Jean-Marc Ginoux
email: ginoux@univ-tln.fr

SELF-EXCITED OSCILLATIONS : FROM POINCAR É TO ANDRONOV

Till recently, the young Russian mathematician Aleksandr' Andronov was considered by many scientists as the first to have applied the concept of limit cycle, introduced almost half a century before by Henri Poincaré, in order to state the existence of self-sustained oscillations.

Consequently, if the discovery of a series of "forgotten lectures" given by Poincaré at the École Supérieure des Postes et Télégraphes (today Sup'Telecom ParisTech) in 1908 proves that he had applied his own concept of limit cycle to a problem of Wireless Telegraphy preceding thus Andronov of twenty years, it reopens the discussion of Poincaré's French legacy in Dynamical System Theory or, more precisely in Nonlinear Oscillations Theory.

Poincaré's "forgotten lectures" will be presented in the section B and their reception in France before the First World War and in the 1920's in the French engineers community in section C. Then, a special attention will be paid to the role of Jean-Baptiste Pomey who asked Élie Cartan to solve a problem of sustained oscillations in an electrotechnics oscillator and, to the particular case of Alfred Liénard who proved, under certain conditions, existence and uniqueness of a periodic solution for such an oscillator without having regarded this periodic solution as a "Poincaré's limit cycle".

Starting from 1929, Andronov's note at the Comptes-Rendus seems to have become the reference in terms of connection with Poincaré's works. So, the reception of Andronov's results by the French scientific community will be analyzed in section D in considering a selection of works published in France between 1929 to 1943.

In the last section, the fact that Poincaré's lectures have been forgotten as well as the fact that neither Cartan nor Liénard have made any connection with Poincaré's works although they have obviously used some of them, will be discussed. Moreover, the role of Jacques Hadamard in the diffusion in France of the works of the Russian schools and of Poincaré's methods will be also pointed out. Thus, the question of Poincaré's legacy will appear in a new perspective.

1. See Petit & Bouthillon [9, p. 128].

Poincaré's forgotten lectures on Wireless Telegraphy

On July 4 th 1902 Poincaré became Professor of Theoretical Electricity at the École Supérieure des Postes et Télégraphes (today Sup'Télecom) in Paris where he taught until 1910. The director of this school, Édouard Éstaunié , also asked him to give a series of conferences every two years in May-June from 1904 to 1912. He told about Poincaré's first lecture of 1904 :

≪ Dès les premiers mots, il apparut que nous allions assister au travail de recherche de cet extraordinaire et génial mathématicien . . . À chaque obstacle rencontré, une courte pause marquait l'embarras, puis d'un coup d'épaule, Poincaré semblait défier la fonction gênante . . . ≫

1

In 1908, Poincaré chose as the subject : wireless telegraphy. The text of his lectures was first published weekly in the journal La Lumière Électrique [START_REF] Poincaré | Sur la télégraphie sans fil[END_REF] before being edited as a book the year after [START_REF] Poincaré | Conférences sur la télégraphie sans fil[END_REF]. In the fifth and last part of these lectures entitled : Télégraphie dirigée : oscillations entretenues (Directive telegraphy : sustained oscillations) Poincaré stated a necessary condition for the establishment of a stable regime of sustained oscillations in the singing arc (a forerunner device of the triode used in Wireless Telegraphy). More precisely, he demonstrated the existence, in the phase plane, of a stable limit cycle.

1.1. The singing arc equation. Starting from the following diagram (see Fig. 1), Poincaré [7, p. 390] explained that this circuit consists of an Electro Motive Force (E.M.F.) of direct current E, a resistance R and a self-induction, and in parallel, a singing arc and another self-induction L and a capacitor. In order to provide the differential equation modeling the sustained oscillations he calls x the capacitor charge and i the current in the external circuit. Thus, the current in the branch (ABCD) comprising the capacitor of capacity 1/H may be written : x ′ = dx/dt. The current intensity i a in the branch (AFED) comprising the singing may be written while using Kirchhoff's law : i a = i+x ′ . Then, Poincaré established the following second order nonlinear differential equation for the sustained oscillations in the singing arc :

(1)

Lx ′′ + ρx ′ + θ (x ′ ) + Hx = 0

He specified that the term ρx ′ corresponds to the internal resistance of the self and various damping while the term θ (x ′ ) represents the E.M.F. of the arc which is related to the intensity by a function, unknown at that time.

1.2. Stability condition for sustained oscillations and limit cycles. Then, Poincaré established, twenty years before Andronov [START_REF] Andronov | Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues[END_REF], that the stability of the periodic solution of the above equation depends on the existence of a closed curve, i.e. of a stable limit cycle in the phase plane he has defined in his memoirs ≪ Sur les Courbes définies par une équation différentielle ≫ [?, p. 168]. He posed : Let's notice that this closed curve is only a metaphor of the solution since Poincaré does not use any graphical integration method such as isoclines.

Then, Poincaré explained that if y = 0 then dy/dx is infinite and so, the curve admits vertical tangents. Moreover, if x decreases x ′ i.e. y is negative. He concluded that the trajectory curves turns in the direction indicated by the arrow (see Fig. 2) and wrote : "Stability condition. -Let's consider another non-closed curve satisfying the differential equation, it will be a kind of spiral curve approaching indefinitely near the closed curve. If the closed curve represents a stable regime, by following the spiral in the direction of the arrow one should be brought back to the closed curve, and provided that this condition is fulfilled the closed curve will represent a stable regime of sustained waves and will give rise to a solution of this problem."

Then, it clearly appears that the closed curve which represents a stable regime of sustained oscillations is nothing else but a limit cycle as Poincaré [2, p. 261] has introduced it in his own famous memoir "On the curves defined by differential equations" and as Poincaré [3, p. 25] has later defined it in the notice on his own scientific works.

But this, first giant step is not sufficient to prove the stability of the oscillating regime. Poincaré had to demonstrate now that the periodic solution of equation ( 1) (the closed curve) corresponds to a stable limit cycle.

1.3. Possibility condition of the problem : limit cycle stability. In the following part of his lectures, Poincaré gave what he calls a ≪ condition de possibilité du problème ≫. In fact, he established a stability condition of the periodic solution of equation ( 1), i.e. a stability condition of the limit cycle under the form of inequality. After multiplying equation ( 2) by x ′ dt Poincaré integrated it over one period while taking into account that the first and fourth term vanish since they correspond to the conservative part of this nonlinear equation. He obtained :

ρ ∫ x ′2 dt + ∫ θ (x ′ ) x ′ dt = 0
Then, he explained that since the first term is quadratic, the second one must be negative in order to satisfy this equality. So, he stated that the oscillating regime is stable iff :

(3)

∫ θ (x ′ ) x ′ dt < 0
As exemplified below, Poincaré's approach is identical to which will be used by Alfred Liénard twenty years later.

The reception of Poincaré's lectures on Wireless Telegraphy in France

The discovery of these Poincaré's "forgotten lectures" on Wireless Telegraphy implied the analysis of their influence on the French engineers community. (1910)(1911)(1912)(1913)(1914). During this period (1910)(1911)(1912)(1913)(1914) one scientific reference to these conferences could be found. It is in the book by Gaston Émile Petit and Léon Bouthillon entitled La Télégraphie Sans Fil published in 1910 in which one can read 1 : ≪ Le problème de la direction des ondes est donc résoluble. Les ondes peuvent théoriquement être concentrées en faisceau comme les rayons lumineux par des dispositifs appropriés ( 1 ). ≫ Unfortunately, this unique quotation is very disappointing since it does not refer to the part of Poincaré's lectures concerning the sustained oscillations. Nevertheless, since Petit (ESPT, 1906) The year after, Gaston Darboux (1842Darboux ( -1917) ) in his historical praise recalled 3 : ≪ Les conférences qu'il a données à l' École de Télégraphie nous montrent également combien il se tenait près de l'expérience, et quels services il a rendus aux praticiens.

Before the First World War

L'équation, dite des télégraphistes, nous fait connaître, comme on sait, les lois de la propagation d'une perturbation électrique dans un fil. Poincaré intègre cette équation par une méthode générale qui peut s'appliquer à un grand nombre de questions analogues. Le résultat varie suivant la nature du récepteur placé sur la ligne, ce qui se traduit mathématiquement par un changement dans les équations aux limites, mais la même méthode permet de traiter tous les cas.

Dans une seconde série de conférences, Poincaré a étudié le récepteur téléphonique ; un point qu'il a mis particulièrement en évidence, c'est le rôle des courants de Foucault dans la masse de l'aimant.

Enfin, dans une troisième série de conférences, il a traité les diverses questions mathématiques relatives à la télégraphie sans fil : émission, champ en un point éloigné ou rapproché, diffraction, réception, résonance, ondes dirigées, ondes entretenues ( 1 ). ≫

( 1 ) Ces Conférences ont été publiées dans la collection des cours de 1'école et dans la revue L' Éclairage électrique.

After the First World War (1919-1928).

One might think that his lectures were forgotten because they dealt with an application to a device for Wireless Telegraphy : the singing arc which had become obsolete after the war. electrical-mechanical analogy, Janet [START_REF] Janet | Sur une analogie électrotechnique des oscillations entretenues[END_REF] provided the nonlinear differential equation characterizing the oscillations sustained by the singing arc and by the triode :

Janet and the electrical-mechanical analogy (1919

(4) L d 2 i dt 2 + [R -f ′ (i)] di dt + k 2 K i = 0
By using the electrical-mechanical analogy and by posing

H → k 2 / K and θ (x ′ ) = f ′ (i) di
dt it is easy to show that Eq. ( 1) and Eq. ( 4) are completely analogous. Nevertheless, there is no reference to Poincaré in this note. ≪ Pour que des oscillations soient engendrées spontanément et s'entretiennent, il ne suffit pas que l'on ait un mouvement périodique, il faut encore que ce mouvement soit stable. ≫ Then, he provided the nonlinear differential equation of the singing arc :

Pomey and the singing arc equation (1920). On

(5)

Lx ′′ + Rx ′ + 1 C x = E 0 + ax ′ -bx ′3 By posing H = 1/C, ρ = R and θ (x ′ ) = -E 0 -ax ′ + bx ′3
it is obvious that Eq. ( 1) and Eq. ( 5) are completely identical 4 . Moreover, it is striking to observe that Pomey has used exactly the same variable x ′ as Poincaré to represent the current intensity. Here again, there is no reference to Poincaré. This is very surprising since Pomey was present during the last lecture of Poincaré at the École Supérieure des Postes et Télégraphes in 1912 whose he had written the introduction. So, one can imagine that he could have attended the lecture of 1908. (1925). On September 28 th 1925, Pomey wrote a letter to the mathematician Élie Cartan (1869Cartan ( -1951) ) in which he asked him to provide a condition for which the oscillations of an electrotechnics device analogous to the singing arc and to the triode whose equation is exactly that of Janet (see Eq. ( 4)) are sustained. Within ten days, Élie Cartan and his son Henri sent an article entitled : ≪ Note sur la génération des oscillations entretenues ≫ in which they proved the existence of a periodic solution for Janet's equation [START_REF] Poincaré | Sur les courbes définies par une équation différentielle[END_REF]. In fact, their proof is based on a diagram (see Fig. 3) which corresponds exactly to a "first return map" diagram introduced by Poincaré à i 1 , par suite le minimum suivant e à -i 1 , etc. On peut maintenant voir facilement que toute solution tend vers une solution périodique 5 . ≫ Obviously, Fig. 3 exhibits an application of Poincaré's method although there is no reference to his works. Moreover, Cartan didn't recognize in this periodic solution a limit cycle of Poincaré.

