likely, the reason is that in extending Krieger and Dougherty's relation to nonlinear viscous fluids through Eq. ͑14͒ we implicitly assumed that m does not depend on s . But, as outlined above, a slight variation of m can have an important impact on dissociation. It would have been possible to fit a function m( s ) in order to improve the correlation between model and experiment. However, our objective here is essentially to show that, even with only few adjustable rheological parameters ͑essentially K, k, m͒, it is possible to describe this squeezing experiment quite reasonably well.

VI. CONCLUSION

In this paper, we have shown indirectly that particle and binder migration occurs when a paste is deformed beyond a certain critical value of the equivalent shear strain. This phenomenon has been analyzed and modeled in the framework of the mixture theory. This theory is usually difficult to handle because of the difficulty related to the experimentation. In particular, it is a priori difficult to solicit the different phases of a heterogeneous sample separately. To cope with this difficulty, we have shown that it is possible, instead of deforming the two phases separately, to deform them either together ͑i.e., at large velocities͒ or independently ͑at low velocities͒. Thus, using a numerical program, we were able to identify the different rheological parameters of our model, which seems to lead to reasonable predictions of the overall behavior of the paste.

The paste flow can be modeled by approaches that are intermediate between non-Newtonian fluid rheology and soil mechanics. Today, the numerical difficulties of this type of model have been eliminated, at least partly, thanks to recent progress in the numerical solution of hyperbolic equations made in the last few years. Thus, mixture theory models which, a few years ago, led to difficulties in the identification of parameters are now much more practical. Quemada, D., ''Rheology of concentrated disperse systems and minimum energy dissipation principle-iviscosity, concentration, relationship,'' Rheol. Acta 16, 82-94 ͑1977͒. Racineux, G., ''Rhe ´ologie des pa ˆtes mine ´rales: cas du me ´lange TiO2-HNO3 destine ´a `l'extrusion de supports de catalyseurs,'' The `se, Ecole Normale Supe ´rieure de Cachan, 1999. Racineux, G., F. Chinesta, and A. Poitou, ''Simulation de l'e ´coulement des pa ˆtes mine ´rales en vue de l'identification de leur comportement,'' in Proceedings of the 4th Colloque National en Calcul des Structures, Teknea, 1999, Vol. II, pp. 617-622.

FIG. 1 .

 1 FIG. 1. SEM photomicrograph of T i O 2 powder ͓Jabbes ͑1995͔͒.

FIG. 2 .FIG. 3 .

 23 FIG. 2.Force vs gap size.

FIG. 5 .

 5 FIG. 5.Tests with P800 abrasive papers ͑initial gap h 0 ϭ 2.5 mm͒.

  FIG. 7.Force as a function of velocity for a few fixed gap widths.

FIG. 9 .

 9 FIG. 9.Influence of the initial gap size h 0 .

with k 0

 0 FIG. 10.The solid phase behavior is identified with a power law.

FIG. 11 .

 11 FIG. 11. Comparison between experiments and calculations.

FIG. 13 .

 13 FIG. 13.Influence of m on V c .


Augustin Fruchard

Reinhard Schäfke

Classification of resonant equations

Keywords: Ackerberg-O'Malley resonance, Gevrey asymptotics, singular perturbation. AMS Classification: 34E

Consider a singularly perturbed equation of the form εy -f (x, ε)y + g(x, ε)y = 0 ,

) is called resonant in the sense of Ackerberg-O'Malley, if there is a solution, analytic for x in some neighborhood of 0 and ε in some sector, which tends to a non-trivial solution of the reduced equation f (x, 0)y = g(x, 0)y as ε → 0.

The article presents a classification of such resonant equations with respect to analytic transformations ỹ = a(x, ε)y + b(x, ε)εy . First of all, f 0 (x) = f (x, 0) is a formal invariant considered fixed below. Furthermore, to each resonant equation are associated three formal series in ε, which are Gevrey of order 1 and invariant under analytic transformations. It is shown that this correspondence between equivalence classes of resonant equations and triples of Gevrey series is essentially bijective, and that each equivalence class contains an equation of a particular form: f (x, ε) = f 0 (x) and g(x, ε) = f 1 (x) + εf 2 (x) with f 1 (0) = 0.

Introduction

In this article we consider the singularly perturbed equation [START_REF] Ackerberg | Boundary layer Problems Exhibiting Resonance[END_REF] with the above assumptions. Throughout the whole article, the prime will denote differentiation with respect to x.

Let V be a neighborhood of x = 0 and S a sector in ε with vertex 0, with a finite radius and containing real positive numbers.

A resonant solution of (1) on V × S is a function y = y(x, ε) analytic and bounded on V × S that is a solution of (1) for all ε ∈ S and tends to a non trivial solution of the reduced equation f (x, 0)y = g(x, 0)y [START_REF] Balser | Formal power series and linear systems of meromorphic ordinary differential equations[END_REF] uniformly on V as ε → 0, ε ∈ S. Equation ( 1) is called resonant if it possesses a resonant solution. It is known [START_REF] Callot | Bifurcation du portrait de phase pour des équations différentielles linéaires du second ordre ayant pour type l'équation d'Hermite[END_REF][START_REF] Fruchard | Overstability and resonance[END_REF] that a necessary condition for (1) to be resonant is that n = g(0, 0)/f (0, 0) is a nonnegative integer; this is an important quantity for the formal series (3): the multiplicity of x = 0 as a zero of its leading term y 0 (x). As shown in section 7, the derivative y (n) satisfies a second order equation analytically equivalent to [START_REF] Ackerberg | Boundary layer Problems Exhibiting Resonance[END_REF] such that this integer is zero. These equations will be called 0-resonant; for them, y 0 (0) = 0.

It is well known, too [START_REF] Fruchard | Overstability and resonance[END_REF][START_REF] Lin | The sufficiency of Matkowsky-condition in the problem of resonance[END_REF], that equation ( 1) is resonant if and only if it admits a formal solution ŷ(x, ε)

= n≥0 y n (x)ε n (3)
with coefficients y n analytic in a neighborhood of 0. Such a formal solution will also be called resonant. The key idea to prove this latter result is to show that the series ŷ (except for a factor that is a formal series in ε, i.e. independent of x) is of Gevrey order 1, i.e. its coefficients y n satisfy

∃A, C, ρ > 0 ∀n ∈ IN ∀|x| < ρ, |y n (x)| ≤ A C n n! . (4)
Unfortunately the only known explicit examples of resonant equations are particularly simple equations having (non-trivial) convergent formal solutions. In [START_REF] Fruchard | Overstability and resonance[END_REF], a method is given which allows to construct, using the fixed point theorem, resonant equations with divergent (of Gevrey order 1) formal solutions. It is therefore natural to link resonant equations and Gevrey-1 series.

The purpose of the present article is to characterize equivalence classes of resonant equations by means of some "invariants". A weak and a strong equivalence relation are considered, depending on whether all transformations analytic near x = ε = 0 are allowed or only those which reduce to the identity for x = 0. To each resonant equation [START_REF] Ackerberg | Boundary layer Problems Exhibiting Resonance[END_REF], we associate as invariants the function f 0 (x) = f (x, 0) and a certain triple of Gevrey-1 series. We show that two 0-resonant equations are strongly equivalent if and only if they have the same invariants and that two general resonant equations are weakly equivalent if and only if their invariants satisfy a simple equivalence relation.

We hope that our work will contribute to a better description of resonant equations; it shows that formal solutions of these equations are naturally divergent and that the framework of Gevrey theory reintroduced by Ramis [START_REF] Ramis | Dévissage Gevrey[END_REF] is the natural framework for the local theory of singular perturbations.

Only resonant equations are considered in this article. A classification of all second order linear equations with turning point would be important. We hope that the study of the special case of resonant equations will give ideas for the more general problem.

We would like to mention a different (unpublished) approach to the classification of resonant equations with f (x, 0) = 2x by Sibuya [8]: It involves the Stokes matrices for a collection of transformations defined on ε-sectors reducing [START_REF] Ackerberg | Boundary layer Problems Exhibiting Resonance[END_REF] to its normal form εy -2xy + 2ny = 0; the ε-sectors have to be a good covering (see subsection 3.1) of a neighborhood of ε = 0.

The structure of the article is as follows. In the next section, we present the main results. Section 3 introduces two auxiliary tools. The first one is the construction of a right inverse of the variation operator; this operator was already used in [START_REF] Fruchard | Overstability and resonance[END_REF] in a special context. The second one is due to Y. Sibuya; roughly speaking, it says that if a (vectorial) formal series solution of a singularly perturbed system of first order differential equations converges for one value of x, then it converges for any x. In the following three sections, we consider the special case of 0-resonant equations (1) with f (x, ε) = 2x. Section 4 contains several preliminary results concerning the invariants and characterizing weak and strong equivalence. The proofs of the main statements in the special case are done in sections 5 and 6. In section 5, we show that in this case the triples of Gevrey-1 series we introduced are indeed invariant with respect to analytic transformations and that such equations having the same invariants are equivalent. We prove in section 6 that any (admissible) triple of Gevrey-1 series can be attained as invariant of some resonant equation with f (x, ε) = 2x. In section 7 finally, we extend the results to general 0-resonant and resonant equations.

Statement of the main results

Before presenting the main results in detail, we make a preliminary reduction and introduce some notation. First of all, substitution of ε = cε allows to reduce to the case f (0, 0) = 2; this will be assumed throughout the article. We denote by R the set of resonant equations of the form (1), by R 0 the set of 0-resonant equations of the form (1) (i.e. with g(0, 0) = 0). By identifying a resonant equation to its coefficient functions (f, g), we identify R to a subset of the space H 2 0 , H 0 := C I {x, ε}, of pairs of germs of holomorphic functions in a neighborhood of (0, 0) ∈ C I 2 .

Using matrix notation, equation (f, g) can be rewritten (R) εy = Ry ,

with y = y εy and R : (x, ε) → 0 1 -εg(x, ε) f (x, ε)
.

Remark: For simplicity, we will often omit the variables x and ε, and write for instance

R = 0 1 -εg f
. Another way to read this kind of formula is to see an identity between functions, the letters x, ε being the canonical projections and 1 being the constant function.

