M M Bateni
email: bateni2000@gmail.com

S Eslamian

S F Mousavi
email: mousavi@cc.iut.ac.ir

E Zia Hosseinipour
email: zia.hosseinipour@ventura.org

Application of a Localization Scheme in Estimating Groundwater Level using Deterministic Ensemble Kalman Filter

Keywords: Kalman Filter, Groundwater, Najafabad, Water table, Stochastic modeling

Sequential data assimilation methods such as Kalman filters (KF) help the modeler yield more accurate results, especially when correction approaches (like localization) are used. In this paper, the groundwater model of Najafabad Aquifer (central Iran) is used to predict water table levels. The results of the model are assumed to be true and considered as observation data for the filter. Deterministic Ensemble Kalman Filter (DEnKF, a newly released filter) assimilated observations from true run into the model with 2, 5, and 10 times of calibrated values of hydraulic conductivity. This filter was run with/without covariance localization assuming three length-scale parameters. The results showed that DEnKF with inaccurate model parameter values has greatly improved the outputs of the inaccurate model mainly in first few time steps. Localization of the filter causes a change in secondary rising trend of error and makes it a falling trend. The amount of improvement due to localization, based on length-scale parameters, was varied and showed no obvious relationship to the accuracy of the model parameters.

Introduction

Numerical modeling of groundwater flow has become increasingly attractive, due to development of modern computational tools in the past decades. However, the model parameters are not precisely known due to high soil heterogeneity. Application of data assimilation methods, e.g. Kalman filter (KF) enhances the results of the model, considering the observational data. The precision can be improved with some simple tricks, e.g. error covariance localization. In this paper, using a novel type of Ensemble Kalman filters (EnKF), which is a version of KF suitable for large scale systems, called Deterministic Ensemble Kalman Filter (DEnKF), the realistic inaccurate model results are amended and the effect of covariance localization to filter performance is investigated.

Background

In order to use observational information more efficiently in flow modeling, [START_REF] Evensen | Sequential data assimilation with a nonlinear quasigeostrophic model using Monte Carlo methods to forecast error statistics[END_REF] introduced the principles of EnKF. [START_REF] Burgers | Analysis scheme in the ensemble Kalman filter[END_REF] proved the abnormal fact that using synthetic noise for observations will increase EnKF

performance. This was a potential source of error, and so, it became a starting point in search for other EnKF versions that do not need artificial noise and resulted in new square root EnKF. These efforts have been summarized by [START_REF] Tippett | Ensemble square-root filters[END_REF] where the authors survey three types of square root ensemble filters developed by [START_REF] Bishop | Adaptive sampling with the ensemble transform Kalman filter Part I: Theoretical aspects[END_REF], [START_REF] Anderson | An ensemble adjustment Kalman filter for data assimilation[END_REF] and [START_REF] Whitaker | Ensemble data assimilation without perturbed observations[END_REF].

The most important weakness of square root EnKF, that was a general limitation of EnKF, was still remaining: the use of a finite ensemble size to approximate the error covariance matrix introduces sampling errors. [START_REF] Ott | A local ensemble Kalman filter for atmospheric data assimilation[END_REF] presented a detailed discussion on the use of local analysis calculations that was introduced by [START_REF] Evensen | The ensemble Kalman filter: Theoretical formulation and practical implementation[END_REF]. In another paper, [START_REF] Hunt | Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter[END_REF] gave a detailed review of implementation of the Local Ensemble Transform Kalman. Their approach to obtain analytical equations and the localization method differs from what have been done before. They showed that this is a good way to handle problems concerning small ensembles. But in both these two works, localization is not implemented easily.