Élie and Henri Cartan and the existence of a periodic solution

Liénard and the existence and uniqueness of a periodic solution (1928).

Three years after, on May 1928, the engineer Alfred Liénard (1869-1958) published an article entitled ≪ Étude des oscillations entretenues ≫ in which he studied the solution of the following nonlinear differential equation :

(6) d 2 x dt 2 + ωf (x) dx dt + ω 2 x = 0
Such an equation is a generalization of the well-known Van der Pol's equation and of course of Janet's equation [START_REF] Poincaré | Sur les courbes définies par une équation différentielle[END_REF]. Under certain assumptions on the function F (x) = ∫ x 0 f (x) dx less restrictive than those chosen by Cartan [START_REF] Cartan | Note sur la génération des oscillations entretenues[END_REF] and Van der Pol [START_REF] Van Der Pol | On "relaxation-oscillations[END_REF], Liénard [START_REF] Liénard | Étude des oscillations entretenues[END_REF] proved the existence and uniqueness of a periodic solution of Eq. ( 6). Then, Liénard [17, p. 906] plotted this solution (see Fig. 4) and wrote :

≪ On se rend ainsi compte que la courbe intégrale décrit une sorte de spirale tendant asymptotiquement vers la courbe fermée D. Pour les courbes intégrales extérieures à la courbe fermée, c'est OA 2 qui 5. See Cartan [14, p. 1199].

devient inférieur à OA 1 . La courbe se rapproche encore de la courbe D, mais par l'extérieur. En raison de ce fait que toutes les courbes intégrales, intérieures ou extérieures, parcourues dans le sens des temps croissants, tendent asymptotiquement vers la courbe D on dit que le mouvement périodique correspondant est un mouvement stable. ≫ The Liénard [17, p. 906] explained that the condition for which the "periodic motion" is stable is given by the following inequality :

(7) ∫ Γ F (x) dy > 0
By considering that the trajectory curve describes the closed curve clockwise in the case of Poincaré and counter clockwise in the case of Liénard, it is easy to show that both conditions (3) and ( 7) are completely identical 6 and represents an analogue of what is now called "orbital stability". Again, one can find no reference to Poincaré in Liénard's paper. Moreover, it is very surprising to observe that he didn't used the terminology "limit cycle" to describe its periodic solution. The case of Liénard is really a riddle that we will discussed below.

6. For more details see Ginoux [START_REF] Ginoux | Analyse mathématiques des phénomènes oscillatoires non linéaires[END_REF] and Ginoux [START_REF] Ginoux | The first "lost" International Conference on Nonlinear Oscillations[END_REF].

The reception of Andronov's results in France

After the Poincaré's death in 1912, the French mathematician Jacques Hadamard (1865-1963) succeeded him at the Academy of Sciences. According to Maz'ya et Shaposhnikova [33, p. 181] Hadamard was elected corresponding member of the Russian Academy of Sciences in 1922 and, foreign member of the Academy of Sciences of the USSR in 1929. That's probably the reason why he was asked to review 7 and to present at the Comptes-Rendus on October 14 th 1929 a note from the young Russian mathematician Aleksandr' Aleksandrovich Andronov (1901Andronov ( -1952) ) entitled :

≪ Les cycles limites de Poincaré et la théorie des oscillations autoentretenues. 8 ≫ In this short paper (only three pages), Andronov [START_REF] Andronov | Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues[END_REF] stated a correspondence between the periodic solution of self-oscillating systems representing for example the oscillations of a radiophysics device used in Wireless Telegraphy and the concept of stable limit cycle introduced by Poincaré [2,p. 261].

In order to analyze the reception of Andronov's result, a selection of works published in France and elsewhere during the period (1930-1943) will be briefly presented.

Van der Pol's relaxation oscillations and limit cycles (1930).

Looking at the famous plot of the graphical integration of the triode oscillator exhibited by Van der Pol [15, p. 983] (see Fig. 5), one might think that he has immediately recognized that such a periodic solution was a stable limit cycle of Poincaré. This is not the case. It is just after the publication of Andronov's note at the Comptes-Rendus that Van der Pol realized that the periodic solution he has plotted (see Fig. Let's notice that although he quotes Poincaré, he makes reference to Andronov. Moreover, he also quotes in the following, the papers of Cartan [START_REF] Cartan | Note sur la génération des oscillations entretenues[END_REF] and that of Liénard [START_REF] Liénard | Étude des oscillations entretenues[END_REF] but it does not seem that he has ever used their works. During his visits in Paris, Van Moreover, his contribution for the understanding of the processes of development of the nonlinear oscillations theory is fundamental since he recalls the essential steps that would surely have fallen into oblivion without his intervention. Nevertheless, it is striking to notice that in his famous lecture given in Paris at the Conservatoire National des Arts & Métiers on May 6 th and on May 7 th 1931, Le Corbeiller [23, p. 4] quoted very few Andronov and never Poincaré :

≪ Mais c'est un physicien hollandais, M. Balth. van der Pol, qui, par sa théorie des oscillations de relaxation (1926) a fait avancer la question d'une manière décisive. Des savants de divers pays travaillent actuellement à élargir la voie qu'il a tracée ; de ces contributions, la plus importante nous paraît être celle de M. [START_REF] Liénard | Étude des oscillations entretenues[END_REF]. Des recherches mathématiques fort intéressantes sont poursuivies par M. Andronow, de Moscou. ≫ On April 22 nd 1932, Le Corbeiller gave a lecture in Paris at the École Supérieure des Postes et Télégraphes where he has been student many years before (ESPT, 1914). Let's recall that Poincaré taught in this school from 1902 till 1912 where he also gave many lectures and in particular his "forgotten lectures" on Wireless Telegraphy.

Discussing the graphical integration of the triode oscillator proposed by Van der Pol [15, p. 983] (See Fig. 5. above), Le Corbeiller [24, p. 708-709] wrote :

≪ La théorie générale de ces courbes intégrales fermée, ou cycles limites, a été faite par H. Poincaré ( 2 ) [START_REF] Poincaré | Sur les courbes définies par une équation différentielle[END_REF] ; la démonstration de l'existence d'un cycle limite unique, dans ce cas actuel, est due à M. Liénard [START_REF] Andronov | Predel nye cikly Puankare i teori kolebani i[END_REF].

( 2 ) D'une manière tout à fait générale, l'équation dx X (x, y) = dy Y (x, y) équivaut aux deux suivantes :

d 2 x dt 2 -y ( Y X + x ) + x = 0, y = dx dt .
Le mémoire cité de Poincaré équivaut donc à l'étude du système oscillant conservatif d 2 x dt 2 + x = 0, soumis à des forces de dissipation et d'entretien dont la résistance est une force quelconque de x et de dx dt It is very interesting to remark that Le Corbeiller made reference to the original paper of Poincaré and that he also gave many mathematical details on the way of writing the differential equation characterizing the oscillations of a dissipative system.

: ≫ F ( x, dx dt ) = y Y X + x. [1] H. POINCAR É,

The Liénard riddle (1931). As recalled above, in his first paper entitled

≪ Étude des oscillations entretenues ≫, Liénard [START_REF] Liénard | Étude des oscillations entretenues[END_REF] proved the existence and uniqueness of a periodic solution of a generalized Van der Pol's equation without making any connection with Poincaré's works. Then, less than one year after the presentation of Andronov's notes at the Comptes-Rendus, Liénard participated with Le Corbeiller and Van der Pol to the Third International Congress of Applied Mechanics held in Stockholm from 24 th to 29 th August 1930 where he presented an article entitled : ≪ Oscillations auto-entretenues ≫. According to the title, one might have thinking that, in this second and last publication on this subject, Liénard would have taken account of Andronov's result and that he would have established a connection between the periodic solution and a Poincaré's limit cycle. But, surprisingly not.

In this work, Liénard [START_REF] Liénard | Oscillations auto-entretenues[END_REF] first summarized his previous results and then he generalized another result established by Andronov and Witt [START_REF] Andronov | Sur la théorie mathématique des auto-oscillations[END_REF] in their second and last note at the Comptes-Rendus in which they studied the "Lyapunov stability" of the periodic solution, i.e. the stability of a limit cycle or "orbital stability". In order to extend Andronov and Witt's proposition, Liénard [21, p. 176] made use of "variational equations", i.e. of a method introduced by Poincaré [2, p. 162] in the first volume of his famous ≪ Méthodes Nouvelles de la Mécanique Céleste ≫ and which corresponds to what is today known under the name of the computation of "characteristics exponents". To do that, he modified his own equation ( 6) and replaced it by the following which is now know as "Liénard's equation" :

(8) d 2 x dt 2 + ωf ( x, dx dt ) + ω 2 x = 0
Then, he wrote :

≪ Si l'équation (8) admet une solution périodique, de période T, la condition pour que cette solution soit stable est que l'intégrale pendant une période de ∂f (x, x ′ ) ∂x ′ dt soit positive. La proposition, établie par Messieurs ANDRONOW et WITT ( 11) dans le cas particulier où la fonction f (x, x ′ ) est très petite se généralise immédiatement. ≫ Thus, Liénard [START_REF] Liénard | Oscillations auto-entretenues[END_REF] generalizes the result of Andronov and Witt [START_REF] Andronov | Sur la théorie mathématique des auto-oscillations[END_REF] for the stability of a periodic solution, i.e. of a limit cycle according to Poincaré's method of "characteristics exponents" but without quoting Poincaré's works and without using the terminology "limit cycle" for describing the stable periodic solution 12 . However, this expression appears in the very first pages of the article of Andronov and Witt [19, p. 256] was invited, the question of periodic solutions of type limit cycle has been much discussed.

3.4. The first "lost" International Conference on Nonlinear Oscillations (1933). From 28 to 30 January 1933 the first International Conference of Nonlinear Oscillations was held at the Institut Henri Poincaré (Paris) organized at the initiative of the Dutch physicist Balthasar Van der Pol and of the Russian mathematician Nikolaï Dmitrievich Papaleksi. The discovery of this event, of which virtually no trace remains, was made possible thanks to the report written by Papaleksi at his return in USSR. This document has revealed, on the one hand, the list of participants who included French mathematicians : Alfred Liénard, Élie and Henri Cartan, Henri Abraham, Eugène Bloch, Léon Brillouin, Yves Rocard, ... and, on the other hand the content of presentations and discussions. The analysis of the minutes of this conference highlights the role and involvement of the French scientific community in the development of the theory of nonlinear oscillations 13 .

According to Papaleksi [25, p. 211], during his talk, Liénard recalled the main results of his study on sustained oscillations : "Starting from its graphical method for constructing integral curves of differential equations, he deduced the conditions that must satisfy the nonlinear characteristic of the system in order to have periodic oscillations, that is to say for that the integral curve to be a closed curve, i.e. a limit cycle."