Definition 1 . -We say that (R) and ( R) are weakly equivalent, and we write (R) ∼ ( R), if there exists a two by two matrix T with entries in H 0 whose determinant does not vanish identically such that the change of unknowns ỹ = T (x, ε)y transforms equation (R) into equation ( R), i.e. ỹ is a solution of ( R) if y is a solution of (R). We say that (R) and ( R) are strongly equivalent, and we write (R) ≈ ( R), if moreover T satisfies T (0, ε) = 1 (for all ε in a neighborhood of 0), where 1 is the identity matrix of M(2, C I ).

If ỹ = T (x, ε)y transforms (R) into ( R), then for some fundamental solution Y of (R), the matrix function

Z(x, ε) = T (x, ε)Y (x, ε
) is a solution of ( R). Thus, if Ỹ denotes a fundamental solution of ( R), there exists a matrix D(ε) of functions analytic at ε = 0 independent of x such that Z(x, ε) = Ỹ (x, e)D(ε). Therefore, if e

x 0 f (t,0)dt/ε w(x, ε) and e x 0 f (t,0)dt/ε w(x, ε) denote the Wronskian determinants of the equations (w, w ∈ H 0 and w(0, 0), w(0, 0) = 0), we must necessarily have f (x, 0) = f (x, 0) and also det

T (x, ε) = w(x,ε) w(x,ε) det D(ε). Hence S(x, ε) = det D(ε)(T (x, ε)) -1
has entries in H 0 and transforms ( R) into (R). This shows that ∼ is indeed symmetric and that f (x, 0) is an invariant, i.e. the same for all weakly equivalent equations. We call a scalar second order equation (f, g) weakly or strongly equivalent to another one ( f , g) if the corresponding systems (R) and ( R) are weakly or strongly equivalent. Thus an equivalence between two scalars equations is realized by change of unknowns of the form ỹ = a(x, ε)y + b(x, ε)εy ; the remaining entries of the corresponding T can then be determined. Clearly, a, b resp. T have to satisfy restrictive properties, because ( f , g) resp. ( R) must be exactly of the same form as (f, g) resp. (R). In the case of f (x, ε) = f (x, ε) = 2x, g(0, 0) = g(0, 0) = 0, this will be studied in details in subsection 4.2.

A priori for the weak equivalence, it is possible that det T (x, ε) = 0 on some nonempty subset of a neighborhood of (0, 0), but we will see in subsection 4.2 that T (0, 0) is necessarily invertible in the case f (x, ε) = f (x, ε) = 2x, g(0, 0) = g(0, 0) = 0.

For any equivalence relation, it is a natural problem to look for invariants; we first focus on invariants for the relation ≈ on the set R 0 of 0-resonant equations. As a first invariant besides f (x, 0), we choose a formal series in ε:

I(ε) = ŷ (0, ε)
where ŷ is the only formal resonant solution of (f, g) which satisfies ŷ(0, ε) = 1. To simplify notation, we omit the hats on the invariants. As ŷ(x, ε) is known to be Gevrey-1 uniformly with respect to x, Cauchy's formula implies that this invariant is also Gevrey of order 1.

Our second invariant, denoted by J, is the expansion in powers of ε 1/2 of the formal expression

J(ε) = n≥0 αn (ε) δi 0 e F (x,ε)/ε x n dx where F (x, ε) = x 0 f (t, ε) dt, δ > 0 is sufficiently small, ŷ is the above formal resonant solution and αn ∈ C I [[ε]] are given by ŷ(x, ε) -2 = n≥0 αn (ε)x n . ( 5
)
The choice of this second invariant is motivated by the solution of (f, g), denoted by Zy, which satisfies Zy(0, ε) = 0, (Zy) (0, ε) = 1; this solution can be expressed using any resonant solution y with y(0, ε) = 1 and J is the right hand side of the asymptotic expansion of the value of Zy(x, ε)/y(x, ε) at x = iδ for δ sufficiently small. See subsection 4.1 for details. Furthermore, it will be shown in subsections 4.1, 7.1, 7.2 that J = J 1 + ε 1/2 J 2 where J 1 , J 2 are Gevrey-1 series. The series I, J 1 , J 2 are the three Gevrey-1 series mentioned in the abstract and in the introduction. Of course, we have to prove that these are actually invariant with respect to ≈. This will be done in section 5 together with subsections 7.1, 7.2.

We denote by G 1 the set of Gevrey-1 series in ε (this is sometimes written

G 1 := C I [[ε]] 1 ), by G the set G := C I {x} × G 1 × (G 1 + ε 1/2 G 1 ) and by K the set K := (h, I, J) ∈ G ; h (0) = 2, J(ε) = i 2 √ πε + O(ε) . (6)
Theorem 2 . -Denote by Φ 0 the mapping

Φ 0 : R 0 → K, (R) → (f (x, 0), I, J) , (7)
which associates to each 0-resonant equation its invariants. Then Φ 0 induces a bijection Φ ≈ between R 0 / ≈ and K; in other words:

(i) two 0-resonant equations are strongly equivalent if and only if they have the same invariants;

(ii) for all (h, I, J) in K there exists a 0-resonant equation with invariants h, I and J.

This theorem classifies the ≈-equivalent classes of 0-resonant equations by their invariants. For a general resonant equation (f, g) with g(0, 0) = 2n, n ∈ IN, we define its invariants to be the invariants of the 0-resonant equation satisfied by y (n) ; this equation is weakly equivalent to (f, g) (cf. subsection 7.3). We denote by Φ(f, g) the vector of invariants of a general resonant equation (cf. [START_REF] Ramis | Dévissage Gevrey[END_REF]). The weak equivalence ∼ between equations yields an equivalence relation between invariants (h, I, J), denoted by the same ∼ sign; it turns out to be the following, see again section 5:

Definition 3 .
-Two elements (h, I, J) and ( h, Ĩ, J) of K are called weakly equivalent if h = h and there are four functions A, B, C, D holomorphic in a neighborhood of ε = 0

∈ C I satisfying AD -εBC ≡ 1, A(0) = D(0) = 1, such that Ĩ = C + DI A + εBI , J = (A + εBI) 2 J -εB(A + εBI) , (8)
where ε has been again omitted (see remark above definition 1).

Remark: 1. As a straightforward calculation shows, this is indeed an equivalence relation. 2. Observe that for two elements (h, I, J) and (h, Ĩ, J) of K, there are uniquely determined formal series A(ε), B(ε), C(ε), D(ε) satisfying the equalities of the definition; the elements are weakly equivalent if all these series are convergent. Indeed, separating

J = J 1 + ε 1/2 J 2 and J = J1 + ε 1/2 J2 with J 1 , J 2 , J1 , J2 ∈ C I [[ε]]
, we obtain four equations for the four series A, B, C, D. Omitting again the argument ε, these are

(A + εBI) Ĩ = C + DI, AD -εBC = 1, J1 = (A + εBI) 2 J 1 -εB(A + εBI), J2 = (A + εBI) 2 J 2 . ( 9
)
The last line uniquely determines A and B; then the first one is a 2 by 2 system of linear equations with determinant A + εBI = 0 for C and D. Using this equivalence relation for the "invariants", we present a result similar to theorem 2 concerning the weak equivalence of resonant equations. Furthermore, the method used to prove surjectivity yields a more precise result, which is stated below. Theorem 4 . -The mapping Φ defined above definition 3 induces a bijection Φ ∼ between R/∼ and K/∼. Moreover, each equivalence class of the relation ∼ contains a 0-resonant equation (f, g) with f independent of ε and g linear in ε. More precisely, given (h, I, J) ∈ K, there exist ϕ = ϕ(ε) and g 0 = g 0 (x), g 1 = g 1 (x) holomorphic in a neighborhood of 0 ∈ C I such that the equation (h, g 0 + εg 1 ) is 0-resonant and has invariants (I + ϕ, J).

Remarks: 1 -By misuse of language, we call invariants the vectors (h, I, J) associated to an equation (f, g), even if they are not properly invariant under ∼.

Auxiliary tools

In this section, we introduce several tools that will be useful in subsequent sections.

A right inverse of the variation operator

The first tool is a continuous right inverse of the ∆-operator described below (see [START_REF] Sibuya | A theorem concerning uniform simplification at a transition point and the problem of resonance[END_REF]). It will be used in section 6. This inverse was already used in [START_REF] Fruchard | Overstability and resonance[END_REF] in a simpler context.

We denote the sector of direction θ, opening |S| := 2δ and radius r by

S(θ, δ, r) = {ε ∈ C I ; 0 < |ε| < r, | arg ε -θ| < δ}
Given r, µ > 0 small enough, we consider, for j ∈ {0, ..., 2h}, the sectors

S j = S j 2π 2h+1 , π 2h -µ, r .
where µ is chosen in such a way that these sectors form a good covering of 0, i.e. intersections D j := S j ∩ S j+1 (with S 2h+1 := S 0 ) of consecutive sectors are non empty and triple intersections D j ∩ D j+1 (with D 2h+1 := D 0 ) are empty. We choose

µ := π 4h(2h + 1)
.

In this way, |S j |/2 = π 2h+1 + µ and |D j |/2 = µ. Let U be the Banach space of (2h + 1)-tuples y = (y 0 , ..., y 2h ) of holomorphic and bounded functions y j : S j → C I , endowed with the norm

|| y|| = max(||y 0 || ∞ , ..., ||y 2h || ∞ ).
Denote by D the Banach space of (2h + 1)-tuples d = (d 0 , ..., d 2h ) of holomorphic functions d j : D j → C I such that d j (ε)/ε is bounded, with the norm

|| d|| = max j∈{0,...,2h} sup ε∈D j d j (ε)
ε .