In the first Hydrogeologic application of EnKF, [START_REF] Chen | Data assimilation for transient flow in geologic formations via Ensemble Kalman filter[END_REF] estimated water table level and hydraulic conductivity values simultaneously using the stochastic type EnKF that was introduced by [START_REF] Burgers | Analysis scheme in the ensemble Kalman filter[END_REF]. In two examples, with two and three-dimensional analysis, they modeled groundwater flow using MODFLOW with known variogram and used the resulting data as actual observations. These researchers examined the effect of ensemble size, initial sampling and different frequency of observation times. Drécourt et al. (2006a) examined the bias-aware filters. They used a onedimensional groundwater flow model with two bias-aware filter algorithms, one with feedback and another without feedback. Filters with feedback (especially the type with colored noise), even if exposed to drifting bias, have good results; but filters without feedback are only acceptable in the case of a constant bias. Drécourt et al. (2006b) used the colored noise KF with a new sampling method that preserves statistical population mean. They parameterized the observation noise covariance matrix and calibrated these parameters. In this study, the results of a two-dimensional aquifer model with known hydraulic parameters and inputs were treated as the actual data and filter performance with these calibrated covariances was evaluated. The results, except in the immediate neighborhood of observations, were acceptable. [START_REF] Sakov | A deterministic formulation of the ensemble Kalman filter: An alternative to ensemble square root filters[END_REF] presented a simple and efficient linear approximation of the square root EnKF that because of its deterministic characteristic and similarity to the Kalman filter algorithm is called Deterministic Ensemble Kalman Filter (DEnKF). [START_REF] Sun | Comparison of deterministic ensemble Kalman filters for assimilating hydrogeological data[END_REF] compared four types of square root EnKF to estimate hydraulic conductivity and found that DEnKF outperforms and is more robust than others, even in small ensemble sizes.

Groundwater flow equation

Groundwater flow equation in a phreatic isotropic aquifer can be written as:

(1)

t h S q N z h k z y h k y x h k x S ∂ ∂ - ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧
Where k is the hydraulic conductivity ( -1 Lt ), h is the potentiometric head ( L ), N is the recharge in volumetric flux per unit volume ( -1 t ), q is the discharge in volumetric flux per unit volume( -1 t ), S S is the specific yield and t is time ( t ) [START_REF] Krešić | Quantitative solutions in hydrogeology and groundwater modeling[END_REF].

Modeling System and Observation Equation

Groundwater flow model which is derived by spatial and temporal discretization of groundwater flow equation (Eq. 1) is a time-discretized linear state equation. In system theory, time forwarding of state of a system can be modeled by a difference equation:

k k k Bu AX X + = -1
Where ( X ) is groundwater levels at every mesh points in the aquifer, A is the transition matrix (groundwater flow law), u is the external variable that represents recharge (due to rainfall or irrigation) and discharge of groundwater, and B is its coefficient matrix. k is the time index of the all equations. The length of the system state vector ( X ) and u are equal to the numbers of mesh point ( n ); where A and B are n n× matrices.

By adding a stochastic term to the equation, Uncertainties can be modeled and the state equation redefined as follows:

k k k k W Bu AX X + + = -1
Where, k W is the system noise vector with size 1 × n and can be realized as a Gaussian random process, which is usually considered white (not auto-correlated in time) with zero mean. The covariance matrix of k W is shown as k Q . It is assumed that the initial system state vector is a random process with mathematical expectancy equal to 0 X and covariance matrix equal to 0 P [START_REF] Van Geer | Application of Kalman filtering in the analysis and design of groundwater monitoring networks[END_REF].

Observation equation of the filter is:

k k k k V X M Z + =
Where, k Z is the vector that contains observational data and k V which is named observation noise is a Gaussian process vector with covariance equal to k R . k M is also observation matrix and its elements are between zero and one depending on the location of observation wells. Observation vector ( k Z ) and the observation noise have the length equal to the number of observation ( p ) where observation matrix is a n p× matrix. It should be noted that covariance matrices k Q , k R and 0 P are all semi-positive definite like all other covariance matrices.