This statement on Liénard must be considered with great caution. Indeed, one must keep in mind that Papaleksi had an excellent understanding of the work of Andronov [START_REF] Andronov | Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues[END_REF] and that his report was also intended for members of the Academy of the USSR to which he must justified his presence in France at this conference in order to show the important diffusion of the Soviet work in Europe. Despite the presence of MM. Cartan, Lienard, Le Corbeiller and Rocard it does not appear that this conference has generated, for these scientists, a renewed interest in the problem of sustained oscillations and limit cycles. However, although the theory of nonlinear oscillations does not seem to be in France at that time a research priority, it is the subject of several Ph-D theses and monographs discussed below.

3.5. The French Ph-D theses. During the period (1936)(1937)(1938)(1939)(1940)(1941)(1942)(1943) several Ph-D theses were defended in France on a subject strongly related to nonlinear oscillations. Two of them are briefly recalled below 14 .

The first is that of R. Morched-Zadeh 15 who defended a Ph-D thesis at the Faculté des Sciences de l'Université de Toulouse in October 1936 entitled : ≪ Étude des oscillations de relaxation et des différents modes d'oscillations d'un circuit comprenant une lampe néon ≫. In the introduction of his study Morched-Zadeh [27, p. 3] wrote :

≪ Au point de vue théorique, les cycles limites de H. POINCAR É prennent une grande place dans la théorie des oscillations autoentretenues comme l'ont démontré A. ANDRONOW et A. WITT. ≫ 13. For more details see Ginoux [START_REF] Ginoux | Analyse mathématiques des phénomènes oscillatoires non linéaires[END_REF] and Ginoux [START_REF] Ginoux | The first "lost" International Conference on Nonlinear Oscillations[END_REF]. 14. For instance four Ph-D thesis have been found during this period and completely analyzed by Ginoux [START_REF] Ginoux | Analyse mathématiques des phénomènes oscillatoires non linéaires[END_REF].

15. No biographic information could be found concerning this student except the fact that he was Iranian but not parent with Lotfi Morched-Zadeh (personal communication).

In this case it is very surprising to observe that Morched-Zadeh made reference to the second and last note of Andronov and Witt [START_REF] Andronov | Sur la théorie mathématique des auto-oscillations[END_REF] at the Comptes-Rendus and not to the first which seemed to be better known and most quoted.

The aim of his work is an experimental study of relaxation oscillations of a neon lamp submitted to various oscillating regimes comprising of course the case of selfsustained oscillations. This led him to take the very first pictures of a limit cycle on a cathode ray tube oscilloscope (see Fig. The second is that of Jean Abelé (1886Abelé ( -1961) ) who was physicist, philosopher and writer 16 . He defended his Ph-D thesis entitled : " Étude d'un système oscillant entretenu à amplitude autostabilisée et application à l'entretien d'un pendule élastique " at the Faculté des Sciences de l'Université de Paris in front of a jury comprising Yves Rocard. In the introduction of his work Abelé [29, p. 18] wrote :

≪ À un mouvement périodique stable correspond une courbe intégrale fermée dont s'approchent asymptotiquement en spirales, de l'intérieur et de l'extérieur, pour t croissant, les solutions voisines. Un des problèmes fondamentaux de la théorie non linéaire consiste dans la recherche de ces courbes fermées, dites cycles limites 17 . ≫ [29, p. 18] 16. For biographic details see for example Ginoux [35,p. 325]. 17. Abelé quotes Andronov [START_REF] Andronov | Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues[END_REF].

Although, the definition of a stable limit cycle exactly corresponds to that given by Poincaré himself, Abelé quotes Andronov.

3.6. Rocard's textbooks. During the Second World War, the physicist Yves Rocard (1903-1992) published two manuscripts. If the title of the first one ≪ Théorie des Oscillateurs ≫ is very close to that of Andronov and Khaikin [START_REF] Andronov | Teori kolebani i (Théorie des oscillations)[END_REF] published in Russian and entitled : "Teori kolebani i 18 " the content is quite different. Rocard [START_REF] Rocard | Théorie des oscillateurs[END_REF] proposes a synthesis of several works done in this area as well as a summary of the article of Van der Pol [START_REF] Van Der Pol | On "relaxation-oscillations[END_REF] on relaxation oscillations with figures including the Fig. 5 that he commented thus : This is the single occurrence of the terminology limit cycle in the whole textbook, which is given without any reference 19 .

In 1943, Rocard [START_REF] Rocard | Dynamique générale des vibrations[END_REF] published his ≪ Dynamique Générale des Vibrations ≫ which has been wrongly considered as a textbook on nonlinear oscillations. In fact, in this book which comprises sixteen chapters only three deals with this subject. In chapter XV, Rocard [31, p. 220] recalls the results of Liénard [START_REF] Liénard | Étude des oscillations entretenues[END_REF] and plots the following figure : Then, he explained : ≪ . . . on constate (comme autrefois H. Poincaré l'a démontré) que la courbe intégrale s'enroule un certain nombre de fois et tend vers une courbe fermée dite cycle limite, qui dans cette représentation correspond au régime d'oscillations permanent. ≫ Here again, there is no reference neither to Poincaré nor to Andronov.

Discussion

This study has shown that two major contributions in the development of nonlinear oscillation theory had occurred in France during the first half of the XX th century. The first is the correspondence between the concept of limit cycle and the existence of a stable regime of sustained oscillations in Wireless Telegraphy established by Henri Poincare in 1908 in these "forgotten lectures" at the École Supérieure des Postes et Télégraphes, and the second is the same kind of correspondence, established twenty years later in a more general context by the Russian mathematician Aleksandr' Andronov in his famous note at the Comptes-Rendus.

If, the "discovery" of these "forgotten lectures" demonstrated that Poincaré has stated that the periodic solution of a nonlinear differential equation characterizing the nonlinear oscillations of a particular radiophysics device named singing arc is nothing else but a stable limit cycle, it has really produced no reaction on the French scientific community.

Nevertheless, the analysis of the influence of Andronov's note on this scientific community from 1929 to 1943 has shown that it was nearly the same. Liénard, for unknown reasons didn't make use of the terminology "limit cycle" neither before 1929 nor after. Moreover, although he became probably aware of Andronov's correspondence during the first "lost" International Conference on Nonlinear Oscillations in 1933 he has not pursued his research in this area. But, in 1933, Liénard was 64 years old and near to retirement. This was not the case for Le Corbeiller who has been one of the first to establish a deep connection with Poincaré's works. However, when the Second World War was declared he went to the USA and became a Professor in Harvard. Concerning Rocard, who made only few allusions to Poincaré's concept of limit cycle in his textbooks, he turned to nuclear research immediately after WWII.

Thus, it seems that although France has been a kind of crossroads for the development of nonlinear oscillations theory, nobody has succeeded in unifying French scientists around a research program in this area.

In fact, the mathematician Jacques Hadamard has been deeply involved in this task at various levels. First, he has presented during the 1930's many notes at the French Academy of Sciences on this subject coming from USSR : that of Andronov, of Andronov and Witt but also that of Kryloff and Bogoliouboff. But, he has also presented the works of Poincaré during his seminar at the Collège de France from 1919 to 1932. Unfortunately it didn't provide any reaction from the French scientific community.

So, although this community has produced many fundamental results necessary for the development of the nonlinear oscillation theory such as that of Liénard for example there has been no research program like in USSR or in USA and so, no "School of nonlinear" in France. The terrible impact of the WWI and WWII is probably responsible for such a lack of organization. Three days later, the physicist Nicolas Minorsky [START_REF] Minorsky | Influence d'Henri Poincaré sur l'évolution moderne de la théorie des oscillations non linéaires[END_REF] presented a lecture to civil engineers entitled : ≪ Influence d'Henri Poincaré sur l'évolution moderne de la théorie des oscillations non linéaires ≫ in which he began with these words :

≪ La répercussion des travaux d'Henri POINCAR É s'est fait sentir dans presque tous les domaines des sciences appliquées, mais c'est surtout dans la théorie des oscillations qu'elle a provoqué de tels changements que cette théorie est aujourd'hui passablement différente de ce qu'elle soit. ≫ Then, he concluded by this sentence which shows that he didn't have knowledge of Poincaré's "forgotten lectures" : ≪ Il est difficile de trouver dans l'histoire de la Science un autre exemple de théorie mathématique développée sans aucune relation aux applications . . . qui ait présenté une base aussi parfaite pour l'étude des phénomènes innombrables qui se sont révélés depuis lors, sans qu'il y ait presque rien a changer à cette théorie un demi-siècle plus tard. ≫

Figure 1 .

 1 Figure 1. Circuit diagram of the singing arc, Poincaré [7, p. 390]

  Ly dy dx + ρy + θ (y) + Hx = 0 Poincaré [7, p. 390] stated then that : "Sustained oscillations correspond to closed curves if there exist any." and he gave the following representation for the solution of equation (2) :

Figure 2 .

 2 Figure 2. Closed curve solution of Eq. (2),Poincaré [1908, p. 390]

( 1 )

 1 Poincaré, Conférences sur la Télégraphie sans fil faites à l'école professionnelle supérieure des Postes et Télégraphes de Paris, 1908, p. 25.

  June 28 th 1920, a textbook written by the engineer Jean-Baptiste Pomey (1861-1943) entitled : " Introduction à la théorie des courants téléphoniques et de la radiotélégraphie " was published in France. A former student of the École Supérieure des Postes et Télégraphes, Pomey (ESPT, 1883) became Professor of Theoretical Electricity in this school alongside Henri Poincare and then director from 1924 to 1926. In Chapter XIX of his book, devoted to the generation of sustained oscillations Pomey [13, p. 375] wrote :

Figure 3 .

 3 Figure 3. "First return map" diagram, Cartan [1925, p. 1199]

Figure 4 .

 4 Figure 4. Closed curve solution of Eq. (6), Liénard [1928, p. 905].

≪ 1 )

 1 5) was a limit cycle. It was during a lecture given in Paris at the École Supérieure d' Électricité on March 10 th and 11 th 1930. Van der Pol [20, p. 294] reproduced the figures of his original paper of 1926 (comprising the Fig. 5) and he said : On remarque sur chacune de ces trois figures une courbe intégrale fermée ; c'est un exemple de ce que Henri POINCAR É a appelé un cycle limite ( 1 ), parce que les courbes intégrales s'en rapprochent asymptotiquement. ≫ ( Voir par exemple : A. Andronow, Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues, C. R. 189, 559 (1929).

Figure 5 . 3 . 2 .≪

 532 Figure 5. Graphical integration of the triode oscillator, Van der Pol [15, p. 983]

10 .

 10 Van der Pol.

≪ ( 2 )

 2 in footnote : Pour la définition des auto-oscillations et la discussion du cas d'un degré de liberté voir A. ANDRONOW, Les cycle limites de Poincaré et la théorie des oscillations auto-entretenues (Comptes rendus, 189, 1929, p. 559). ≫ Therefore, it seems very difficult to explain the attitude of Liénard especially since at the first International Conference on Nonlinear Oscillations, to which he 11. Liénard quotes Andronov and Witt [1930].12. For more details see Ginoux[35, p. 210].