The space D can be identified to the space of holomorphic functions d on D 0 ∪ ... ∪ D 2h that are bounded with respect to the norm sup ε |d(ε)/ε|. We consider the variation operator ∆ defined by

∆ y = d := (y 1 -y 0 , ..., y 2h -y 2h-1 , y 0 -y 2h ) . (10)
This defines a mapping ∆ :

U 0 → D, where U 0 is the subset of all y ∈ U such that ∆ y ∈ D, i.e. (y j+1 -y j )(ε) = O(ε) as ε → 0 for all j ∈ {0, ..., 2h}. Given d ∈ D, to find y = (y 0 , ..., y 2h ) ∈ U 0 such that ∆ y = d, it seems natural to use the classical formula y k : S k → C I , ε → 1 2πi 2h j=0 ε j 0 d j (z) dz z -ε (11)
with

ε j := re iα j ∈ D j and α j := (2j + 1)π 2h + 1 ,
where the path of integration is the straight line from 0 to

ε j if j = k and j = k -1, on the left of ε if j = k and on the right of ε if j = k -1.
Actually it is well-known that the above y satisfies ∆ y = d with y k analytic in S k . However, this does not yield an operator from D into U, because y k has logarithmic singularities at ε = ε k and ε = ε k-1 . We overcome this difficulty by considering some average of integrals similar to [START_REF] Wasow | Asymptotic expansions for ordinary differential equations[END_REF], namely, with δ := µ/2:

Σ d (ε) = 1 2δ δ -δ y θ (ε)dθ (12)
with y θ = (y θ,0 , ..., y θ,2h ), y θ,k :

S k → C I , ε → 1 2πi 2h j=0 ε j e iθ 0 d j (z) dz z -ε
, ε j and the paths of integration as above. Since D j = S(α j , µ, r), ε j e iθ remains on D j and sufficiently far away from the sides of D j .

Theorem 5 . -1 -The operator Σ : D → U 0 given by ( 12) is bounded and satisfies

∆Σ = id : D → D .
2 -We have ker ∆ = H ε 0 , the space of holomorphic and bounded functions on D(0, ε 0 ).

3 -For all g ∈ U 0 we have Σ∆ g -g ∈ H ε 0 .

4 -Consider the operator L defined from D into U 0 by

L d := εΣ( d ) -Σ(ε d ) . (13)
Then for all d ∈ D, the function L d is constant with respect to ε. Thus we use L as mapping D into C I . . In order to establish an upper bound of Σ , we first estimate y θ defined below [START_REF] Wasow | Linear Turning Point Theory[END_REF] in a way similar to the proof of lemma 20 in [START_REF] Fruchard | Overstability and resonance[END_REF]. We could refer to [START_REF] Fruchard | Overstability and resonance[END_REF] for more details, but we prefer include a proof for completeness.

Let us look at the first coordinate y θ,0 ; the estimate will be the same for the other ones. We furthermore focus on ε ∈ S 0 with arg ε ∈ [0, α 0 + 2δ]; the case arg ε < 0 is similar.

If arg ε ≤ α 0 -2δ we simply use

|z -ε| ≥ |z| sin δ (14)
for any z on any segment [0, ε j e θi ] and obtain

|y θ,0 (ε)| ≤ 2h+1 2π r sin δ d .
If arg ε > α 0 -2δ (notice that only the integral involving d 0 poses a problem) we distinguish two cases: In the case arg ε ≤ α 0 + δ we use a path of integration γ θ arbitrarily close to the segment [0, e i(α 0 +2δ) ] and the arc of circle C θ = (e i(α 0 +2δ) , e i(α 0 +θ) ). On the first piece, the same inequality (14) holds. On the remaining one, we have

C θ d 0 (z) dz z -ε ≤ d 0 (z) C θ |z| |z -ε| |dz| . (15)
In the case arg ε > α 0 + δ we first use Cauchy's formula which shows that

γ θ d 0 (z) dz z -ε = 2πid 0 (ε) + γθ d 0 (z) dz z -ε
where now the path of integration is on the right of ε, arbitrarily close to [0, e i(α 0 -2δ) ] and the arc (e i(α 0 -2δ) , e i(α 0 +θ) ). We still have ( 14) on the segment and (15) on the arc.

Therefore it remains to show that

I = δ -δ dθ C θ |z| |z -ε|
|dz| is bounded, where C θ is either the arc (e i(α 0 +2δ) , e i(α 0 +θ) ) if arg ε ≤ α 0 + δ, or the arc (e i(α 0 -2δ) , e i(α 0 +θ) ) if arg ε > α 0 + δ. This is done exactly as in [START_REF] Fruchard | Overstability and resonance[END_REF]: by splitting the exterior integral in two parts at θ = arg ε -α 0 , we show that the double integral assumes its maximum for arg ε = α 0 . Hence it suffices to estimate 2 2 -If y satisfies ∆ y = 0, then y is a single valued function, hence holomorphic on D(0, ε 0 ) \ {0}. As this function is also bounded in a neighborhood of 0, the singularity at ε = 0 is removable and y is holomorphic at ε = 0. Conversely, a function holomorphic on D(0, ε 0 ) is clearly identified to a triple of U with differences equal to zero.

3 -For g ∈ U 0 , we have ∆(Σ∆ g -g) = (∆Σ)∆ g -∆ g = 0; then use item 2.

4 -Denote by S : D → U 0 the operator given by

S( a(µ)) := 1 2δ δ -δ 2h j=0 1 2πi ε j e iθ 0 a j (µ)dµ dθ , δ = π 8h(2h+1) , ε j = re iα j , α j = (2j+1)π 2h+1 . We have Σ( a)(ε) = S a(µ) µ-ε , therefore (13) reads L d (ε) = S (ε d (µ) -µ d (µ)) 1 µ -ε = -S( d (µ)) .
This shows that L d (ε) does not depend on ε.

5 -Let L n : D → D, d → L n d = ε n Σ( d )-Σ(ε n d ) (
here the letter ε denotes multiplication by the variable). We have

L n ( d ) = εL n-1 ( d ) + L(ε n-1 d
) with L given at item 4; hence the statement follows by induction.

Convergence of formal solutions

The second tool is due to Y. Sibuya [START_REF] Sibuya | On the convergence of formal solutions of systems of ordinary differential equations containing a parameter[END_REF]. It will be used in section 5 (proof of proposition 12). For completeness, a proof is included.

Let Ω be a domain (i.e. connected open subset) of C I and ε 0 > 0. We denote by D = D(0, ε 0 ) the closed disk of radius ε 0 around the origin and by H the Banach space of holomorphic bounded functions on Ω × D endowed with the usual norm. Let d and h be positive integers and A ∈ M(d, H) be a d×d-matrix with entries in H. Theorem 6 . -Suppose that the differential equation

ε h y (x, ε) = A(x, ε)y(x, ε) , y ∈ C I d (16)
admits a non trivial formal solution ŷ = n≥0 z n (x)ε n whose coefficients z n are analytic on Ω.

If there exists x 0 ∈ Ω such that ŷ(x 0 , ε) converges for ε ∈ D, then ŷ converges for all x ∈ Ω and ε ∈ D.

A := sup (x,ε)∈Ω×D |A(x, ε)| .
Let R > 0 be such that D(x 0 , R) ⊂ Ω. We first claim that it suffices to prove that ŷ converges for |x -

x 0 | < ρ := min ε h 0 A , R and |ε| ≤ ε 0 . (17)
Indeed, for all x in Ω, there exists a finite sequence of points x j , j = 1, ..., N such that x N = x and x j+1 ∈ B(x j , ρ j ) with ρ j = min{ ε h 0 A , dist(x j , ∂Ω)} for j = 0, ..., N -1. We conclude by induction that ŷ converges for x.

Without loss of generality, we assume that x 0 = 0. Here we prove even more.

Lemma 7 Suppose that (16) admits a nontrivial formal solution ŷ

∈ C I [[x, ε]] such that ŷ(0, ε) converges for ε ∈ D. Then ŷ(x, ε) converges for |x| < ρ and ε ∈ D.
The idea is to expand A and ŷ in powers of x :

A(x, ε) = n≥0 A n (ε)x n , A n ∈ M(d, C I {ε}) and ŷ(x, ε) = n≥0 y n (ε)x n , y n ∈ C I [[ε]] d .
First, Cauchy's inequalities yield that A n (ε) are holomorphic and bounded in D and

∀n ∈ IN, ∀ε ∈ D, A n (ε) ≤ A R -n ≤ A ρ -n .
Identification of the coefficients of x n in (16) yields

ε h (n + 1)y n+1 (ε) = n k=0 A k (ε)y n-k (ε) =: ϕ n (ε) . ( 18
)
Lemma 8 . -For all n ≥ 0, y n converges for |ε| ≤ ε 0 .

Proof. By induction on n. For n = 0, the series y 0 (ε) = ŷ(0, ε) converges for |ε| ≤ ε 0 , by assumption of the lemma.

If the statement of the lemma is valid for all k ≤ n, then ϕ n converges in D(0, ε 0 ). Now we already know that y n+1 is a formal series in ε, hence the left hand side of (18), ε h (n + 1)y n+1 , is a series with valuation at least h, hence ϕ n has a zero of order at least h at ε = 0. Therefore, y n+1 = 1 n+1 S h 0 ϕ n , where S h 0 is the h-th iterate of the shift operator S 0 defined by S 0 ϕ(ε) = 1 ε (ϕ(ε) -ϕ(0)). This operator preserves the radius of convergence.

Lemma 9 . -For all n ≥ 0 and all |ε| ≤ ε 0 , we have ||y n (ε)|| ≤ M ρ -n , with M := sup{||ŷ(0, ε) ; |ε| ≤ ε 0 } and ρ given by (17).

Proof. Once again by induction on n. It is obvious for n = 0 since y 0 = ŷ(0, •). If it is true for k ≤ n, then (18) gives

||ε h y n+1 (ε)|| ≤ 1 n + 1 n k=0 A ρ -k M ρ k-n = M A ρ -n .
For |ε| = ε 0 , taking into account that ρ ≤ In this section and the following two, we will only consider 0-resonant equations (f, g) with f (x, ε) = 2x; this simplifies many considerations and it seems easier to extend the results to general equations later. As f is fixed, we denote the equation (f, g) simply by (g) and the invariants simply by (I, J). As will be shown is subsection 4.1, I and J have to satisfy the relation

J(ε) = i 2 √ πε + I 0 ε + O(ε 3/2
) in this case. Thus instead of the sets R, R 0 and K introduced above theorem 2, we use R o = {g ∈ C I {x, ε} ; g(0, 0) = 0} and

K o = {(I, J) ∈ G 1 × (G 1 + ε 1/2 G 1 ) ; J(ε) = i 2 √ πε + I 0 ε + O(ε 3/2
)}, instead of the mappings Φ 0 , Φ of theorems 2 and 4, we use Φ o : R o → K o , (g) → (I, J).