Filter algorithm

Recursive equations to obtain mean ( ) | ( l k X

) and covariance matrix ( ) | ( l k P

) of probability density of the state at time k ( k X ) conditioned on the observations till time l is summarized as:

Time updating 1 :

(2)

k k k k k k k u B X F X ) ( + ) 1 - | 1 - ( ) ( = ) 1 - | ( ˆ T T k k k k k k k k k ) ( ) ( ) ( + ) ( ) 1 - | 1 - ( ) ( = ) 1 - | ( G Q G F P F P Measurement updating 2 : (3) ( ) ) 1 - | ( ) ( - ) ( + ) 1 - | ( = ) | ( ˆk k k k k k k k k X M Z K X X ( ) k k k k k k k k Z K X M K I X ) ( + ) 1 - | 1 - ( ) ( ) ( - = ) | ( ˆ (4) ( ) ) 1 - | ( ) ( ) ( - = ) | ( k k k k k k P M K I P
Where, Kalman gain matrix is:

(5)

( ) 1 ) ( ) ( ) 1 | ( ) ( ) ( ) 1 | ( ) ( - + - - = k k k k k k k k k T T R M P M M P K
And this is a weighting factor between the predicted and the measured value.

Ensemble Kalman filter -Prediction and Analysis in EnKF

If an Initial ensemble with size N is defined, for each member of the ensemble there is a state vector ( i x ) for N i ,..., 1 = . State vector for each ensemble member has the dimension n . Ensemble mean is defined as:

∑ 1 = 1 = N i i N x x
Anomalies matrix with Dimensions N n× defined as:

1 Results of this update is called prior results 2 Results of this update is called analyzed or updated results ( )

x x x x x x X - - - - = ′ N N L 2 1 1 1
Prediction equation of the EnKF is similar to time update equation in the KF (Eq.

2). The only difference is that each ensemble member (column of the ensemble matrix) is operated on individually. By putting members together again forecast ensemble matrix ( f X ) is made. The forecast ensemble mean can be updated to the analysis ensemble mean by a relationship similar to the measurement update of KF (Eq. 3) only with different gain matrix.

Analysis error covariance ( a e P ) is derived using a relationship similar to KF (Eq. 4). For measurement update of ensemble anomalies matrix several methods have been proposed. Different Ensemble Kalman Filters are tried to derive analysis ensemble matrix from forecast ensemble matrix by various schemes -with known analysis ensemble mean and covariance-. The key point is how to make the ensemble that is consistent with analysis distribution.

Deterministic Ensemble Kalman Filter (DEnKF)

Each realization of the ensemble with covariance matrix a e P is Just one linear combination of -forecast vectors

N x x x , , , 2 1 K
.So analysis ensemble will be:

A X X f a =
Where, A is a matrix containing weighting coefficients for the linear combination; A is called Ensemble Transformation Matrix (ETM). The matrix A is defined to satisfy the following:

T e a f T f T a a = = X AA X X X P
An answer for A can be obtained simply by replacing f e P with anomaly matrices in Eq. 4 with algebraic manipulations and factorizing. This solution result in A so that:

( ) 2 1 1 2 1 f T T f X M S M X I D A ′ ′ - = = -
Use of Taylor series expansion and approximating with only two first bigger terms yields:

) ′ ′ ( 2 1 - = 1 f T T f X M S M X I A
This ETM which is a new filter and because of its deterministic characteristic and similarity to the Kalman filter algorithm, is called Deterministic Ensemble Kalman Filter (DEnKF) [START_REF] Sakov | A deterministic formulation of the ensemble Kalman filter: An alternative to ensemble square root filters[END_REF].

Problems Concerning Small Ensembles

Using small ensembles for calculating error covariance matrix can introduce sampling errors as false correlations between distant locations or between points that should be uncorrelated. With a false update corresponding to these false correlations ensemble variance has an inevitable reduction that during the filter time causes underestimation of actual variance significantly [START_REF] Petrie | Localization in the ensemble Kalman Filter[END_REF]). This situation is called inbreeding. It is notable that the stochastic ensemble Kalman filter that needs synthetic noise of observations is more talented for inbreeding than the square root one.

If forecast error covariance is underestimated, The Kalman gain matrix weights a little to observations based on more Uncertainty of them versus state predictions from the model and filter will not be able to correct the forecast values. This status is called filter divergence.

One approach to attenuate negative ramifications of small ensemble size is covariance localization.

In this method, localization is implemented by setting false elements of forecast error covariance matrix to zero which are generated due to small ensemble size. If there is prior information about structure of the covariance matrix, elements out of this structure can be simply set to zero without rebuttal of positive semi-definiteness of the covariance.