Figure 6 .

 6 Figure 6. Limit cycle of the neon lamp oscillator, Morched-Zadeh [1936, p. 127]

Figure 7 .

 7 Figure 7. Limit cycle of relaxation oscillator, Rocard [31, p. 220]18. "Theory of oscillations".19. The question of the absence of references in Rocard's book has been much discussed. Interviewed on this issue, Rocard said that because of the war he was unable to access these documents. In fact it has been shown in Ginoux[35, p. 263] that it was not true.

  During the commemoration of the centenary of Poincaré's birth, the newspaper Le Monde published on May 15 th , 1954 an article entitled ≪ Les conferences de Henri Poincaré à l' École Supérieure des P.T.T. ≫ (See Fig. 8) in which we learn that Eugène Reynaud-Bonin, a former student of this school (ESPT, 1911), has attended to Poincaré's "forgotten lectures" in 1908 and 1910. His interview was reproduced in Le Monde (see Fig. 8 below).

Figure 8 .

 8 Figure 8. Le Monde, May 15 th , 1954

  ). Of course, during the First World War the triode invented on January 15 th 1907 by Lee de Forest (1873-1961) had supplanted the singing arc. However, a French engineer named Paul Janet (1863-1937) published in 1919 a note at the Comptes-Rendus in which he established an analogy between the triode and the singing arc. In this paper, entitled ≪ Sur une analogie électrotechnique des oscillations entretenues ≫ and, by using the classical

2. See Blondel [10, p. 100]. 3. See Darboux [11, p. 37].

  in his memoir ≪ Sur les Courbes définies par une équation différentielle ≫ [2, p. 251]. They wrote : Les points H 1 , H 2 , H 3 où la courbe coupe la bissectrice correspondent à des solutions périodiques (oscillations entretenues) dont l'existence est ainsi démontrée. Elles sont périodiques parce qu'en partant d'un minimum donné -i 1 , le maximum suivant est égal 4. For more details see Ginoux & Lozi [36].

≪

  der Pol was hosted by a young French engineer named Philippe Le Corbeiller (1891-1980) who helped him to translate his talks 9 . Le Corbeiller who was probably present at the École Supérieure d' Électricité on March 10 th and 11 th 1930 was invited to give a lecture at the Third International Congress of Applied Mechanics held in Stockholm from 24 th to 29 th August 1930. He was accompanied by Van der Pol himself and by Alfred Liénard.

	7. The original of this note is presented in Ginoux [35, p. 494-496].
	8. Andronov [18]. It has been pointed out by Ginoux [35, p. 176] that this note has been
	preceded by a presentation at the Congress of Russian Physicists between 5 th and 16 th August
	1928. See Andronov [16].
	9. See Van der Pol [20, p. 312].

  Mémoire sur les courbes définies par une équation différentielle, Deuxième partie, Journal de Mathém. Pures et app. 8, 251, 1882 ; et OEuvres, T. 1, p. 44. [16] A. LI ÉNARD, Étude des oscillations entretenues, Rev. gén. d'électr., 901 et 946, 1926.

≪ On voit au fur et à mesure que ε croît, se déformer le cycle limite et apparaître les harmoniques. ≫


Andrea Cattini
email: andrea.cattini@unimore.it

Devis Bellucci

Antonella Sola

Lech Pawlowski

Valeria Cannillo

Lech Pawłowski

Microstructural design of functionally graded coatings composed of suspension plasma sprayed hydroxyapatite and bioactive glass

Keywords: bioactive glass, calcium phosphate(s), in vitro

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

Microstructural design of functionally graded coatings composed of suspension plasma sprayed hydroxyapatite and bioactive glass

INTRODUCTION

Calcium phosphates in general and, in particular, hydroxyapatite (HA; Ca 10 (PO 4 ) 6 (OH) 2 ) are good biomaterials for orthopedic prostheses because of their similarity to the mineral component of bone, which is a mixture of various salts, such as carbonated apatite, calcium-deficient apatite and whitlockite, and numerous impurities such as sodium, potassium, magnesium, fluoride, and chloride. [START_REF] De Groot | Calcium phosphate coatings for medical implants[END_REF] As HA is brittle, it is unsuitable for structural applications. Accordingly, the implants for bone repair usually include a HA coating on a metal substrate, which provides the required mechanical strength. [START_REF] Chevalier | Ceramics for medical applications: A picture for the next 20 years[END_REF] Unfortunately, there are some drawbacks in this design. The most important of them is the fact that crystalline HA is more stable than other calcium phosphates in aqueous media for pH values between 4.2 and 8.0. This implies that HA is relatively insoluble at neutral pH and therefore it reacts so slowly in the human body that it may touch a limit of applicability in specific clinical applications. [START_REF] Juhasz | Bioactive ceramics: Processing, structures and properties[END_REF] However, several results in the literature indicate that the reaction rate of HA coatings is deeply conditioned by their porosity, defectiveness, and degree of crystallinity. [START_REF] Sun | Material fundamentals and clinical performance of plasma-sprayed hydroxyapatite coatings: A review[END_REF][START_REF] Stiegler | High-velocity suspension flame sprayed (HVSFS) hydroxyapatite coatings for biomedical applications[END_REF] Finally, from a technological point of view, the deposition of HA-based coatings is still a challenging issue, despite the progress achieved in several coating deposition meth-ods, especially in thermal spraying. [START_REF] Cheang | Addressing processing problems associated with plasma spraying of hydroxyapatite coatings[END_REF] The main concern is represented by the thermal decomposition of HA, which results in the uncontrolled development of secondary phases, such as tricalcium phosphate (TCP), tetracalcium phosphate (TTCP), and amorphous calcium phosphate (ACP), whose reactivity is hard to predict. Some decomposition phases, such as CaO, may reveal unsuitable for prostheses. [START_REF] Liao | Thermal decomposition and reconstitution of hydroxyapatite in air atmosphere[END_REF][START_REF] Lu | Factors influencing phase compositions and structure of plasma sprayed hydroxyapatite coatings during heat treatment[END_REF] As a possible alternative to calcium phosphates, bioactive glasses may be successfully used to coat prosthetic devices. The bone-bonding ability was first detected for specific glasses belonging to the Na 2 O-CaO-SiO 2 -P 2 O 5 system, with appropriate proportions between the constituent oxides. [START_REF] Hench | Bioceramics: From concept to clinic[END_REF] Among them, the Bioglass V R 45S5 (24.5 wt % Na 2 O, 24.5 wt % CaO, 6 wt % P 2 O 5 , 45 wt % SiO 2 ), formulated by Prof. L. L. Hench at the end of the 1960s, is currently considered to be the reference standard. [START_REF] Hench | The story of BioglassV R[END_REF] Bioactive glasses offer several advantages because they possess the highest degree of bioactivity and their reaction rate may be controlled by their composition. [START_REF] Jones | Review of bioactive glass: From Hench to hybrids[END_REF] Particular formulations are even able to interact with soft tissues. [START_REF] Rahaman | Bioactive glass in tissue engineering[END_REF] Moreover, bioactive glasses do not decompose at high temperature. However, it is worth noting that the coating deposition by thermal spraying implies the achievement of extremely high temperatures, and bioglasses are prone to devitrification and some of their components may evaporate. [START_REF] Lefebvre | Sintering behaviour of 45S5 bioactive glass[END_REF][START_REF] Bretcanu | Sintering and crystallisation of 45S5 BioglassV R powder[END_REF] The development of crystalline phases is known to alter and retard the bone-bonding ability of the original glass. [START_REF] Boccaccini | Sintering, crystallisation and biodegradation behaviour of BioglassV Rderived glass ceramics[END_REF][START_REF] Bellucci | An overview of the effects of thermal processing on bioactive glasses[END_REF] As a consequence, the introduction of bioactive glass coatings may represent an improvement with respect to conventional systems, but a careful setting of the spraying parameters is required.

In a previous contribution, [START_REF] Cattini | Functional bioactive glass topcoats on hydroxyapatite coatings: Effects on microstructure and in-vitro bioactivity[END_REF] the effectiveness of a thin glass topcoat was evaluated to promote the performance of a standard plasma-sprayed HA coating. The glass topcoat was produced by means of the suspension plasma spraying (SPS) technique, a new deposition process which works on suspensions instead of dry powders, thus making it possible to handle even fine particles. [START_REF] Pawłowski | Suspension and solution thermal spray coatings[END_REF] The specific aim of the current study was to produce various glass/HA coatings configurations, and then to verify the effect of the phase distribution on the final properties, especially on the apatite-forming ability in a simulated body fluid (SBF). [START_REF] Kokubo | How useful is SBF in predicting in vivo bone bioactivity?[END_REF] The so-called BG_Ca was selected as the bioactive glass (in wt %: 4.7 Na 2 O, 42.3 CaO, 6P 2 O 5 , 47 SiO 2 ) because it offers both good bioactivity and a limited tendency to crystallize at high temperature. This is due to its formulation, which is relatively rich in CaO and poor in Na 2 O with respect to conventional bioactive glasses based on the standard bioglass family. [START_REF] Lockyer | NMR investigation of the structure of some bioactive and related glasses[END_REF][START_REF] Sola | Heat treatment of Na2O-CaO-P2O5-SiO2 bioactive glasses: Densification processes and postsintering bioactivity[END_REF] Figure 1 shows three different types of coating designs that were produced and compared in this study. The composite coatings were obtained by spraying a mixture of the two raw materials and therefore the constituent phases were randomly distributed throughout the coating. The duplex coatings consisted of a double-layered structure: BG_Ca on the surface and HA in contact with the substrate. Finally, the graded coatings included both components varying gradually from pure HA to pure BG_Ca. The latter resulted from the subsequent deposition of many layers with progressively different compositions. Because the volume fraction of each constituent was the same in the three types of coatings, the study was specifically focused on the effect of the phase distribution on the coating behavior.