The second invariant

Denote by Zy the solution of (g) that satisfies Zy(0, ε) = 0, (Zy) (0, ε) = 1. The variation of constant formula gives:

Zy(x, ε) = y(x, ε) x 0 e ξ 2 /ε y(ξ, ε) -2 dξ .
where y is any resonant solution of (g) with y(0, ε) = 1. Define

J := Zy(iδ, ε)/y(iδ, ε) = iδ 0 e x 2 /ε y(x, ε) -2 dx
where δ > 0 is an arbitrary constant. J is defined from a resonant solution and an arbitrary constant, but its expansion in powers of ε 1/2 (as ε → 0) depends only on the formal resonant solution ŷ ; we define J as being this expansion. Namely, if we denote by ŷ(x, ε) -2 = n≥0 αn (ε)x n the expansion of ŷ-2 in powers of x, where each αn represents a formal series in ε, we put

J(ε) := n≥0 αn (ε) 1 2 i n+1 Γ( n+1 2 ) ε n+1 2
.

We will use in sections 5 and 6 the operator

K : C I [[x, ε]] → C I [[ε 1/2 ]] which, to a formal series u(x, ε) = m,n≥0
a m,n x m ε n , associates the formal expansion

Ku(ε) := m,n≥0 a m,n Γ( m+1 2 ) 1 2 i m+1 ε m+1 2 +n .
With this notation, the second invariant of (g) is

J = K(ŷ -2 ). Now we have ŷ(x, ε) = 1 + xŷ (0, ε) + O(x 2
), hence the first terms of ŷ-2 are given by α0 (ε) = 1 and α1 (ε) = -2I(ε). Therefore J(ε)

= i 2 √ πε + I 0 ε + O(ε 3/2 ). Furthermore, as ŷ is analytic in x and Gevrey-1 in ε then ŷ-2 is Gevrey-1, too, hence αn (ε) = m≥0 α m,n ε m satisfies ∃A, C ≥ 1 ∀m, n ∈ IN, |α m,n | ≤ A C m+n m! .
Writing J in the form J = J 1 + √ εJ 2 :

J(ε) = ν≥1    m+ n+1 2 =ν α m,n 1
2 i n+1 Γ n+1 2    ε ν + √ ε ν≥0   m+ n 2 =ν α m,n 1
2 i n+1 Γ n+1 2   ε ν and using Γ n+1 2 ≤ n+1 2 ! if n is odd, Γ n+1 2 ≤ 2 n 2 ! if n is even, and m+n =ν m!n ! ≤ 3ν! ,
it follows that J 1 and J 2 are Gevrey-1 in ε.

Auxiliary statements

In terms of the scalar equations, the equivalence relations of definition 1 can be expressed as follows: We have (g) ∼ (g) if and only if there exist two functions a and b ∈ H 0 (i.e. holomorphic in a neighborhood of (0, 0) ∈ C I 2 ) such that the change of unknowns ỹ = a(x, ε)y + εb(x, ε)y transforms equation (g) to equation (g). The corresponding matrix transformation is given by

T = a b εc d , c = a -gb, d = a + εb + 2xb . ( 19
)
We have (g) ≈ (g) if moreover the functions a and b satisfy

a(0, ε) = 1, a (0, ε) = b(0, ε) = b (0, ε) = 0
for all ε in a neighborhood of 0.

In addition to these equivalence relations, we will use the following intermediate relation. We write (g) ∼ 1 (g) if (g) ∼ (g) and if moreover the functions a and b which realize the equivalence satisfy b(0, ε) = b (0, ε) = 0 and a(0, ε) = 1.

We write (I, J) ∼ 1 ( Ĩ, J) if there exists ϕ holomorphic in a neighborhood of ε = 0 such that Ĩ = I + ϕ and J = J, in other words if (I, J) ∼ ( Ĩ, J) with A = D ≡ 1 and B ≡ 0 (C = ϕ).

We will show together with the analogues of theorems 2 and 4 for

R o , K o that Φ o induces a bijection Φ o ∼ 1 : R o / ∼ 1 → K o / ∼ 1 .
Observe that the last sentence of theorem 4 now actually means that there exists an equation that is linear in ε in each equivalence class of K o / ∼ 1 .

We now present two elementary results which will be very useful for the sequel. The first one expresses the equivalences between 0-resonant equations in terms of the functions a, b of (19). In the second one we construct functions a, b satisfying these conditions with prescribed initial values at x = 0. Proposition 10 . -Let (g) and (g) be two 0-resonant equations. Recall that H 0 is the space of functions holomorphic in a neighborhood of (0, 0) ∈ C I 2 and that we require g(0, 0) = g(0, 0) = 0.

1 -We have (g) ∼ (g) if and only if there exist two functions a and b in H 0 that satisfy both relations below at any point (x, ε) (in a neighborhood of (0, 0)) :

(W ) a 2 + ε(ab -a b) + 2xab + εgb 2 = 1 , (20)
(T ) εa -εg b -2εgb -ag + ag -2xa = 0 , (21)
as well as a(0, 0) = 1. As a consequence a (0, 0) = -b(0, 0). 2 -We have (g) ∼ 1 (g) if and only if there exist a, b ∈ H 0 satisfying (W ) and (T ) at any (x, ε) small enough and such that a(0, ε) = 1, b(0, ε) = b (0, ε) = 0 for all ε small enough. As a consequence a (0, 0) = 0.

3 -We have (g) ≈ (g) if and only if there exist a, b ∈ H 0 satisfying (W ) and (T ) for all x, ε small enough and such that a(0, ε) = 1, b(0, ε) = b (0, ε) = a (0, ε) = 0 for all ε small enough.

Remarks: 1 -The letters x and ε in formulae (W ) and (T ) denote the canonical projections as remarked above definition 1.

-

The above statement has the advantage to avoid the use of the solutions y and ỹ. Of course the equivalence between the equations is given by ỹ = ay + εby .

-The transformation T of (19) satisfies the differential equation:

εT = RT -T R . ( 22
)
Proof of proposition 10. 1 -Assume that (g) ∼ (g) and denote by a, b the functions that realize the equivalence. We can assume that a(x, 0) and b(x, 0) do not both vanish identically; otherwise one can realize the equivalence also by 1 ε a and 1 ε b. Equation ( 22) yields

         a = c + gb εb = d -a -2xb εc = gd -ga + 2xc d = -gb -c (23)
The first two equations give c = a -gb and d = a + 2xb + εb . With c and d replaced by their expressions, the third equation yields (T ); the last one gives εb + 2xb + (g -g + 2)b + 2a = 0 .

(24)

Multiplying (24) by a and (T ) by b, the difference of both equations is 2aa + ε(aba b) + 2ab + 2xa b + 2xab + εg b 2 + 2εgbb = 0, which yields by integration

a 2 + ε(ab -a b) + 2xab + εgb 2 = w(ε), (25)
where w depends only upon ε. This is equation (W ) except that 1 is replaced by w(ε).

Actually, this equation can be directly obtained from the wronskians of (g) and (g). Both wronskians are multiples of exp(x 2 /ε), hence necessarily det T depends only on ε.

Now equation (T ) with ε = 0 implies that a(x, 0) satisfies some linear first order differential equation. As g(0, 0) = g(0, 0) = 0, we find that a(x, 0) = C exp x 0 g(t,0)-g(t,0) 2t dt with some constant C. We claim that C = 0. Otherwise, a(x, 0) = 0 for all x and (24) with ε = 0 implies as above that b(x, 0) = D

x exp x 0 g(t,0)-g(t,0) 2t dt with some constant D. Now D = 0 would imply that also b(x, 0) = 0 for all x which had been excluded at the beginning of the proof and D = 0 would imply that b(x, 0) has a pole at x = 0 which contradicts our definition of equivalence. Thus we have shown that C = 0 and hence a(0, 0) = C = 0.

Putting x = ε = 0 in (25) , we find that w(0) = C 2 = 0 and therefore there is a function t(ε) analytic at ε = 0 with t(0) = C such that w(ε) = t(ε) 2 . Dividing a, b by t(ε) finally yields one direction of statement 1.

Conversely, if a and b satisfy the assumptions of the theorem, then they satisfy (T ) and ( 24), hence the elements a, b, c, d of the matrix defined by ( 19) satisfy (23) and T satisfies (22). Hence T realizes a transformation between the matrix equations associated to (g) and (g), with T (0, 0) invertible.

The statement a (0, 0) = -b(0, 0) follows by differentiation of (W ) and evaluation at x = ε = 0 (and also directly from (24)).

The statements 2 and 3 follow from the first and the conditions imposed on T resp. a, b.

The following proposition will be useful in the next section for finding 0-resonant equations equivalent to a given one. Proposition 11 . -Suppose that A, B, C, D analytic in a neighborhood of ε = 0 are given and that A(0) = D(0) = 1, C(0) = -B(0) and AD -εBC ≡ 1. Let g be a function analytic near x = ε = 0 with g(0, 0) = 0. Then there exist a, b analytic in a neighborhood of x = ε = 0 satisfying condition (W ) of proposition 10 and

a(0, ε) = A(ε), a (0, ε) = C(ε) + g(0, ε)B(ε) b(0, ε) = B(ε), b (0, ε) = 1 ε (D(ε) -A(ε)) . (26)
Proof. Let us assume first that a, b exist that satisfy (W ) and (26). Then the function q = b a has to satisfy q(0, •) = B A and q (0,

•) = ab -ba a 2 (0, •) = L with L := 1 A 2 1 ε A(D -A) -B(C + RB) , (27)
where R(ε) = g(0, ε). Therefore we are led to put

q(x, ε) := B A (ε) + xL(ε) + x 2 2 Q(ε)
where the coefficient Q of the "quadratic" term will be determined later. Division of equation (W ) by a 2 (non zero in a neighborhood of 0), yields

1 + εq + 2xq + εgq 2 = a -2 , (28)
which leads to set a := (1 + εq + 2xq + εgq 2 ) -1/2 and b := aq. Now let us check that a, b defined above satisfy the conditions we want for suitably chosen Q. (W ) is obviously satisfied. Concerning (26) :

• Formula (28) yields a(0, •)

-2 = 1 + εL + εR B A 2 = 1 A 2 (A 2 + A(D -A) -εB(C + RB) +εRB 2 ) = 1 A 2 (AD -εBC) = A -2
, hence a(0, •) = A since both these holomorphic functions take the same value 1 at ε = 0.