A feasible approach is that correlations which are within a physical neighborhood of each point should be taken into account and correlations outside that which are false should be omitted. If selected size of the neighborhood is too large so that all dynamic relations are recorded completely, many false correlations will remain and if it is too small some significant correlations will be lost. Radius of the neighborhood in both localization methods is called length-scale parameter.

Correlation matrix is multiplied to the covariance matrix element-wise. Every zero element of the correlation matrix causes the correspondent element of the product matrix to be zero. So, the problem is to find a semi-positive definite correlation matrix that its elements a distance away from each other are zero3 . To determine elements of such a correlation matrix, correlation or tapering functions are used. One of the most widely used of such functions is compactly supported 5th order piecewise rational function which is approximately coincident with the Gaussian distribution. The function is [START_REF] Gaspari | Construction of correlation functions in two and three dimensions[END_REF]: Where, r is the (positive) distance between points and c is a parameter that controls the length-scale over which the correlations go to zero. The function is used to determine the elements of the correlation vector.

⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ + + + + + + =

Application example

The studied region is Najafabad sub-basin shown in Figure 1. Najafabad is located between 50˚ 57˝ to 51˚ 44' 26˝ E and 32˚ 20'13˝ to 32˚49'21˝ N. The area of the site is approximately 1720.23 km 2 , and the aquifer area is about 1142.67 km 2 . Mean annual evapotranspiration is 1500 mm, and mean annual precipitation is only 158 mm, which occurs mostly in the winter months.

According to the past studies [START_REF] Alijanian | Conjunctive use and management of surface water and groundwater with fuzzy logic[END_REF][START_REF] Bahreini | Surface and groundwater interaction modeling under uncertainty with reliability analysis[END_REF], the aquifer is being recharged through Karvan and Lenjanat aquifers and is a source for Isfahan-Borkhar and North Mahyar aquifers.

Najafabad aquifer (Figure 1) is located in Najafabad sub-basin. Important sources of recharge for the aquifer, in order of importance, are river, irrigation percolation-due to the water taken from groundwater wells and modern irrigation systems-and precipitation. The area is served by three modern irrigation networks: Right Nekoabad, left Nekoabad, and Khamiran. Withdrawals from the aquifer are for agricultural use in almost all cases (Isfahan Regional Water Authority 2007).

Figure 1. Najafabad aquifer divided into five zones for calibration and its wells

The iso-level lines of groundwater within the Najafabad aquifer domain in the first month of the modeling and unit hydrograph of the aquifer for whole modeling period taken from Isfahan regional water authority are shown in Figure 2.

Modeling details

The model is a Mesh-centered finite difference which is implemented in Matlab ©

R2009a. It has cell size of 500 ×500 m, monthly stress periods and time span from October 2000 to September 2007.

Because nearly all uses of water in the region is agricultural, all extractions from or recharge to the aquifer, except precipitation, are intensified to green surfaces. Each green surface relates to an irrigation network and was imported to GIS.

According to the available discharge information, monthly aquifer extractions are considered with three distinct values in each green surface. The recharge due to irrigation losses as a percentage of water incoming to the networks and 0.3 of withdrawal values of each surface was exerted equally in the cells of each of these three surfaces. The percentage of incoming water of the irrigation networks recharging the aquifer are selected based on the previous studies [START_REF] Alijanian | Conjunctive use and management of surface water and groundwater with fuzzy logic[END_REF][START_REF] Bahreini | Surface and groundwater interaction modeling under uncertainty with reliability analysis[END_REF]. Monthly rainfall amounts based on Najafabad station records are distributed equally in all cells of the aquifer. This assumption is acceptable because almost all the lands in the region are almost flat or with gentle slope. modeling studies [START_REF] Alijanian | Conjunctive use and management of surface water and groundwater with fuzzy logic[END_REF][START_REF] Bahreini | Surface and groundwater interaction modeling under uncertainty with reliability analysis[END_REF]. Constant Hydrogeologic coefficients for each zone were considered (Figure 2).