MATERIALS AND METHODS

Feedstock materials

Following the procedure defined in previous articles specifically addressed to the production and deposition of bioactive glass coatings by SPS, the BG_Ca glass was obtained by the conventional melt-quenching method, then it was dry milled in an agate jar, and, to conclude, attrition-milled in ethanol with 2 wt % of Beycostat C213. [START_REF] Lockyer | NMR investigation of the structure of some bioactive and related glasses[END_REF][START_REF] Sola | Heat treatment of Na2O-CaO-P2O5-SiO2 bioactive glasses: Densification processes and postsintering bioactivity[END_REF][START_REF] Cattini | Suspension plasma sprayed bioactive glass coatings: Effects of processing on microstructure, mechani-cal properties, and in-vitro behaviour[END_REF] The obtained BG_Ca powder had a mono-modal distribution with a mean diameter of d 50 5 4.7 mm, as determined with a Partica LA-950V2 (Horiba) granulometer. The feedstock suspension was formulated by dispersing 20 wt % of solid phase in 80 wt % of ethanol. Ethanol was chosen as the milling and suspending medium because bioactive glasses are waterreactive. [START_REF] Zeng | Fabrication of Al2O3-A/W bioglass bioactivity tapes by tape casting[END_REF] The HA initial powder was a commercial product by Tomita (Japan) and had the mean diameter of d 50 5 120 mm. The commercial powder was attrition-milled using 0.8 mm zirconia balls and adding the dispersant Beycostat C213 (2 wt % of the dry powder). The final powder had a slightly bimodal distribution with a mean diameter of d 50 5 4.6 mm (mean peak at 4.8 mm, secondary peak at 0.2 mm). The feedstock suspension was formulated by dispersing 20 wt. % of solid phase in 40 wt % of ethanol and 40 wt % of water. The mixture of water and ethanol was chosen instead of pure ethanol (as previously described for the glass powder suspension) because it is known from the literature that water vapor may prevent or slow down the thermal decomposition of HA during plasma spraying. [START_REF] Li | Effect of steam treatment during plasma spraying on the microstructure of hydroxyapatite splats and coatings[END_REF] In particular, because it was necessary to co-spray the HA powder and the BG_Ca one to obtain HA-glass composite layers, the processing conditions had to be chosen in an optimal way for each of the constituent material. [START_REF] Cattini | Suspension plasma sprayed bioactive glass coatings: Effects of processing on microstructure, mechani-cal properties, and in-vitro behaviour[END_REF][START_REF] Kozerski | Two zones microstructure of suspension plasma sprayed hydroxyapatite coatings[END_REF] Generally speaking, the deposition of the BG_Ca glass requires more power input in the plasma jet than that of HA, so the high power applied to obtain HA-glass biphasic systems was expected to promote the decomposition of HA. Therefore, the addition of water, which is known to cool down the plasma, [START_REF] Pateyron | Influence of water and ethanol on transport properties of the jets used in suspension plasma spraying[END_REF] aimed to preserve the structure of HA. On the other hand, it is worth noting that, if the HA powder were dispersed in pure water, the enthalpy of the plasma would be reduced to such an extent that the glass particles could not be properly melted anymore. Consequently, the use of an ethanol-water mixture appeared to be the best compromise solution to deal with HA and glass powders at the same time.

Coatings deposition

TheSG-100 torch (Praxair, S.T., Indianapolis, IN) was used to produce the coatings. The two feedstock suspensions were separately supplied by means of peristaltic pumps described elsewhere. [START_REF] Tomaszek | Microstructure of suspension plasma sprayed multilayer coatings of hydroxyapatite and titanium oxide[END_REF] In addition, two nonreturn valves were mounted to avoid any undesired mixing and reflux phenomena. During spraying, the suspensions were continuously stirred to prevent sedimentation and agglomeration. Because the pumping systems were independent, the suspensions mixed together only at the entrance of the injector (for biphasic glass/HA layers only). An injector with a diameter of 500 lm was used. This injection of the suspension was internal to the torch at 16 mm from its exit.

Numerous tests were made to find the optimal deposition parameters that could combine a minimal decomposition of HA with a good cohesion of the BG_Ca particles. The final spray parameters are listed in Table I.

All the coatings were obtained through eight spraying runs, each of which corresponded to four torch passages. Before each session, the sample surface was left to cool down to the temperature below 80 C.

To obtain the different microstructures of the composite, duplex, and graded coatings, the feed rates of the two suspensions were properly regulated by means of the peristaltic pumps. The values are listed in Table II. The planned compositions are schematically shown in Figure 1.

Substrates were 316 L stainless steel discs (25 mm in diameter, thickness of 8 mm). They were sandblasted using corundum powder having a size of 250 lm at the pressure of 5 bar. The resulting roughness was Ra 5 3.5 mm. Before spraying, the substrates were ultrasonically washed with ethanol and acetone.

During the deposition, an optical IN 5 Plus pyrometer (Impac) was used to record the surface temperature. The mean values of the temperature were calculated for each spray run, and they are reported in Table III.

The coatings of pure BG_Ca and of pure HA were sprayed using the same parameters applied for the biphasic systems. These coatings were considered as the references.

In vitro tests

The samples were tested in vitro by soaking them in an SBF solution. [START_REF] Kokubo | How useful is SBF in predicting in vivo bone bioactivity?[END_REF] According to the standard protocol defined by Kokubo and Takadama, the SBF volume-to-coating surface ratio has to be 20 mL/cm 2 . [START_REF] Kokubo | How useful is SBF in predicting in vivo bone bioactivity?[END_REF] Therefore, the samples were cut to obtain a surface area of 1 cm 2 , and each sample was immersed in 20 mL of SBF. The samples were removed from the SBF after time periods of 1, 7, and 14 days. During the tests, the SBF was refreshed every 2 days to simulate the dynamic conditions of the human body. The temperature was maintained constant at 37 C and the sealed containers were kept in a water-vapor saturated environment to avoid any undesired fluid evaporation. Once extracted from the SBF, the samples were carefully washed with distilled water and dried at room temperature.

Characterization

The samples were analyzed before and after soaking in SBF using different techniques. The surface of the coatings was investigated by means of X-ray diffraction (XRD). The instrument was an X'pert PRO diffractometer (PANalytical, Almelo, The Netherlands) equipped with an X'Celerator detector. The diffraction patterns were collected in the 10 -70 2u range (step size: 0.017 , step time: 71.12 s).

The microstructural analysis was performed on the coatings' surface and polished cross-section by means of an environmental scanning electron microscope (ESEM; ESEM Quanta 200-FEI Company, Eindhoven, The Netherlands), equipped with an X-EDS microanalysis system (Inca, Oxford Instruments, UK). The microscope was operated in high vacuum (pressure: 10 25 Pa) and in low-vacuum mode (pressure: 60 Pa) according to the specific sample to analyze.

Scratch tests were performed on the surface of the asproduced biphasic coatings (Open platform, CSM Instruments, equipped with 100 mm Rockwell diamond tip). The applied load was linearly increased from 20 mN to 30 N over a scratch length of 3 mm, with a loading rate of 10 N/ min. The critical loads were determined by optical analysis. Three scratches were performed on each sample.

RESULTS AND DISCUSSION

Regardless of their microstructural design, the obtained biphasic coatings were continuous and well-adhering to the substrates. However, the surfaces of the samples, especially the duplex and the graded ones, were rough, as shown in Figure 2. The roughness was mainly due to the presence of semi-spherical structures, which resulted from sintering and swelling phenomena that occurred for the glass particles. In fact, the surfaces of the duplex and graded samples [Figure 2(b,c)], which were extremely rich in glass, were similar to that of the pure BG_Ca coating [Figure 3(a)], with a presence of many semi-spherical features. Inversely, the morphology of the composite coating [Figure 2(a)], having a surface rich in HA, was similar to that of the pure HA coating [Figure 3(b)], without any semi-spherical structures. At a higher magnification (Figure 4), both flattened splats and crystallized zones could be observed. The crystals were thin and possessed an acicular morphology. As previously observed for semi-spherical structures, also the crystallizations were more visible in the glass-rich surfaces of the duplex and graded deposits.

To verify whether the obtained microstructures matched the expected ones, the polished cross-sections were observed and analyzed by means of X-EDS analysis. In particular, the compositional maps were acquired to visualize the distribution of silicon, which was selected as the marker of the BG_Ca glass, and that of phosphorus, which was chosen to indicate HA. As shown in Figure 5, a good correspondence could be generally detected. More in detail, the microstructure of the composite sample [Figure 5 , the composition smoothly changed across the thickness of the coating. However, it was apparent that the relative volume fraction of the glass in the final samples was greater than expected. Most likely, this discrepancy was because of the higher deposition efficiency of the bioactive glass with respect to that of HA. This explanation is confirmed by the comparison between the cross-sections of the mono-component coatings, which is presented in Figure 6. In fact, the pure BG_Ca coating was twice as thick as the pure HA one, although both of them were produced using the same processing parameters.

The XRD spectra of the coatings are shown in Figure 7. For comparison, Figure 7 incorporates the X-ray spectra of the mono-component coatings. A clear similarity between the patterns of the duplex and the graded samples can be easily observed. They are also similar to the spectrum of the pure BG_Ca coating. In the spectra, the broadband between 25 and 35 (2u) was due to the amorphous glassy phase, because it is typical of silicate glasses, 28 and the peaks of pseudowollastonite (CaSiO 3 ) could be detected as a result of a limited devitrification of the original glass. Instead, the pattern of the composite sample was more similar to that of the pure HA coating. This is not surprising because, in this sample, a high amount of HA was present on the surface. In the spectrum of the composite coating, the peaks of HA and the band of glass were both present, but the development of pseudowollastonite was not evident. It is worth noting that the diffraction pattern of the pure HA coating, besides the peaks of the HA itself, showed some secondary peaks attributable to the decomposition phases such as TTCP, TCP, and CaO. Because the composite, duplex, and graded coatings were sprayed using the same parameters as the pure HA one, it is probable that also the HA present in biphasic coatings could have decomposed, even if the peaks of the secondary phases were not visible because of the presence of the bioactive glass, which surrounded (in the composite coating) or covered (in the duplex and graded coatings) the HA grains.

The mechanical characterization of the samples was made using the scratch test. The critical load was defined as the load at which the metal substrate became detectable. The critical load values were mainly determined by means of optical observation, but also frictional force and acoustic emission were useful for its detection (Figure 8). The critical load (mean value 6 standard deviation) was 27.1 6 0.8 N for the composite coating and 21.2 6 1.7 N for the duplex one. During the scratch test of the graded sample, the maximum applied load (30 N) was not enough to reach the substrate. Therefore, the graded microstructure provided the best scratch resistance. Probably, the double-layered structure of the duplex sample, with an abrupt interface between the glass and the HA, promoted the generation of residual stresses. The stresses could have been strongly reduced by the progressive change in composition of the graded coating. [START_REF] Cannillo | Microscale computational simulation and experimental measurement of thermal residual stresses in glass-alumina functionally graded materials[END_REF] With regards to the in vitro tests, the SEM analysis revealed that, already after 1 day of immersion in SBF, the surface morphology of the biphasic samples changed; indeed, diffused "cauliflower-like" features could be detected [Figure 9(a-c)], which are characteristic of the hydroxycarbonate-apatite deposits grown in vitro. [START_REF] Li | Characterization of calcium phosphates precipitated from simulated body fluid of different buffering capacities[END_REF][START_REF] Pattanayak | Apatite wollastonite-poly methyl methacrylate bio-composites[END_REF] All the biphasic samples followed a similar evolution. Instead, if the mono-component coatings are considered, the surface of the BG_Ca sample showed the same behavior of the biphasic coatings, with the development of dome-like features [Figure 10(a)], whereas the apatite-forming ability of the pure HA coating was less visible. In fact, as shown in Figure 10(b), the surface of the HA coating did not undergo any significant change as a result of the immersion in SBF.

Further details on the in vitro behavior could be deduced by the analysis of the polished cross-sections, which are presented in Figures 111213. Generally speaking, the duplex and the graded coatings evolved in a similar way and they both reproduced the transformations occurring on the pure BG_Ca coating; the composite sample, instead, was slightly different because HA particles were dispersed throughout the coating thickness, thus a layered microstructure was not recognizable. Finally, the pure HA coating remained almost unaltered by the SBF.