• We have b(0, •) = a(0, •)q(0, •) = A B A = B. • Differentiation of (28) yields -2a -3 a = εq + 2xq + 2q + εg q 2 + 2εgqq , therefore -2A -3 a (0, •) = εQ + 2 B A + εg (0, •) B A 2 + 2εR B A L which is of the form εQ + P with P (0) = 2B(0). Thus a (0, •) = C + RB is satisfied, if we choose Q := 1 ε -2A -3 (C + RB) -P
which is holomorphic, since A(0) = 1, -2C(0) = 2B(0) = P (0) and R(0) = 0.

• Finally, b = aq gives b = a q + aq , hence b (0,

•) = (C + RB) B A + AL. Replacing L by its value (cf. (27)), this gives b (0, •) = 1 A (C + RB)B + 1 ε A(D -A) -B(C + RB) = 1 ε (D -A).
This completes the verification of (26).

Formal and analytic classification for the special case

In this section we will show that the quotient functions

Φ o ∼ : R o / ∼→ K o / ∼, Φ o ∼ 1 R o / ∼ 1 → K o / ∼ 1 and Φ o
≈ : R o / ≈→ K o analogous to those of theorems 2, 4 and just above proposition 10 in the case of f (x, ε) = 2x are well defined and injective. This justifies the definitions of ∼ and ∼ 1 for the invariants. In the proof, we will use the formal equivalence ≈ related to ≈.

For the formal equivalence, g and g are still holomorphic at (0, 0) (and satisfy g(0, 0) = g(0, 0) = 0), but the functions a and b which realize the equivalence are only assumed to be in

C I [[x, ε]].
Indeed, thanks to lemma 7 we have the following.

Proposition 12 .

-If (g) and (g) are two analytic 0-resonant equations with (g) ≈(g), then (g) ≈ (g). In other words, if there exists T ∈ GL(2, C I [[x, ε]]) with T (0, ε) = 1 such that ỹ = T y transforms (g) in (g), then there exist x 0 , ε 0 > 0 such that T converges for all |x| < x 0 and |ε| < ε 0 .

Proof. As already mentioned (see remark 3 below proposition 10), T is a solution of the linear singularly perturbed equation ( 22), considered as a system in C I n 2 and rewritten below:

εT = RT -T R
with the initial condition T (0, ε) = 1. Hence the statement follows from lemma 7.

This will be useful in order to prove the following result.

Theorem 13 . -Let (g) and (g) be two analytic 0-resonant equations with invariants respectively (I, J) and ( Ĩ, J).

1 -We have (g) ≈ (g) if and only if (I, J) = ( Ĩ, J).

2 -We have (g) ∼ (g) if and only if (I, J) ∼ ( Ĩ, J).

3 -We have (g) ∼ 1 (g) if and only if (I, J) ∼ 1 ( Ĩ, J).

Before beginning the proof of this result, we need some further preliminaries.

As before, ŷ denotes the only formal resonant solution of (g) with ŷ(0, ε) = 1. Now we set

Z ŷ(x, ε) = e x 2 /ε ŷ(x, ε)v(x, ε) where v(x, ε) = n≥0 v n (x)ε n is the only formal solution of εv + 2xv = εŷ -2 .
This solution is determined recursively by v 0 = 0 and 2xv n+1 = z n -v n for n = 0, 1, ..., where z n are given by ŷ-2 (x, ε) = n≥0 z n (x)ε n . The function v n has a pole of order at most 2n at x = 0. This formal expression has the following analytic meaning. Given an analytic resonant solution y of (g), a second (analytic) solution Zy of (g) is given by

Zy(x, ε) = y(x, ε) x 0 e ξ 2 /ε y(ξ, ε) -2 dξ .
It is easily shown that v is the asymptotic expansion of e -x 2 /ε Zy(x, ε)/y(x, ε) as ε → 0 in the domain Re(x 2 ) > 0.

We recall the operator

K : C I [[x, ε]] → C I [[ε 1/2 ]
] which, to a formal series f in x, ε,

associates the formal expansion in ε 1/2 of iδ 0 e x 2 /ε f (x, ε)dx. Precisely, if f (x, ε) = m,n≥0 a m,n x m ε n , then K f is given by K f (ε) = m,n≥0 a m,n Γ( m+1 2 ) 1 2 i m+1 ε m+1 2 +n .
Recall that the second invariant of (g) is J = K(ŷ -2 ). The following result will be useful in the proof of theorem 13.

Lemma 14 . -1 -Given f ∈ C I [[x, ε]], we have K f = 0 if and only if there exists ĝ ∈ C I [[x, ε]] such that ĝ(0, ε) = 0 and εĝ + 2xĝ = f . 2 -More generally, for all f ∈ C I [[x, ε]] there exist unique formal series ĝ ∈ C I [[x, ε]] and â ∈ C I [[ε]] such that εĝ + 2xĝ = f -â .
3 -If f , â, ĝ are as above then we have

K f = i 2 √ πε â -εĝ(0, •). Proof. 2 -Let f ∈ C I [[x, ε]]. Writing f (x, ε) = n≥0 f n (x)ε n , ĝ(x, ε) = n≥0 g n (x)ε n , with f n , g n ∈ C I [[x]
], and â(ε) = n≥0 a n ε n , we obtain the equations

2xg 0 (x) = f 0 (x) -a 0 and for n ≥ 1, g n-1 (x) + 2xg n (x) = f n (x) -a n .
The condition

g n ∈ C I [[x]
] uniquely determines a n , and we obtain a n and g n recursively:

a 0 = f 0 (0) , g 0 (x) = 1 2x (f 0 (x) -a 0 ) ,
and for n ≥ 1,

a n = f n (0) -g n-1 (0) , g n (x) = 1 2x f n (x) -a n -g n-1 (x) .
3 -Straightforward computation shows that K(εĝ + 2xĝ) = -εĝ(0, ε) if ĝ(x, ε) = x m for some m ∈ IN. As K apparently is C I [[ε]]-linear and continuous w.r.t. the topology of formal series in x, ε, this proves the formula for all ĝ (one could also use the fact that

(K f )(ε) is the formal expansion of iδ 0 e x 2 /ε f (x, ε)dx). Now Kâ(ε) = i 2 √ πε â(ε) and hence K f (ε) = -εĝ(0, ε) + i 2 √
πε â(ε) follows from the the preceeding statement. Statement 1 follows obviously from the 2 and 3.

Proof of theorem 13. Recall that ŷ is the unique formal solution of (g) with ŷ(0, ε) = 1 and that the equivalence relation ỹ = ay + εby between the equations implies (cf. also ( 22) and ( 23)) that also ỹ = (a -gb)y + (a + 2xb + εb )y .

(

) 29
Of course, ẑ = aŷ + εbŷ is a non-trivial formal resonant solution of g, but it does not necessarily satisfy ẑ(0, ε) = 1 in the cases of ∼, ∼ 1 .

We first prove that (I, J) is indeed an invariant for the relation ≈. If (g) ≈ (g) then we have a(0, ε) = 1, a (0, ε) = b(0, ε) = b (0, ε) = 0, hence ỹ = aŷ + εbŷ is the formal resonant solution of (g) with ỹ(0, ε) = 1. Thus Ĩ = ỹ (0, ε) = ŷ (0, ε) = I. In order to show that J = J, it suffices to prove that the formal expression K f vanishes, where

f := ỹ-2 -ŷ-2 = (aŷ + εbŷ ) -2 -ŷ-2 .
In order to show this, put ĝ := b ŷ ỹ = b ŷ(aŷ+εbŷ ) . A short calculation shows that εĝ + 2xĝ = f , hence by lemma 14 we have

K f (ε) = -εĝ(0, ε) = -εb(0, ε)/ỹ(0, ε) = 0. Note that the formula K(ỹ -2 -ŷ-2 )(ε) = -εb(0, ε)/ỹ(0, ε) (30)
is still valid for relations ∼ and ∼ 1 .

Suppose now that (g) ∼ 1 (g). Then we still have a(0, ε) = 1, b(0, ε) = b (0, ε) = 0 but not necessarily a (0, ε) = 0. Hence we have Ĩ(ε) = ỹ (0, ε) = a (0, ε) + I(ε). On the other hand, we still have ( J -J)(ε) = K(ỹ -2 -y -2 )(ε) = 0. Thus we obtain ( Ĩ, J) ∼ 1 (I, J) in this case. Observe that proposition 10 yields that a (0, 0) = 0; this also follows from (I, J) ∼ 1 ( Ĩ, J) and (I, J), ( Ĩ, J) ∈ K o .