Hydrogeological coefficients of these five zones were derived through calibration with the index of sum of squared differences between monthly values of 32 observation wells data (of total number of 49 observation wells) since October 2000 to March 2005 (54 months). The results are summarized in Table 1 and are consistent with previous studies results [START_REF] Alijanian | Conjunctive use and management of surface water and groundwater with fuzzy logic[END_REF][START_REF] Bahreini | Surface and groundwater interaction modeling under uncertainty with reliability analysis[END_REF]. The aquifer was modeled with these calibrated parameters from April 2005 to September 2007. Modeling results are compared with 17 observation well records and the model showed a good performance. It should be mentioned that because of the assumption that the calibrated model results as true data, the calibration is not effective in evaluation of filter performance.

Filtering details

Simulation results for all grid nodes within the aquifer for 84 months were recorded. The results were treated as actual data in assessment of the filter performance.

Records of 45 randomly selected cells in the simulation were treated as observation data for use in filtering process. Observational error is assumed as a Gaussian stochastic process with zero mathematical expectancy (white noise) and variance of 0.01 m 2 and added artificially to the observed values. This variance value was recommended by the local water authority experts and is based on the precision of measuring instruments for the observation wells. We made the effort to select 45 cells so that they contain all types of discharge and recharge regimes existing in the region. The position of all observation wells and points selected as the observation wells in the filtering procedure, are shown in

Filter performance evaluation

For evaluating performance of the filter that consists of a model with inaccurate parameters with and without localization, the best criteria can be defined as square root of mean squared differences between filter results and true model values. In other words, if X ˆis the vector of filter results and t X is the vector of true model results, then the statistical criteria RMSE can be defined as:

∑ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = n i i t i x x n RMSE 1 2 1
where i

x ˆ is the i th element of vector, X ˆ and t X are the i th element of vector t X and n is the size of both vectors which is equal to the dimension of the system.

Summary of Results

As expected, RMSE values for cases of inaccurate parameters model have lower variance when compared with accurate parameter model values and were generally lower. This point highlights the fact that the deterministic ensemble Kalman filter can make a significant contribution for adjusting the results of any inaccurate model. The RMSE values for all 6 inaccurate models are shown in Figure 3.

Filter Performance without Localization

In the case of inaccurate hydraulic conductivity values (with factors of 2, 5 or 10 times of the accurate ones), the RMSE values for the filter without localization are small and reasonable. The RMSE values for these three non-localized filters have severe reduction in the first few steps and then they show smooth curves and a considerable drop is not observed. Figure 4 shows the results. For some of the ensemble sizes smaller than 100, trends of RMSE values don't form smooth and stable curves within the period. To show the reasonableness of the results and also to satisfy the paper length requirements only the results of the ensemble size of 100 are reported here. When their square root or other forms exist, applying the localization is not directly possible. Therefore, localization often leads to computational difficulties. It is very unlikely that localization can be is easily performed due to the presence of If the length-scale for localization is larger than the optimum value, it cannot set spurious correlations to zero for relatively far distances (in error covariance matrix). In other words, false correlations remain in the range of the length-scale in this case. If the selected length-scale is smaller, the filter is not effective enough to adjust the model due to setting some real dependencies to zero artificially.

Figure 2 :

 2 Figure 2: Iso-level lines of groundwater within Najafabad aquifer domain (October 2000) and unit hydrograph of the aquifer (Oct. 2000-Sep. 2007)

Figure 2 .

 2 Figure 2. Hydraulic conductivity values of the aquifer for the five zones were adjusted by factors of 2, 5 and 10 respectively, in three different simulations. A set of observations from the simulation with the calibrated values of the coefficients are available for comparisons. The results of the simulation with inaccurate parameters assimilated within the ensemble Kalman filter with 100 ensemble members with and without localization

Figure 3 :

 3 Figure 3: RMSE values for inexact models with inexact hydraulic conductivity

  . On the other hand, as mentioned before, localization provides better filter results in all cases. Therefore, localization (with this tapering function) is always recommended.