After 1 day of immersion, the cross-sections of the duplex [Figure 11(a)] and graded [Figure 12(a)] samples showed a layered structure. Both samples had a surface layer rich in calcium and phosphorus, and a deeper layer mainly composed of silicon; below them, the deepest part of the coating was still unchanged. On account of the data available in the literature, it is possible to conclude that the surface layer was the result of the precipitation and growth of HA induced by the reaction between the glass and the SBF, mediated by the development of a silica gel layer. 32,[START_REF] Sola | Bioactive glass coatings: A review[END_REF] As previously mentioned, the same behavior was observed on the cross-section of the BG_Ca coating, which displayed a HArich layer on the surface and a silica gel zone underneath. In this sample, the silica gel layer grew quickly, such that the coating was almost completely converted to silica gel already after 1 week of immersion in SBF (data not shown).

In addition, the composite coating [Figure 13(a)] was able to promote the development of a thin reaction deposit, which was rich in Ca and P. Moreover, localized silica gel zones and unreacted glass areas could be detected. Nevertheless, in this case, there was not a clearly stratified structure, but only the single spots dispersed throughout the coating thickness. This was due to the initial microstructure of the composite coating in which areas of BG_Ca were interspersed with HA zones. In fact, the reactions in SBF primarily involved the BG_Ca particles that could be reached by the SBF percolating through the residual pores and possible cracks present in the coating, whereas the HA-rich zones remained almost unchanged. As previously mentioned, the limited apatite-forming ability of the HA phase is even more evident if the pure HA coating is considered. In fact, the areas with different Ca/P ratios could be observed on the cross-section only after 1 week in SBF (these data are not shown). In particular, a Ca-deficient layer was detected on the surface, which may correspond to the development of a Ca-poor ACP, as described by Kim et al. [START_REF] Kim | Process and kinetics of bonelike apatite formation on sintered hydroxyapatite in a simulated body fluid[END_REF] After 2 weeks in SBF, the duplex [Figure 11(b)] and graded [Figure 12(b)] samples had an unusual microstructure. In fact, they both showed the highest concentration of silicon in the deepest part of the cross-section, close to the interface with the metal substrate. This behavior is commonly observed in bioactive glasses, [START_REF] Sola | Bioactive glass coatings: A review[END_REF] because the reaction with SBF starts at the glass surface and then it moves inward. Accordingly, in bioactive glass coatings, the deepest layers in contact with the substrate are usually the last part of the coating that undergoes chemical changes to silica gel first and then to HA. However, such phenomenon is difficult to explain for the duplex and graded samples, because in the original unreacted coatings, there was no glass at the interface with the substrate, but HA (Figure 5). One possible explanation is that, at the beginning of the reaction process with the SBF, the BG_Ca-rich layers on the surface quickly reacted to form silica gel and HA. In a second step, the reaction proceeded further, involving the underlying HA-rich layers. In this way, the deepest layers of the coatings, consisting of HA, were dissolved by the SBF, and the precipitation of new HA did not take place in the same point, but on the surface of the preexisting silica gel and HA layers (derived from the precocious reaction of the BG_Ca-rich fractions). This involved a gradual migration of the silicon-rich zone downward in the direction of the substrate. The silicon-rich zone close to substrate was not observed in the composite coating [Figure 13(b)] in which silicon-rich particles were still present throughout the thickness of the coating even after 2 weeks of immersion in SBF. Again, the different behavior of the composite coating depends on both its "zoned" (not layered) microstructure and the different reaction kinetics of bioactive glass and HA in a physiological environment.

The XRD analysis basically confirmed the reaction model previously outlined according to the SEM observations. As far as the composite coating is concerned, the diffraction spectra (Figure 14) should be considered with caution, because the HA peaks were already present in the XRD diagram of the as-produced sample, thus confusing the identification of the peaks associated with the in vitro grown HA. Nevertheless, it is possible to see a gradual change in the spectra, because the broad halo in the 25 -35 2u range, reflecting the presence of the BG_Ca (silicate) glass, progressively disappeared as a result of the glass dissolution-HA precipitation process. As already seen for the ESEM images, also comparing the evolution of the XRD spectra after in vitro tests, a similar behavior could be observed for the duplex (Figure 15) and the graded (Figure 16) samples. The peaks associated with the partial crystallization of the glassy phase (pseudowollastonite) completely disappeared after 1 week of immersion; at the same time, the main peaks of HA became visible. A broad halo in the 20 -25 2u range was due to the growth of a silica gel layer, whereas the peaks of calcium carbonate, coprecipitated with HA, could be observed occasionally. [START_REF] Kokubo | How useful is SBF in predicting in vivo bone bioactivity?[END_REF][START_REF] Cannillo | Potassium-based composition for a bioactive glass[END_REF] After 2 weeks in SBF, the peaks in the diffraction spectra of the duplex and graded samples were exclusively associated with HA. However, the peaks were still broad, thus confirming the defective and microcrystalline nature of the HCA developed in vitro. [START_REF] Vitale Brovarone | Macroporous bioactive glass-ceramic scaffolds for tissue engineering[END_REF] The XRD spectra of the pure BG_Ca coating showed the same trend (data not shown). However, in this case, after 2 weeks of immersion, the silica gel was still detectable and the peaks of the precipitated HA looked even broader than in the duplex and graded spectra. This result suggests that the presence of the HA-rich areas underneath the bioactive glass surface layer did not compromise the reaction kinetics in the duplex and graded coatings.

To conclude, the XRD spectra of the pure HA coating showed a limited change, even if the peaks associated with the secondary phases (such as TTCP, TCP, etc.) gradually disappeared starting from the first day of immersion (data not shown). In fact, it is known from the literature that the solubility of both TTCP and TCP is greater than that of (crystalline) HA. [START_REF] Dorozhkin | Calcium orthophosphates in nature, biology and medicine[END_REF]

CONCLUSIONS

In this study, various bioactive glass/HA biphasic coatings were produced by means of SPS. To reveal the importance of the phase distribution, three different layouts were considered: (i) composite coating, that is, randomly distributed constituent phases; (ii) duplex coating with glass top layer onto HA layer; and (iii) graded coating with a gradual changing composition starting from pure HA at the interface with the metal substrate up to pure glass on the surface.

All the biphasic coatings had a good adhesion to the substrate and they were continuous. Moreover, the obtained microstructure met the designed ones, although the glass fraction was greater than the designed one because of the HA deposition efficiency, which was lower than the glass one. However, the deposition technique and the spray setup proved to be proper for producing coatings with specific microstructures.

According to the in vitro tests, all the coatings were able to promote the development of apatite when immersed in an SBF. Nevertheless, the apatite-forming ability was especially evident for those coatings, which presented a glassrich layer on the surface (i.e., duplex and graded). As regards the mechanical performance, the duplex configuration should be avoided, because the abrupt interface between the glass topcoat and the HA layer implies relevant residual stresses, which undermine the scratch resistance.

The graded coating therefore seems to be the most promising solution, because it combines a good apatiteforming ability and a strong adhesion to the substrate, as observed by means of scratch tests.

FIGURE 1 .

 1 FIGURE 1. Designed microstructures of the biphasic coatings: (a) composite, (b) duplex, and (c) graded. Light blue: BG_Ca glass; Dark grey: HA. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

  (a)] presented HA-rich areas dispersed in a glass matrix. The crosssection of the duplex sample [Figure 5(b)] was clearly divided into two layers. Finally, in the graded sample [Figure 5(c)]

FIGURE 2 .

 2 FIGURE 2. ESEM images of the coatings' surface: (a) composite, (b) duplex, and (c) graded.

FIGURE 3 .

 3 FIGURE 3. ESEM images of the reference coatings' surface: (a) pure BG_Ca and (b) pure HA.

FIGURE 4 .

 4 FIGURE 4. High-magnification ESEM images of the coatings' surface: (a) composite, (b) duplex, and (c) graded.

FIGURE 5 .

 5 FIGURE 5. X-EDS maps showing the distribution of Si (in green -representative of the BG_Ca glass) and P (in red -representative of HA) on the cross-sections: (a) composite, (b) duplex, and (c) graded. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

FIGURE 6 .

 6 FIGURE 6. ESEM images of the reference coatings' cross-sections.

FIGURE 7 .

 7 FIGURE 7. X-ray diffraction patterns of the as-produced coatings *: pseudowollastonite [01-089-6485], ~: TTCP [00-025-1137], !: TCP [00-009-0169], : CaO [01-075-0264], • : HA [01-089-6437]. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

FIGURE 8 .

 8 FIGURE 8. Analysis of a representative scratch test (duplex coating): juxtaposition of acoustic emission graph, frictional force graph, and optical analysis. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

FIGURE 9 .

 9 FIGURE 9. SEM images (secondary electrons) of the biphasic coatings' surface after 14 days in SBF: (a) composite, (b) duplex, and (c) graded.

FIGURE 10 .

 10 FIGURE 10. SEM images of the reference coatings' surface after 14 days in SBF: (a) BG_Ca and (b) HA.

FIGURE 11 .

 11 FIGURE 11. Microstructure and X-EDS analysis of the cross-section of the duplex coating after 1 day in SBF (a) and after 14 days in SBF (b).[Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

FIGURE 12 .FIGURE 13 .

 1213 FIGURE 12. Microstructure and X-EDS analysis of the cross-section of the graded coating after 1 day in SBF (a) and after 14 days in SBF (b).[Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

FIGURE 15 .

 15 FIGURE 15. XRD of the duplex coatings after in vitro tests. w: CaCO 3 , • : HA. [Color figure can be viewed in the online issue, which is avail- able at wileyonlinelibrary.com.]

FIGURE 16 .

 16 FIGURE 16. XRD of the graded coatings after in vitro tests. w: CaCO 3 , • : HA. [Color figure can be viewed in the online issue, which is avail- able at wileyonlinelibrary.com.]

FIGURE 14 .

 14 FIGURE 14. XRD of the composite coatings after in vitro tests. w: CaCO 3 [01-085-1108], • : HA. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

TABLE I .

 I Spray Parameters

								ORIGINAL RESEARCH REPORT
								TABLE III. Surface Temperature (Mean Value and Standard
	Parameter				Value	Deviation) Measured During Spraying
	Electric power input				34 kW	Temperature ( C)
	Gases							321 6 10	Composite
	Ar					45 slpm	247 6 53	Duplex
	H 2					7.5 slpm	297 6 46	Graded
	Spray distance				55 mm
	Torch speed				750 mm/s
	Scan step				10 mm
	TABLE II. Suspensions Flow Rates in Different Spray Runs
	(g/min)
		Composite	Duplex		Graded
	Run	BG_Ca	HA	BG_Ca	HA	BG_Ca	HA
	1	15	15	-	30	-	30
	2	15	15	-	30	4	26
	3	15	15	-	30	9	21
	4	15	15	-	30	13	17
	5	1 5	1 5	3 0	-	1 7	1 3
	6	1 5	1 5	3 0	-	2 1	9
	7	1 5	1 5	3 0	-	2 6	4
	8	1 5	1 5	3 0	-	3 0	-

JOURNAL OF BIOMEDICAL MATERIALS RESEARCH B: APPLIED BIOMATERIALS | MONTH 2013 VOL 00B, ISSUE 00

ACKNOWLEDGMENTS

The Vinci Program (Italian-French University) support for Dr. A. Cattini is gratefully acknowledged.