In the case of (g) ∼ (g), proposition 10 still allows us to assume that (W ) holds and a(0, 0) = 1 and a (0, 0) = -b(0, 0). The formula ỹ(x, ε) = a(x, ε)ŷ(x, ε) + εb(x, ε)ŷ (x, ε) no longer gives the formal resonant solution of (g) reducing to 1 at x = 0; it is now given by z(x, ε) = ỹ(x,ε) λ(ε) where λ(ε) = ỹ(0, ε) = a(0, ε) + b(0, ε)I(ε). Using (29), we find

Ĩ(ε) = z (0, ε) = 1 λ(ε) [(a (0, ε) -g(0, ε)b(0, ε)) + (a(0, ε) + εb (0, ε))I(ε)]. With A(ε) = a(0, ε), B(ε) = b(0, ε), C(ε) = a (0, ε) -g(0, ε)b(0, ε) and D(ε) = a(0, ε) + εb (0, ε)
, we obtain the first part of (8). Moreover by (30

), J = K(λ 2 ỹ-2 ) = λ 2 K(ỹ -2 ) = λ 2 (K(ŷ -2 ) -εb(0, ε)/ỹ(0, ε)) = λ 2 (J -εB/λ
), which corresponds to the second part of (8). The conditions for A, B, C, D required in (8) follow from the statements at the beginning of this paragraph. Conversely, let 0-resonant equations (g) and (g) with I = Ĩ and J = J be given. We have to find a transformation T between the corresponding matrix equations (R) and ( R) whose entries are formal series in x, ε that satisfies T (0, ε) = 1. We recall that v denotes the only formal series solution of

εv + 2xv = εŷ(x, ε) -2
(its coefficients may have poles at x = 0). A formal fundamental solution of (R) is given by:

Y = ŷ ẑ εŷ εẑ where ẑ := Z ŷ = e x 2 /ε ŷv. A brief calculation shows that Y = ŷ 0 εŷ εŷ -1 1 v 0 1 1 0 0 e x 2 /ε .
An analogous formula holds for a formal fundamental solution Ỹ of ( R). Hence a transformation between both equations, given by T := Ỹ Y -1 , can be written as

T = ỹ 0 εỹ εỹ -1 1 ṽ -v 0 1 y 0 εy εy -1 -1 .
It remains to show that the coefficients of ε n of the entries of T have no poles at x = 0. Since K(y -2 ) = J = J = K(ỹ -2 ), by lemma 14 there is h

∈ C I [[x, ε]] such that h(0, ε) = 0 and εh + 2xh = ỹ-2 -y -2 . Since g := 1 ε (ṽ -v) satisfies εg + 2xg = ỹ-2 -y -2 , too, we conclude that u := g -h ∈ C I [[x, ε]] is a solution of εu + 2xu = 0, therefore u = 0. It follows that g = h ∈ C I [[x, ε]] and g(0, ε) = 0. We obtain that T = ỹ 0 εỹ ỹ-1 1 g 0 1 y -1 0 -εy y has entries in C I [[x, ε]] . Moreover, T (0, ε) = 1 0 ε Ĩ(ε) 1 1 0 0 1 1 0 -εI(ε) 1 = 1 since I = Ĩ.
This proves that (g) ≈(g) and thus by proposition 12, we obtain g ≈ (g). Now let (g) and (g) two 0-resonant equations such that their invariants satisfy (I, J) ∼ ( Ĩ, J) (resp. (I, J) ∼ 1 ( Ĩ, J)). We show below Lemma 15 Under the above conditions, there exists a 0-resonant equation (g) with invariants ( Ĩ, J) satisfying (g) ∼ (g) (resp. (g) ∼ 1 (g)).

As this new equation (g) and (g) have the same invariants, we have shown above that (g) ≈ (g). A fortiori (g) ∼ (g) and hence (g) ∼ (g) by transitivity. The proof for ∼ 1 is the same.

It remains to prove lemma 15. The hypothesis of the lemma means that there exist A, B, C, D analytic in a neighborhood of ε = 0 satisfying AD -εBC ≡ 1 and A(0) = D(0) = 1 such that ( 8) is satisfied. Observe that this and (I, J), ( Ĩ, J) ∈ K o imply that C(0) = Ĩ0 -I 0 = -B(0). As seen above, it is sufficient to find an analytic equation (g) ∼ (g) such that the equivalence between them is realized by a, b satisfying A(ε) = a(0, ε), B(ε) = b(0, ε), C(ε) = a (0, ε) -g(0, ε)b(0, ε) and D(ε) = a(0, ε) + εb (0, ε) as well as the conditions of proposition 10.

By proposition 11, we can find a, b with the above initial conditions satisfying (W ). Condition (T ) now uniquely determines a function g analytic near x = ε = 0 with g(0, 0) = 0. Thus (g) is an equation equivalent to (g) such that the equivalence is realized by a, b, its invariants are related to the ones for (g) by (8) and hence they are equal to ( Ĩ, J).

The proof for ∼ 1 is analogous.

6 Construction of 0-resonant equations in the special case

In this section, we prove the surjectivity of the quotient functions

Φ o ∼ : R o / ∼→ K o / ∼, Φ o ∼ 1 R o / ∼ 1 → K o / ∼ 1 and Φ o ≈ :
R o / ≈→ K o analogous to those of theorem 2, 4 and below proposition 10 in the case f (x, ε) = 2x. In other words, we prove the existence of 0-resonant equations (g) with a prescribed pair of invariants.

Thus let (I, J) ∈ K o be given, namely a Gevrey-1 series I = I 0 + O(ε) and a series J in ε 1/2 of the form J = J 1 + ε 1/2 J 2 , with J 1 , J 2 Gevrey-1, whose first terms are given by

J(ε) = i 2 √ πε + I 0 ε + O(ε 3/2
). We first prove that there exists a function ϕ, analytic (i.e. convergent) in a neighborhood of ε = 0 with ϕ(0) = 0, and an equation (g) with invariants (I + ϕ, J). In other words, we first prove the surjectivity of Φ o ∼ and Φ o ∼ 1 . The surjectivity of Φ o ≈ : R o / ≈→ K o will then result from lemma 15 of section 5.

Then we show that the equation is linear in ε.

As in section 3 (with h = 1) we consider, for j = 0, 1, 2, the three sectors

S j = ε ∈ C I ; 0 < |ε| < ε 0 , | arg ε -j 2π 3 | < π 2 -µ
where ε 0 , µ > 0 are small enough. U denotes the Banach space of triples y = (y 0 , y 1 , y 2 ) of functions y j : S The following Banach space will also be useful. Fix δ > 0 and let G ρ,δ be the set of all u ∈ U ρ such that there exists L > 0 satisfying

|(u j+1 -u j )(x, ε)| ≤ L exp(-(ρ 2 + δ)/ |ε|) for all j, x ∈ B ρ , ε ∈ D j . equipped with the norm || u|| G ρ,δ = max(|| u|| Uρ , G ρ,δ ( u)) where G ρ,δ ( u) = sup{|(u j+1 -u j )(x, ε)|e (ρ 2 +δ)/|ε| ; j = 0, 1, 2, x ∈ B ρ , ε ∈ D j } . ( 32
)
We recall the operator

K : C[[x, ε]] → C I [[ε 1/2 ]
] that associates, to a series g = n=0 ∞ g n (x)ε n , the expansion in ε 1/2 of the expression

∞ n=0 i∞ 0 e x 2 /ε g n (x)dx ε n .
As indicated in subsection 4.1, Kg can be written Kg = K 1 g + ε 1/2 K 2 g, where K 1 g and K 2 g are formal series in ε, K 1 g without constant term. Recall also that, given a 0-resonant equation (g), its second invariant is J = K(ŷ -2 ), where ŷ is a formal 0-resonant solution of (g) satisfying ŷ(0, ε) = 1. By misuse of notation, we apply K also to functions of x, ε having an asymptotic expansion as ε → 0 (and to triples of functions having a common asymptotic expansion, for example elements of G ρ,δ ). Let k denote the restriction of K to functions of x only. The following properties of k and K will be useful:

Lemma 16 . -The operator k maps C I {x} into εC I [[ε]] 1 + √ εC I [[ε]]
1 and is bijective. The linear operator k -1 K maps G ρ,δ into H ρ and satisfies

k -1 K u ≤ ρ 2 δ + 1 2 G ρ,δ ( u) + 1 + 2 ρ 2 ε 0 e ρ 2 /ε 0 || u|| Uρ . Proof. Since i∞ 0 e x 2 /ε x n dx = 1 2 Γ n+1 2 i n+1 ε n+1 2 , we have k   n≥0 g n x n   = ν≥0 g 2ν+1 2 Γ(ν + 1)(-1) ν+1 ε ν+1 + √ ε   ν≥0 g 2ν 2 Γ ν + 1 2 i(-1) ν ε ν  
which clearly shows that the correspondence between convergent power series in x and pairs of Gevrey-1 series in ε is one-to-one. In order to prove the second statement, we first have to find a relation between a triple u ∈ G ρ,δ and the right hand side û(x, ε) = ∞ n=0 ũn (x)ε n of its common asymptotic expansion. This is done by the well known Cauchy-Heine formula (cf. [START_REF] Sibuya | A theorem concerning uniform simplification at a transition point and the problem of resonance[END_REF]); here we find ũn

(x) = 1 2πi 3 j=1 T j T j-1 u j (x, z) z n+1 dz + 1 2πi 3 j=1 T j 0 u j+1 (x, z) -u j (x, z) z n+1 dz (33)
where T j ∈ S j ∩ S j+1 have modulus ε 0 , and the paths from 0 to T j are line segments, those from T j-1 to T j are close to the circular arcs between the two points.

A straightforward estimate implies for n ≥ 1

|ũ n (x)| ≤ || u|| Uρ ε -n 0 + 3 2π (n -1)!(ρ 2 + δ) -n G ρ,δ ( u) (34)
as well as |ũ 0 (x)| ≤ || u|| Uρ + 3 2π G ρ,δ ( u)) for all x ∈ B ρ if ε 0 ≤ ρ 2 + δ. Denote now by S : H ρ → H ρ the operator given by S(ϕ)(x) := -2x x 0 ϕ(ξ)dξ and by S n its n-th iterate. With this notation, a short calculation shows that

k -1 K(ε n x m ) = (-1) n Γ m+1 2 Γ m+1 2 + n x 2n+m = S n (x m ) . Hence k -1 K(û(x, ε)) = k -1 K( n≥0 u n (x)ε n ) = n≥0 S n (u n )(x). Another short calcula- tion shows that for all v ∈ H ρ , n ≥ 1 |S n (v)(x)| ≤ Γ( 1 2 ) Γ(n+ 1
2 )

|x| 2n v ≤ 2 (n-1)! ρ 2n ||v|| . (35)
Together with (34) we obtain

k -1 K(û) ≤ || u|| Uρ + 3 2π G ρ,δ ( u) + n≥1 2 (n-1)! ρ 2n || u|| Uρ ε -n 0 + 3 2π (n -1)!(ρ 2 + δ) -n G ρ,δ ( u) ≤ 1 + 2 ρ 2 ε 0 e ρ 2 /ε 0 || u|| Uρ +   n≥1 ρ 2 ρ 2 +δ n + 1 2   G ρ,δ ( u) .
This finally yields the wanted estimate.