Figure 4 :

 4 Figure 4: RMSE values for filter a) with 2×K exact b) with 5×K exact c) with 10×K exact

Table 1 :

 1 Calibrated values for hydrogeological coefficients

		Hydraulic conductivity	Specific yield
	Zone number	(m/day)	(%)
	1	0.88	0.25
	2	0.73	0.44
	3	0.61	0.49
	4	1	0.31
	5	1	0.31

semi-positive definiteness of both matrices to be multiplied guaranteed semi-positive definiteness of the product matrix[START_REF] Petrie | Localization in the ensemble Kalman Filter[END_REF].


Heidi Strømskag
email: heidi.stromskag@ntnu.no

A model for instructional design in mathematics implemented in teacher education

Keywords: teachers' and students' practices at university level, novel teaching approaches, epistemological model, adidactical situation, milieu

This paper presents a model for instructional design in mathematics and reports from student teachers' experimentation with it in secondary school. The model is framed within the theory of didactical situations in mathematics. It contributes to the research field through the notion of an epistemological model, which provides an enhanced understanding of the stage of 'conception and a priori analysis' of didactical engineering. Findings show how the model is instrumental in creating student teachers' awareness of the impact of the milieu on the nature of the knowledge developed by the pupils.

INTRODUCTION

The mathematical performances that represent the ambitions of the mathematics classroom and curriculum are constituted through teacher and student participation in activities stimulated by mathematical tasks designed (or selected) by the teacher for the realisation of an instructional purpose [START_REF] Clarke | Mathematical tasks and the student[END_REF]. The centrality of tasks as instruments in mathematics classrooms is reported in the TIMSS 1999 Video Study: in the eighth-grade classrooms that were investigated (in seven countries), at least 80% of lesson time, on average, was spent on solving mathematical tasks [START_REF] Hiebert | Teaching mathematics in seven countries: Results from the TIMSS 1999 video study[END_REF]. In this paper, I present a model for instructional design in mathematics, where tasks are embedded in situations that preserve meaning for the target knowledge. Further, I report from student teachers' utilising the model to teach an optional piece of mathematical knowledge in secondary school. The research question addressed in the paper is: What do student teachers learn from using the model for instructional design?

The model for instructional design is applicable when there is an intention of teaching someone some particular mathematical knowledge. It is rooted in the theory of didactical situations in mathematics, TDS [START_REF] Brousseau | The theory of didactical situations in mathematics: Didactique des mathématiques, 1970-1990[END_REF], the main concepts of which I present briefly in the next section.

A BRIEF INTRODUCTION TO TDS

TDS provides a systemic framework for investigating teaching and learning processes and for supporting didactical design in mathematics, where the particularity of the knowledge taught plays a significant role. In the following I explain concepts of TDS (based on [START_REF] Brousseau | The theory of didactical situations in mathematics: Didactique des mathématiques, 1970-1990[END_REF] that are central to the model for instructional design.

The milieu represents the elements of the material and intellectual reality with which the students interact when solving a problem. An adidactical situation is a situation in which the students as a group try to solve the problem given to them on the basis of features of the milieu, without significant help from the teacher. The milieu of an adidactical situation is called an adidactical milieu. An appropriate adidactical milieu provides feedback to the students, whether their responses are adequate with respect to the knowledge at stake. The teacher has three main roles in the broader didactical situation: first is the devolution of an adidactical situation to the students, which means to present the adidactical situation and the problem to be solved and make the students accept this transfer of ownership; a second role is the regulation, which means to handle the evolution of the adidactical situation and its milieu; and, a third role is the institutionalisation of the knowledge developed in the adidactical situation, which means to transform the contextualised responses produced by the students into scholarly knowledge aimed at by the institution.

The didactical contract refers to the phenomenon that the interaction between the teacher and students in a didactical situation is regulated by rules related to the knowledge at stake. These rules form a set of implicit reciprocal obligations and mutual expectations. In devolution, the teacher (implicitly) negotiates a contract that involves a temporary transfer of responsibility for the knowledge at stake, from the teacher to the students.