T Chanier

On the origin of the different Mayan Calendars

The Maya were known for their astronomical proficiency. This is demonstrated in the Mayan codices where ritual practices were related to astronomical events/predictions. Whereas Mayan mathematics were based on a vigesimal system, they used a different base when dealing with long periods of time, the Long Count Calendar (LCC), composed of different Long Count Periods: the Tun of 360 days, the Katun of 7200 days and the Baktun of 144000 days. There were two other calendars used in addition to the LCC: a civil year Haab' of 365 days and a religious year Tzolk'in of 260 days. Another cycle was also in use: a 3276-day cycle (combination of the 819-day Kawil cycle and the 4 directions-colors). Based on the hypothesis that Mayan astronomers had a good knowledge of naked-eyed astronomy, we propose here an explanation of the origin of the LCC, the Haab', the Tzolk'in and the 3276-day cycle. This study shed more light on the connection between astronomy, arithmetics and religion in Maya civilization.

I. INTRODUCTION

Mayan astronomy and religion have always been intertwined. This can be seen in the various codices where Mayan astronomers-priests were calculating the commensuration of astronomical and ritual cycles to determine the date of religious and civil ceremonies in a complex set of calendars and cycles. To date these ceremonies over long period of time and pinpoint historical events, the Maya used the Long Count Calendar (LCC), counting the number of day elapsed since the beginning time, the mythical date of creation. The LCC is organized as follow. The smallest unit of time is the day (Kin); 20 Kin form a Winal, 18 Winal form a Tun (360 Kin), 20 Tun form a Katun (7200 Kin), and 20 Katun form a Baktun (144000 Kin). The LCC represents a date D as a set of coefficients (C i , ..., C 3 , C 2 , C 1 , C 0 ) such that: 20), C 1 = int(mod(D,360)/20) and C i = int(mod(D, 18 × 20 i )/(18 × 20 i-1 ) for i > 1. The day count usually restarts when C 4 reaches 13, such as the date is given as a set of 5 coefficients: D ≡ mod(D, 13 × 144000) = C 4 .C 3 .C 2 .C 1 .C 0 . Along with the LCC, the Maya used two other independent calendars: a religious year (Tzolk'in), a civil year (Haab'). One Tzolk'in of 260 days comprised 13 months (numerated from 1 to 13) containing 20 named days (Imix, Ik, Akbal, Kan, Chicchan, Cimi, Manik, Lamat, Muluc, Oc, Chuen, Eb, Ben, Ix, Men, Cib, Caban, Etznab, Cauac, and Ahau). This forms a list of 260 ordered Tzolk'in dates from 1 Imix, 2 Ik, ... to 13 Ahau. 1 One Haab' of 365 days comprised 18 named months (Pop, Uo, Zip, Zotz, Tzec, Xul, Yaxkin, Mol, Chen, Yax, Zac, Ceh, Mac, Kankin, Muan, Pax, Kayab, and Cumku) with 20 days each (Winal) plus 1 extra month (Uayeb) with 5 nameless days. This forms a list of 365 ordered Haab' dates from 0 Pop, 1 Pop, ... to 4 Uayeb. 2 The Tzolk'in and the Haab' coincide every 73 Tzolk'in or 52 Haab' or a Calendar Round (CR). A CR corresponds to the least common multiple (LCM) of 260 and 365: 73 × 260 = 52 × 365 = 18980 days. In the Calendar Round, a date is represented by αXβY with the religious month 1 ≤ α ≤ 13, X one of the 20 religious days, the civil day 0 ≤ β ≤ 19, and Y one of the 18 civil months, 0 ≤ β ≤ 4 for the Uayeb. According to the Goodman-Martinez-Thompson (GMT) correlation to the Gregorian calendar, which is based on historical facts, the Mayan Calendar began on 11 August 3114 BC or 0 (13).0.0.0.0 4 Ahau 8 Cumku and ended on 21 December 2012 or 0 (13).0.0.0.0 4 Ahau 3 Kankin. This corresponds to a 13 Baktun cycle (1872000 days) or a period of approximately 5125 years. Another ritual cycle of 3276 days was used. It is formed by the combination of the Kawil, a 819-day cycle, and a 4-day direction-color cycle. [START_REF] Berlin | The 819-day Count and direction-color Symbolism among the Classic Maya[END_REF] The direction-color decreases/changes for each Kawil as mod(n × 819 + 3,4) = 3, 2 ,1 and 0 (East-Red, South-Yellow, West-Black and North-White) for n ∈ N i.e. the 4-Kawil cycle starts at {3, East-Red}.

D = C 0 + C 1 × 20 + n i=2 C i × 18 × 20 i-1 with C 0 = mod(D,
The origin of the Long Count Periods (LCPs) is unknown. A common assumption is the desire of the calendar keeper to maintain the Tun in close agreement with the tropical/solar year of approximately 365.24 days. [START_REF] Aveni | Skywatchers: A Revised and Updated Version of Skywatchers of Ancient Mexico[END_REF] There is no consensus concerning the origin of the Tzolk'in, which has been associated with various astronomical cycles. Three Tzolk'in correspond to Mars synodic period, 16 Tzolk'in equal 11 of Saturn synodic periods (+2 days), and 23 Tzolk'in are equivalent to 15 Jupiter synodic periods (-5 days). [START_REF] Justeson | The ancient Maya Ethnoastronomy: An Overview of Hieroglyphic Sources[END_REF] It has been tentatively connected to the eclipse half-year (173.31 days) because 2 Tzolk'in are very close to 3 eclipse half-years. [START_REF] Aveni | Skywatchers: A Revised and Updated Version of Skywatchers of Ancient Mexico[END_REF] Finally, it has been noted that the Tzolk'in approximates the length of time Venus is visible as a morning or evening star. [START_REF] Bricker | Astronomy in the Maya Codices[END_REF] However, these interpretations fail to link the Tzolk'in to the LCPs. The Kawil cycle has been attributed to the observation of Jupiter and Saturn 8,9 because 19 (6) Kawil correspond to 39 (13) Jupiter (Saturn) synodic period. Why the Calendar Round starts on a 4 Ahau 8 Cumku is still an opened question as is the starting count of the 4-Kawil cycle. Four numbers of possible astronomical significance have been discovered on the walls of a residential structure in Xultun, Guatemala and have been dated from the early 9 th century CE. The Xultun numbers are given in Table I. They are such that X 1 = 365 × 3276 and Based on the assumption that Mayan astronomers observed the periodic movements of the five Planets visible to the naked eye, this paper explains a possible origin of the Long Count Periods, the religious year Tzolk'in and the relationship between the Kawil cycle and the 4 directions-colors. This hypothesis provides also a possible explanation of the date origin of the Calendar Round and the 4-Kawil cycle. The paper is organized as follow.

X 3 = X 2 + 2X 0 . Xi LCC D [
In section II, we derive an important number based on astronomical observations and arithmetical considerations that seems to explain some aspects of Mayan religion and calendars. In section III, we study the various calendar cycles, some of them identified on Mayan Codices and monuments, and examine their relationship with the Planet canonic cycles. The last section is left for conclusion. The Maya were known for their astronomical skills as exemplified by the Dresden Codex, a bark-paper book of the 11 th or 12 th century CE. On page 24 of this Codex is written the so-called Long Round number noted 9.9.16.0.0 in the LCC or 1366560 days, a whole multiple of the Tzolk'in, the Haab', the Calendar Round, the Tun, Venus and Mars synodic periods: LR = 1366560 = 4 × X 0 = 5256 × 260 = 3744 × 365 = 72 × 18980 = 3796 × 360 = 2340 × 584 = 1752 × 780. Only the moon, Mercury, Venus, Earth (solar year), Mars, Jupiter, and Saturn are visible to the naked eye. Their respective canonic synodic periods are given in Table II. Evidences have been found in different Mayan Codices that Mayan astronomers observed the periodic movements in the night sky of Mercury, Venus, and Mars, but it is unclear whether they tracked the movements of Jupiter and Saturn. [START_REF] Bricker | Astronomy in the Maya Codices[END_REF] The periods relevant for the prediction of solar/lunar eclipses are the pentalunex of 148 days (5 Moon synodic periods of 29.53 days) and the lunar semesters of 177 or 178 days (6 Moon synodic periods), which are the time intervals between subsequent eclipse warning stations present in the Eclipse Table on pages 51 to 58 of the Dresden Codex. 12 The LCM of these numbers is N = 20757814426440 days (including the Planet synodic periods and the two lunar semesters) or N † = 768039133778280 days (also including the pentalunex). Table III gives the divisibility of N by a polynomial expression of the type P i = 13 × 73 × ( n 18 × 20 n ).

II. MAYAN ASTRONOMY AND RELIGION

Planet

= 13 × 73 × ( 6-i n=0 18 × 20 n ). Di = int(N /Pi).
N is such that: N is such that the mod(N ,LR) = 341640 days or 2.7.9.0.0 in the LCC. This is the Xultun number X 0 (Table I). It is to be noted that X 0 is a whole multiple of the Tzolk'in, Haab', Tun, Venus and Mars synodic periods: 341640 = 1314 × 260 = 936 × 365 = 949 × 360 = 585 × 584 = 438 × 780. X 0 is also the largest common divisor of LR and N .

Taking into account the ordered Tzolk'in and Haab' date, 1,2 a date D in the CR can be written as D ≡ {mod(D + 160,260);mod(D + 349,365)} from the beginning of the 13 Baktun era 4 Ahau 8 Cumku {160;349}. We remark that mod(N /13/73,260) = 160 = 4 mod 13 = 0 mod 20 or the Tzolk'in date 4 Ahau, associating a number to the day in the Tzolk'in dates list. 1 We remark also that mod(N /13/73,73) = 49 or the Haab' date 8 Zip, associating a number to the day in the Haab' dates list. 2 The date 4 Ahau 8 Zip {160;49} may define the date origin of the CR. If we consider the Mayan conception of time as cyclical, this day 0 (18980) corresponds to the completion of a CR such that: mod(18980,4680) = 260 where 4680 = LCM(260,360) = 13 × 360 is the 13 Tun Wheel. The next date such that mod(D,4680) = 0 is the date 4 Ahau 8 Cumku {160;349}: mod(4680 + 160,260) = 160 and mod(4680 + 49,365) = 349. The Calendar Round began on a 4 Ahau 8 Zip, 4680 days earlier than the Long Count Calendar beginning date 0(13).0.0.0.0 4 Ahau 8 Cumku.