Let us now consider both invariants I, J. From I we construct, by the classical truncated Borel-Laplace transform, a triple I ∈ U (reducing ε 0 if necessary) of functions having I as common asymptotic expansion. Let d := ∆ I. It is known too that d decreases exponentially as ε → 0 (cf. [START_REF] Balser | Formal power series and linear systems of meromorphic ordinary differential equations[END_REF] for details). Precisely there exists γ > 0 such that ∀ε ∈ D

0 ∪ D 1 ∪ D 2 , | d (ε)| ≤ |ε|e -γ/|ε| (36)
(the factor |ε| will be useful to fit with the norm of D).

Assume for the moment that some u ∈ U ρ exists that satisfies

u = f + Σ( d • Z u) -x(Σ d )(0) (37)
with d • Z u := (d j Z u j ) j=0,1,2 and where f ∈ H ρ , the space of holomorphic functions on B ρ = D(0, ρ) (identified to the space of functions of U ρ constant with respect to ε), satisfies f (0) = 1 and f (0) = I 0 , the first term of I. Recall that Zu is defined by Zu(x, ε) = u(x, ε) x 0 e ξ 2 /ε u(ξ, ε) -2 dξ. In that case, we have ∆ u = d • Z u. We then deduce that the function

g(x, ε) := -ε u u (x, ε) + 2x u u (x, ε) (38)
Hence Ψ is a contraction provided that ε 0 is small enough, namely ε 0 < γ-ρ 2 ln(28M ρ) (in the case 28M ρ > 1 only; otherwise no restriction for ε 0 is needed).

Thus for f ∈ F , equation (37) has a unique solution u ∈ U provided ε 0 is small enough. Thus we can consider now the operator P : F → U which associates to any f ∈ F the corresponding solution u ∈ U of (37).

In order to construct a 0-resonant equation with a pair of invariants of the form (I + ϕ, J), it thus suffices to find f ∈ F such that K ( P f ) -2 = J. Using the inverse k -1 introduced above, this amounts to finding f such that

(k -1 K)( P f ) -2 -1/2 = g 0 with g 0 := k -1 J -1/2 . ( 40
)
We will show that N defined by

N f := (k -1 K)( P f ) -2 -1/2
is close to the identity, if ε 0 is sufficiently small and conclude that N f = g 0 has a solution in F . Observe that the admissibility of (I, J) implies that g 0 (0) = 1 and g 0 (0) = I 0 ; thus g 0 ∈ F and even ||g 0 -1|| ≤ 1 16 if ρ is chosen small enough. From now on, we fix ρ, δ > 0 satisfying additionally 2ρ 2 + δ < γ. First as in the proof of lemma 17, we show that P f -f Uρ ≤ 7M ρe (ρ 2 -γ)/ε 0 . Using that u = P f satisfies (37), we find that (using G ρ,δ defined in (32)) G ρ,δ ( P f -f ) = G ρ,δ ( P f ) = sup{|d j (ε)(Zu j )(x, ε)| e (ρ 2 +δ)/|ε| ; j = 0, 1, 2, x ∈ B ρ , ε ∈ D j } and hence by (36) and (39) that G ρ,δ ( P f -f ) ≤ 6ρε 0 e (2ρ 2 +δ-γ)/ε 0 for f ∈ F . As ( P f ) -1 ≤ 2 and ||f -1 || ≤ 4/3 (we consider the multiplicative inverses here), we find ( P f ) -2 -f -2 Uρ ≤ 63M ρe (ρ 2 -γ)/ε 0 and G ρ,δ (( P f ) -2 -f -2 ) = G ρ,δ (( P f ) -2 ) ≤ 96ρε 0 e (2ρ 2 +δ-γ)/ε 0 for f ∈ F . In both estimates, we used the formula b -2 -a -2 = (a -2 b -1 + a -1 b -2 )(a -b). Using lemma 16 and, of course, the fact that K reduces to k on F ⊂ H ρ , this yields

k -1 K( P f ) -2 -f -2
Hρ ≤ e (2ρ 2 +δ-γ)/ε 0 63M ρ 2 ρ 2 ε 0 + 1 + 96ρε 0 ρ 2 δ + 1 2 =: L(ρ, δ, ε 0 ) for f ∈ F . Then, for sufficiently small ε 0 > 0, the right hand side L(ρ, δ, ε 0 ) of the last estimate is smaller than 1/9. As f ∈ F implies ||f -2 -1|| ≤ 7/9, the above estimate shows that k -1 K( P f ) -2 -1 ≤ 8/9 and hence the square root of k -1 K( P f ) -2 can be defined as the principal value. Furthermore, using the inequality b -1/2 -a -1/2 ≤ for all f ∈ F , provided ε 0 is sufficiently small. Consider now F 2 = {f ∈ F ; ||f -1|| Hρ ≤ 1/8} and the mapping ψ : F 2 → H ρ given by ψ(f ) = g 0 -(N f -f ) (cf. (40)). Recall that ||g 0 -1|| ≤ 1 16 if ρ is sufficiently small. Thus ψ maps F 2 into itself because of (41) if ε 0 and ρ are sufficiently small. We will show below that ψ is a contraction and hence it has a unique fixed point in F 2 in this case.

In order to show that ψ is a contraction, fix f ∈ F 2 and h ∈ H ρ with ||h|| Hρ = 1 for a moment and consider the mapping Ψ(t) = ψ(f + th), t ∈ C I , |t| < 1/8. Suppose that ε 0 is small enough such that (41) holds for all g ∈ F . As all the elements f + th considered are in F , (41) implies that ||Ψ(t) -g 0 || Hρ < 1/16 for all t, |t| < 1/8. By Cauchy's inequality, we obtain ||Ψ (0)|| ≤ 1/2, i.e. ||ψ (f )h|| Hρ ≤ 1/2 for all f ∈ F 2 and ||h|| Hρ = 1. Hence ||ψ (f )|| L(Hρ,Hρ) ≤ 1/2 for all f ∈ F 2 and thus ψ is indeed a contraction if ε 0 is sufficiently small.

As we have shown that ψ has a fixed point f ∈ F 2 , we conclude that the equation N f = g 0 has a solution in F 2 ⊂ F if ε 0 and ρ are sufficiently small. With this choice of f , the solution u of the fixed point equation (37) has J as second invariant, and a first invariant of the form I + ϕ. This shows the surjectivity of the mappings Φ o ∼ : R o / ∼→ K o / ∼ and Φ o ∼ 1 : R o / ∼ 1 → K o / ∼ 1 . The surjectivity of Φ o ≈ : R o / ≈→ K o follows easily: given (I, J) ∈ K o , the above proof shows that there exists a 0-resonant equation (g) with invariants ( Ĩ, J) ∼ 1 (I, J). Then by lemma 15, there exists (g) ∼ 1 (g) with invariants (I, J). Thus the proof of the surjectivity of Φ 0 ≈ is also complete. We now prove Proposition 18 . -Let f ∈ H ρ and let u be a solution of equation (37). Then the function g (holomorphic in a neighborhood of 0) defined by (38) is linear in ε, i.e. of the form g(x, ε) = g 0 (x) + εg 1 (x).

Proof. As in the proof of theorem 6, the idea is to expand the ingredients in powers of x and to use induction. We have f (x) = n≥0 f n x n , g(x, ε) = n≥0 g n (ε)x n , u(x, ε) = n≥0 u n (ε)x n , Z u(x, ε) = n≥0 z n (ε)x n with f n ∈ C I , g n convergent, u 0 ≡ 1, z 0 ≡ 0 and z 1 ≡ 1.

The triples u n are in U (triples of holomorphic bounded functions from S j into C I ) but the triples z n are not necessarily bounded near 0. However the formulae for z n show that ε [n/2] z n is bounded as ε → 0. It follows that the products d z n (with d = ∆ I) still decrease exponentially as ε → 0.

Since u 0 = 1, equation (g) yields for n ≥ 0:

-g n = ε(n + 1)(n + 2) u n+2 -2n u n + n-1 j=0 g j u n-j (42) and equation (37) gives u j = f j + Σ( v j ), v j := d • z j .

By induction, if for all j ∈ {0, ...n -1} we have g j (ε) = g j,0 + εg j,1 , g j,0 , g j,1 ∈ C I , then

-g n (ε) = ε(n + 1)(n + 2)(f n+2 + Σ v n+2 (ε)) -2n(f n + Σ v n (ε)) + n-1 j=0
(g j,0 + εg j,1 )(f n-j + Σ v n-j (ε)) .

Replacing the terms of the form εΣ( v k (µ) by Σ(µ v k (µ)) -L( v k (µ)), where L = εΣ -Σε (cf theorem 5, part 4), and grouping together terms of same nature, (43) can be written in the form

-g n (ε) = C 0 + εC 1 + L V + Σ W (ε) (44)
with C 0 , C 1 ∈ C I and V , W ∈ U. By theorem 5, part 4, L V is constant with respect to ε. Applying ∆ to (44) and using that (∆f )(ε) = 0 if f (ε) converges at ε = 0, we obtain 0 = ∆Σ W = W .

5 -

 5 More generally, for all d ∈ D and all n ∈ IN, ε n Σ( d ) -Σ(ε n d ) is polynomial in ε of degree at most n -1. Remark: As above definition 1, ε denotes the identity function in the above theorem. Proof. 1 -By construction, we have (Σ d ) j+1 -(Σ d ) j = d j , j = 0, ..., 2h. The fact that Σ d ∈ U will shown below. It is then clear that Σ d ∈ U 0 since ∆Σ d = d ∈ D.

.