After devolution, four situations follow where the role of the teacher and the status of knowledge change: Situations of action, formulation, and validation are (intentionally) adidactical, whereas the situation of institutionalisation is didactical. The adidactical situations are designed with milieus that are supposed to give feedback to the students, as mentioned above. The situation of action is where the students engage in the situation on the basis of its milieu without the teacher's involvement-that is, they construct an implicit solution (a strategy) that guides them in their decisions. The situation of formulation is where the students' formulations are useful in order to indirectly solve the problem-that is, formulation of an explicit solution that enables somebody else to operate on the material milieu. Here, the teacher's role is to make different formulations "visible" in the classroom. The situation of validation is where the students try to explain a phenomenon or verify a conjecture. Here, the teacher's role is to lead a whole-class discussion and trying to make the students use precise mathematical notions. The situation of institutionalisation is where the teacher's role is to connect the contextualised knowledge built by the students with scholarly forms of knowledge.

A MODEL FOR INSTRUCTIONAL DESIGN IN MATHEMATICS

The model for instructional design introduced here is based on the theoretical analysis presented in [START_REF] Strømskag | A methodology for instructional design in mathematics-with the generic and epistemic student at the centre[END_REF]. It contains four phases: epistemological analysis; development of an epistemological model; implementation; and, institutionalisation. They correspond to design and implementation phases of didactical engineering [START_REF] Artigue | Perspectives on design research: The case of didactical engineering[END_REF]: epistemological analysis corresponds to 'preliminary analyses'; development of an epistemological model corresponds to 'conception and a priori analysis'; and implementation followed by institutionalisation corresponds to 'realization'. The model contributes to the research field through the notion of an epistemological model, which provides an enhanced understanding of the phase of 'conception and a priori analysis' of didactical engineering. Its elements are illustrated in Figure 1 (the dotted curve signifies that the epistemological analysis informs the institutionalisation). Figure 1 shows how TDS concepts are constituents in design and implementation of a didactical situation that aims at some particular mathematical knowledge. For an account of its methodology, see [START_REF] Strømskag | A methodology for instructional design in mathematics-with the generic and epistemic student at the centre[END_REF]. The phases of the model are described briefly in the following sections.

Epistemological analysis

Epistemological analysis of the knowledge at stake involves two components: an analysis of the knowledge itself, and a didactical analysis. The aim of analysing the knowledge is to identify possible epistemological obstacles, to find out what this knowledge is for, what questions that motivated its genesis, and how its validity can be justified. A didactical analysis aims at surveying research on teaching and learning of the knowledge at stake. The purpose of the epistemological analysis is to figure out what conditions must be fulfilled for a situation to implement the target knowledge (i.e., what problem would have the target knowledge as its solution), and by this inform the next phase, development of an epistemological model.

An epistemological model

An epistemological model of the knowledge at stake is a construct that consists of three components: first, a model of the target knowledge-possibly an iconic representation; second, a situation that preserves meaning-involving a problem that can be solved in an optimal manner using the target knowledge; and, third, milieus of situations of action, formulation and validation-designed so as to make students' knowledge progress towards gradually more explicit and mathematical forms, based on an image of students' adaptations to the milieus. The second and third component together can be considered a model of the generic and epistemic student's intended learning-it is developed according to the conditions that must be fulfilled for a situation to implement the knowledge it defines. The point is to design milieus so that the responses produced by the students will become gradually more explicit and mathematical, and ultimately can be institutionalised to become the scholarly knowledge aimed at by the teacher.

Implementation

The third phase involves implementation in the classroom, where the epistemological model is the basis for the teacher's devolution and regulation of an adidactical situation (including a problem) aiming at students' interaction with the milieus of situations of action, formulation and validation. Students' learning during implementation is understood as independent adaptation.

Institutionalisation

The fourth phase involves institutionalisation of the solution to the problem into scholarly and decontextualised forms of knowledge. Students' learning in this phase is understood as acculturation, which is intended to enable them to know the place, importance, and future of the mathematical knowledge reached.

The two processes of learning-adaptation and acculturation-are governed by the didactical contract, and the relationship between them is understood as devolution and institutionalisation. The model presented in Figure 1 displays the teacher's roles in the design and realization of a didactical situation: the process of didactical transposition [START_REF] Chevallard | On didactic transposition theory: Some introductory notes[END_REF] transforms the knowledge at stake into an epistemological model; the process of devolution transforms the epistemological model into a problem embedded in a situation; and, the process of institutionalisation transforms the situated knowledge used to solve the problem into scholarly forms of knowledge aimed at by the institution.