III. MAYAN CALENDAR AND RITUAL CYCLES

We first examine the cycles that can be constructed from the combination of the Tzolk'in (260 = 13 × 20), the Haab' (365 = 5 × 73) and the LCPs (D = 18 × 20 n ). Because 13 is not present in the prime factorizations of the LCPs, we obtain LCM(260,D) = 13 ×D. There are, for example, the 13 Tun Wheel of 4680 days (18 Tzolk'in) and a 13 Katun cycle of 93600 days (360 Tzolk'in) reminiscent of the Katun Wheel. The 13 Baktun cycle comes from the coincidence between the Tzolk'in and the Baktun, corresponding to 1872000 days (400 × 13 Tun Wheel) or approximately 5125 years. Because 73 is not present in the prime factorizations of the LCPs, we obtain LCM(365,D) = 73 ×D. This gives rise to 73 LCP cycles. Moreover, the LCM(260,365,D) = 13 × 73 × D. This is the case of the Xultun number X 0 = 341640 = LCM(260,365,360) which makes coincide the Tzolk'in, the Haab' and the Tun. Various religious cycles can be constructed by taking the LCM of 260 and the Planet synodic periods (Table IV). The most important one in Mayan religion is the Calendar Round, which is the LCM(260,365) = 73 × 260 = 52 × 365 = 18980 days. There is a coincidence between Venus synodic period and the Tzolk'in: the LCM(260,584) = 65 × 584 = 146 × 260 = 104 × 365 = 37960 days (2 Calendar Rounds), the length of the Venus Table on pages 24-29 of the Dresden Codex.

There is also a coincidence between Mars synodic period and the Haab': the LCM(780,365) = 73 × 780 = 219 × 260 = 156 × 365 = 56940 days (3 Calendar Rounds): this is the largest common divisor of the Xultun numbers (Table I). We identify here two other cycles based on the coincidence of the Tzolk'in and the 234-and 338-day periods: the LCM(260,234) = 9 × 260 = 2340 days and the LCM(260,338) = 13 × 260 = 3380 days. Table V gives the coincidence of the 338-day period with the LCPs, as well as the Tzolk'in and the 234-day period. As described earlier, the Tun and 338-day period coincide every 60840 days, or 234 Tzolk'in. The 338-day period also coincides with the Winal every 3380 days, corresponding to 13 Tzolk'in, a cycle that does not coincide with the 234-day period. Rather, the 234-day period coincides with the Winal every 2340 days or 9 Tzolk'in = 9 × 260 = LCM (9,13,20). This 2340-day cycle is present in the Dresden Codex on pages D30c-D33c and has been attributed to a Venus-Mercury almanac because 2340 = 20 × 117 = 5 × 585 is an integer multiple of Mercury and Venus mean synodic periods (+1 day). [START_REF] Bricker | Astronomy in the Maya Codices[END_REF] Another explanation may be of divination origin because 117 = 9 × 13. In Mesoamerican mythology, there are a set of 9 Gods called the Lords of the Night [START_REF] Aveni | Skywatchers: A Revised and Updated Version of Skywatchers of Ancient Mexico[END_REF][START_REF] Boone | Cycles of Time and Meaning in the Mexican Books of Fate[END_REF][START_REF] Lounsbury | Dictionary of Scientific Biography[END_REF][START_REF] Joyce | Mexican Archaeology: An Introduction to the Archaeology of the Mexican and Mayan Civilizations of Pre-Spanish America[END_REF] and a set of 13 Gods called the Lords of the Day. [START_REF] Joyce | Mexican Archaeology: An Introduction to the Archaeology of the Mexican and Mayan Civilizations of Pre-Spanish America[END_REF] Each day is linked with 1 of the 13 Lords of the Day and 1 of the 9 Lords of the Night in a repeating 117-day cycle. This cycle coincides with the Tzolk'in every 2340 days. On Table V figures a date D = LCM(338,7200) = 1216800 days or 8.9.0.0.0 in the LCC or 5 February 219 CE according to the GMT correlation. This date may have marked the beginning of the reign of Yax Moch Xoc, the founder of the Tikal dynasty, in 219 CE. [START_REF] Schele | A forest of Kings[END_REF] We then study the cycle created by the combination of the 819-day Kawil cycle and the 4-day directioncolor cycle, coinciding every 3276 days. The Haab' and the Tzolk'in coincide with the 3276-day cycle every LCM(365,3276) = 365 × 3276 = 260 × 4599 = 1195740 days or 8.6.1.9.0 in the LCC. This is the Xultun number X 1 (Table I). X 1 = LCM (13,9,4,819,260,365) is the time distance between two days of the same Lord of the Day, Lord of the Night, direction-color, Kawil, Tzolk'in and Haab' date. X 0 and X 1 are such that 341640 × 819 = 1195740 × 360 = 2391480 or Y = 16.12.3.0.0 in the LCC. Y represents the same as X 1 at the end of a Tun (0 Winal, 0 Kin). The Kawil coincides with the Tun every 32760 days = LCM(360,819) = 7 × LCM(260,360) = 10 × 3276. Table VI shows the coincidence of the 32760-day cycle with the Planet canonic cycles, giving rise to whole multiples of the Tzolk'in and the 2340-day cycle. We remark that mod(N /32760,4) = 3 which may explain the start of the 4-Kawil cycle. As can be seen in its prime factorization (3276 = 2 2 × 3 2 × 7 × 13), the 4-Kawil cycle commensurates with all the Planet cycles except the Haab' (365 and 3276 are relatively prime numbers).

We have for example the LCM(148,3276) = 37 × 3276 = 121212 days. Finally, we discuss an important religious site in Mesoamerica, the pyramid of Kukulkan built sometime between the 9 th and the 12 th century CE at Chichen Itza (Figure 1) where various numbers in the architecture seem to have been related to calendrical considerations. The pyramid shape may be linked to the Long Count Calendar and the Planet canonic cycles (Table III) which draws a pyramid-like structure. It is constituted of 9 platforms with 4 stairways of 91 steps each leading to the platform temple corresponding to the 3276-day cycle: 3276 = LCM(4,9,91) = 4 × 819, the coincidence of the 4 directions-colors with the Kawil. The dimensions of the pyramid may be of significance: the width of its base is 55.30 m (37 zapal), its height up to the top of the platform temple is 30 m (20 zapal) and the width of the top platform is 19.52 m (13 zapal), taking into account the Mayan zapal length measurement such that 1 zapal ≈ 1.5 m. [START_REF] O'brien | [END_REF] The width of the base and 3276 are such that 37

IV. CONCLUSION

The study presented here describe a possible explanation of the origin of the Long Count Calendar (LCC) and its connection with the Tzolk'in and Haab' calendars. Based on arithmetical calculations on relevant astronomical periods obtained from naked-eyed observation of the Solar System, the LCC gives rise to three different periods of 234 days, 260 days (Tzolk'in) and 338 days. These astronomical calculations explain the various cycles used in Mayan religion and calendars. The 13 Baktun cycle originates from the coincidence between the Tzolk'in and the Baktun. The coincidence of the 234-day period and the Tzolk'in, a 2340-day cycle identified in the Dresden Codex, was certainly used for divination purposes. The coincidence of the 338-day period and the Tzolk'in gives rise to a 13-Tzolk'in cycle that has not yet been identified in the Codices. The 3276-day cycle, the coincidence of the 819-day Kawil and the 4 directions-colors, commensurates with all the Planet canonic cycles. These results indicate that Mayan astronomers calculated the synodic periods of all the Planets visible to the naked eye, as well as the basic cycles for solar/lunar eclipse prediction, the pentalunex and the lunar semesters.

* e-mail: thomas.chanier@gmail.com

3 n=1 18 ×

 318 int(N /13/73/144000) = 338 + 360 + 7200 + 144000. (1) N † gives the same results but divided by 13, 37, 73 and 144000. That defines the Tun, Katun, and Baktun as a polynomial expression of int(N /13/73) of the form 18 × 20 3 × (C 0 + 20 n ) with C 0 = 338. The LCM(338,360) = 234 × 260 = 60840. 338 and 365 are relatively prime numbers: the LCM(338,365) = 338 × 365 = 123370. 234 and 365 are also relatively prime numbers: the LCM(234,365) = 234 × 365 = 85410. On the other hand, the Tzolk'in and the Haab' are commensurate: the LCM(260,365) is 73 × 260 = 52 × 365 = 18980 days or a Calendar Round. That may define the Tzolk'in.

  LCP D [day] L [day] M [day] N [day] L/260 LCoincidence of the 234-, 260-and 338-day periods with the LCPs. L = LCM(D,338), M = LCM(D,260) and N = LCM(D,234).

FIG. 1 .

 1 FIG. 1. Pyramid of Kukulkan during an equinox. The pyramid is situated at Chichen Itza, Yucatan, Mexico.

  × 3276 = LCM(148,3276) = 121212. The pyramid height and the width of the top platform represents the Tzolk'in. The stairways divide the 9 platforms of each side of the pyramid into 18 segments which, combined with the pyramid height, represents the 18 Winal of a Tun. The Haab' is represented by the platform temple making the 365 th step with the 4 × 91 = 364 steps of the 4 stairways. Each side of the pyramid contains 52 panels corresponding to the Calendar Round: 52 × 365 = 73 × 260 = 18980. During an equinox, the Sun casts a shadow (7 triangles of light and shadow) on the northern stairway representing a serpent snaking down the pyramid (Figure 1). A 1820-day cycle or 7 Tzolk'in = 5 × 364 is present on pages 31-32a of the Dresden Codex. This may also correspond to 32760 = LCM(360,819) = 7 × LCM(260,360).

TABLE II .

 II Planet canonic cycles 11,12 and their prime factorizations.

		P [day] Prime factorization
	Mercury	116 2 2 × 29
	Venus	584 2 3 × 73
	Earth	365 5 × 73
	Mars	780 2 2 × 3 × 5 × 13
	Jupiter	399 3 × 7 × 19
	Saturn	378 2 × 3 3 × 7
	Lunar	177 3 × 59
	senesters	178 2 × 89
	Pentalunex 148 2 2 × 37

TABLE III

 III

	Name	i Ci [day]	N /Pi	Di
	-Tun Katun Baktun Pictun Calabtun 5 57600000 N /13/73/ 1 0 18 N /13/73/ 6 0 Ci 18 1 360 N /13/73/ 5 0 Ci 360 2 7200 N /13/73/ 4 0 Ci 7215 3 144000 N /13/73/ 3 0 Ci 144304 4 2880000 N /13/73/ 2 0 Ci 2886428 0 Ci 57866020
	Kinchiltun 6 1152000000 N /13/73/C0	1215186420

. Divisibility of N = 20757814426440 days by a polynomial expression of the type Pi

TABLE IV .

 IV Coincidence of the Planet canonic cycles 11,12 and the Tzolk'in.

	Planet	P [day] N = LCM(P ,260) N /260 N /P
	Mercury	116	7540	29	65
	Venus	584	37960	146	65
	Earth	365	18980	73	52
	Mars	780	780	3	1
	Jupiter	399	103740	399	260
	Saturn	378	49140	189	130
	Lunar	177	46020	177	260
	semesters	178	23140	89	130
	Pentalunex 148	9620	37	65

A. F. Aveni, in Skywatchers: A Revised and Updated Version of Skywatchers of Ancient Mexico (University of Texas Press

, 2001), p. 143. 2 A. F. Aveni, in Skywatchers: A Revised and Updated Ver-