  As the denominator is at least sin s, the theorem of Fubini-Tonelli yields I ≤ 2

  Proof. For C I d , we can use any norm, for instance the maximum norm, for M(d, C I ) we use the associated matrix norm both denoted by |.|. On M(d, H) we introduce the norm

ε h 0 A

 0 , we obtain ||y n+1 (ε)|| ≤ M ρ -(n+1) . Now the function y n+1 is analytic in the open disk D = D(0, ε 0 ), therefore this estimate remains valid on D by the maximum modulus principle. Statement (17) now becomes evident: ŷ converges for |x| < ρ and |ε| ≤ ε 0 . As shown above, the theorem follows 4 0-Resonant equations in the special case : Preliminary results

2 j=0(

 2 j → C I that are holomorphic and bounded, with the norm || y|| = max(||y 0 || 0,∞ , ||y 1 || 1,∞ , ||y 2 || 2,∞ ), ||y|| j,∞ := sup{|y(ε)| ; ε ∈ S j }. With D j := S j ∩ S j+1 (S 3 = S 0 ), D denotes the Banach space of triples d = (d 0 , d 1 , d 2 ) of holomorphic functions d j : D j → C I such that d j (ε)/ε is bounded, with the norm || d|| = max j=0,1,2 sup ε∈D j d j (ε) ε (identified to the corresponding space of holomorphic functions on D 0 ∪ D 1 ∪ D 2 ). ∆ denotes the difference operator ∆ : U 0 → D, y → d with d j = y j+1 -y j . As in section 3, U 0 is the subset of U of all y such that ∆ y ∈ D. We recall the right inverse of ∆ of section 3: Σ : D → U 0 , d → Σ d defined by Σ d (ε) = 12 π is bounded and satisfies ∆Σ = id : D → D. 2 -We have ker ∆ = H ε 0 , the space of holomorphic and bounded functions on D(0, ε 0 ). 3 -For all g ∈ U we have Σ∆ g -g ∈ H ε 0 . 4 -The operator L defined from D into U by L d := εΣ( d ) -Σ(ε d ) (31) takes its values in C I . In other words, for all d ∈ D the function L d is constant with respect to ε. Now we set B ρ = {x ∈ C I ; |x| < ρ}, where ρ > 0 will be determined later. U ρ and D ρ denote the spaces of functions of both variables x and ε analogous to U and D:U ρ = { y = (y 0 , y 1 , y 2 ) ; y j : B ρ × S j → C I holomorphic bounded , j = 0, 1, 2}with the max-sup norm on j, x and ε,D ρ = { d = (d 0 , d 1 , d 2 ) ; d j : B ρ × D j → C I holomorphic, d j (x, ε)/ε bounded , j = 0, 1, 2}with the corresponding norm, and similarly U ρ,0 is the subset of U ρ of all y such that ∆ y ∈ D ρ . The operator Σ naturally induces a bounded linear operator also noted Σ : D ρ → U ρ . The above statements naturally carry over to U ρ and D ρ . For instance, for all d ∈ D ρ the function L d = εΣ( d) -Σ(ε d) is constant with respect to ε.

1 2 (

 2 min 0≤t≤1 |a + t(b -a)|) -3/2 |b -a|, we finally obtain that ||N f -f || Hρ ≤

-In each equivalence class of R/ ∼, there exist several resonant equations (h, g) for which g is linear in ε.Sections 5 and 6 are devoted to the proofs of theorems 2 and 4 in the special case of 0resonant equations with f (x, ε) = 2x; section 7 extends these results to general 0-resonant and resonant equations.

In this subsection denotes differentiation with respect to x, ˙with respect to t.

Here the independent variable is denoted t

is holomorphic in a neighborhood of (x, ε) = (0, 0), i.e. it is single valued with respect to ε and remains bounded as ε → 0. Indeed, we can define functions g j : B ρ × S j → C I by g j (x, ε) := -ε u j u j (x, ε) + 2x u j u j (x, ε) . Then u j satisfy the differential equations εu j -2xu j + g j (x, ε)u j = 0 for ε ∈ S j . On the other hand, we know that Zu j satisfies the same differential equation as u j and thus, for ε ∈ D j , u j and u j+1 = u j + d j (ε)Zu j satisfy the same equation. Hence g j (x, ε) = g j+1 (x, ε) for x ∈ B ρ , ε ∈ D j ; here j = 0, 1, 2. Hence the three functions g j define one analytic function g :

This means, that for any analytic function f , if we find a solution u of (37), then the equation (g), g defined by (38), has as first invariant the series expansion of u (0, ε) which is equal to I(ε) + ϕ(ε) where ϕ ∈ C I {ε} with ϕ(0) = 0.

In order to solve (37), we consider it as a fixed point equation. Let us introduce the following subsets.

Of course, we only consider ρ < 1 4|I 0 | ; otherwise F would be empty. Lemma 17 . -Let γ > 0 satisfy (36) and ρ 2 < γ, then for f ∈ F and ε 0 small enough the operator Ψ :

Proof. For simplicity we omit the arrow on u and d in this proof.

Given u ∈ U , we have u ≤ 3 2 and u -2 ≤ 4, hence

We obtain using (36) that d • Zu D ≤ 6ρe (ρ 2 -γ)/ε 0 and d D ≤ e -γ/ε 0 . If M > 0 denotes a bound for Σ given by theorem 5, then this yields

This gives

Hence g n has the form g n (ε) = g n,0 + εg n,1 (with g n,0 = -C 0 -L V and g n,1 = -C 1 ).

Thus the analoga of theorems 2 and 4 in the case of f (x, ε) = 2x have been completely proved.

7 Extension of the results to the general case

Extension to an intermediate class of 0-resonant equations

Suppose first that a 0-resonant equation ( 1) is given with f (x, 0) = 2x, i.e. f (x, ε) = 2x + εϕ(x, ε). Then the transformation y = a(x, ε)ỹ, a(x, ε) = exp(

. This is also a 0-resonant equation with formal resonant solution ỹ(x, ε) = ŷ(x, ε)/a(x, ε). Hence its invariants are Ĩ = ỹ (0, ε)/a(0, ε) = ỹ (0, ε) = I -1 2 ϕ(0, ε) and J the formal series asymptotic of

and (2x, I, J) ∼ 1 (2x, Ĩ, J). This immediately shows that two equations (2x+εϕ 1 , g 1 ) and (2x+εϕ 2 , g 2 ) are weakly equivalent if and only if their invariants are weakly equivalent. The surjectivity statements of the analoga of theorem 2 and 4 proved in section 6 in the case f (x, ε) = 2x immediately imply those of the present case f (x, 0) = 2x.

Two equations (2x+εϕ 1 , g 1 ) and (2x+εϕ 2 , g 2 ) are strongly equivalent if and only if the corresponding equations (2x, g1 ) and (2x, g2 ) are ∼ 1 -equivalent and some transformation ỹ2 = a ỹ1 + εb ỹ 1 realizing it satisfies a(0

2 ϕ 2 (0, ε); the most convenient way to see this is to go over to matrix notation in a way similar to the beginning of subsection 4.2. According to section 5, the last statement is equivalent to Ĩ2 = Ĩ1 + 1 2 ϕ 1 (0, ε) -1 2 ϕ 2 (0, ε) and J2 = J1 . The above calculation shows that this is equivalent to

Thus theorems 2 and 4 are proved in the case of f (x, 0) = h(x) = 2x.

Extension to all 0-resonant equations

Given some analytic function f 0 (x) vanishing at x = 0 with f 0 (0) = 2, it is easy to find some analytic function ϕ(t

The second equation is 0-resonant if and only if the first one is. Observe that the two equations are not equivalent according to any of our definitions. The change of variables permits, however, to carry over the results of the previous subsection to any 0-resonant equation. First of all, two equations (f 0 + εh, g) and (f 0 + ε h, g) are equivalent via y = a(x, ε)ỹ + εb(x, ε)ỹ if and only if the corresponding equations (2t + εf 1 , g 1 ) and (2t + ε f1 , g1 ) are equivalent via v = a(ϕ(t), ε)ṽ + εb(ϕ(t), ε) φ(t) -1 v. This means that each of our three equivalence relations for (f 0 + εh, g) and (f 0 + ε h, g) corresponds to the same relation for (2t + εf 1 , g 1 ) and (2t + ε f1 , g1 ).

Furthermore, a short calculation shows that both invariants I and J for an equation (f 0 + εh, g) and its corresponding equation (2t + εf 1 , g 1 ) are the same. Thus theorem 2 and the first part of theorem 4 carry over to general 0-resonant equations immediately.

Finally, suppose that a vector (f 0 , I, J) ∈ K is given. From the preceding section, we know that there is a 0-resonant equation 2 (2t, h 0 (t) + εh 1 (t)) having the invariants (I +ψ(ε), J) where ψ(ε) is some convergent series. Using the transformation v = φ(t) -1/2 ṽ with the above ϕ(t), we find an equation of the form (2t+ε φ(t)/ φ(t), h0 (t)+ε h1 (t)) having the invariants I + ψ(ε) + 1 2 φ(0) and J. The above calculations show that the equation obtained using x = ϕ(t), y(ϕ(t)) = ṽ(t) is of the form (f 0 (x), g 0 (x) + εg 1 (x)) with some functions g 0 , g 1 analytic near x = 0 and that its invariants are still I + ψ(ε) + 1 2 φ(0) and J. This proves the second part of theorem 4.

Extension to all resonant equations

Consider now a general resonant equation in its matrix form (R) εy = Ry , with y = y εy and R :

by definition g(0, 0) = 0. Hence y 1 = R(x, ε)y induces a weak equivalence between (R) and some equation

Clearly, this new equation is satisfied by (y , εy ) T if (R) is satisfied by (y, εy ) T and the quantity g 1 (0, 0) = g(0, 0) -2. This procedure can be repeated n = g(0, 0)/2 times and leads to a 0-resonant (matrix) equation (R n ) weakly equivalent to (R) which is satisfied by (y (n) , εy (n+1) ) T if (R) is satisfied by (y, εy ); this means that the corresponding scalar equation has a solution y (n) if the scalar equation (f, g) has a solution y.

By our definition (see above definition 3), the invariants of (R), i.e. (f, g), are those of (R n ). The first part of theorem 4, our only statement concerning general resonant equations, follows immediately.