AN EPISTEMOLOGICAL MODEL EXEMPLIFIED

I give here a brief presentation of an epistemological model, where the target knowledge is the general numerical statement expressing that the sum of the first n odd numbers is equivalent to the square of n, possibly represented by . I have implemented the epistemological model exemplified here in a teacher education programme, a detailed account of which is given in [START_REF] Strømskag | A methodology for instructional design in mathematics-with the generic and epistemic student at the centre[END_REF].

A model of the theorem

A model of the target knowledge is created using a dissection of a square into Lforms consisting of consecutive odd numbers of unit squares (where 1 is represented by one unit square, hence a degenerated L). A generic example is given in Figure 2, illustrating that . It is made of a dissection of the fourth square into the first four odd numbers. The model of the target knowledge is not to be shown to the students. It is a tool for developing a of the students' intended learning, parts of which I describe below. -Finding a method for building a square of side length a natural number, using Lforms from TILEL. -Explaining why the method will work for any natural number.

The milieu

The milieu of action is the material milieu on which the students are supposed to operate. The material milieu consists of ten paper cut-outs with unit squares that represent the first ten odd numbers, as illustrated in Figure 3. The task in the situation of action is to find a method of building a square of side length a natural number up to ten, using L-forms of different sizes. The features of the milieu (common for action, formulation, and validation) are the following: -The material milieu does provide feedback: it is visible for the students whether or not they succeed in building a square of the intended size, using the L-forms.

-There is an obligation to use L-forms of different sizes. This is part of the didactical contract, and is to ensure that the students engage with odd numbers. -It is only the size of the resulting square that matters. This is part of the didactical contract, and means that different configurations of L-forms should not be distinguished (i.e., it is not a combinatorial task). For an account of the milieus of formulation and validation, and the intended knowledge progress based on the TILEL-situation, see [START_REF] Strømskag | A methodology for instructional design in mathematics-with the generic and epistemic student at the centre[END_REF].

STUDENT TEACHERS' EXPERIMENTS IN SECONDARY SCHOOL

Methodical approach

In 2017, within a master's course in mathematics education, I gave a group of 13 student teachers (hereafter 'students') the task of using the presented model to design and implement (with pupils in secondary school) a situation that preserved meaning for a chosen piece of mathematical knowledge. The students were in the 4 th or 5 th year in a 5-year teacher education programme in mathematics and natural sciences.

The preparation at campus for the experiment (spanning 4 lessons, á 90 minutes) entailed first, my introduction of necessary concepts from TDS and presentation of the model itself. Then the students worked on a task on percentages that I had designed (using the design principles of the model) to let them experience and solve a problem embedded in a TDS situation. The next step was my comparison of the a priori and a posteriori analyses of the TILEL-situation, implemented with 20 students in a teacher education programme (this experiment is reported in [START_REF] Strømskag | A methodology for instructional design in mathematics-with the generic and epistemic student at the centre[END_REF]. This was followed by a brief presentation of two other models of mathematical knowledge: models of convergent geometric series, shown in Figures 4 and5. The students discussed what series they were models of, and what the limit of each of them was.

The last lesson comprised students' work in four groups on an assignment that involved using the model for instructional design to design and implement a teaching situation aiming at some particular mathematical knowledge (suitable for the class in which they were to do the experiment). I was present for guidance during the initial design part, where they created an adidactical situation with a milieu and a problem. Data used to explore the utility of the proposed model consist of students' reports from the experiments, which I received their consent to use (texts from the reports are translated into English by H. S.). Even if the students worked in groups to plan and conduct the experiment, they wrote individual reports. A demand was that the report contained a comparison between a priori and a posteriori of the implemented situation-that is, how the situation was suitable to develop the target knowledge.

Results

A general finding from the 13 reports is that the implemented situations did not have a satisfactory adidactical potential. The students described how they-contrary to the ideal roles of the teacher in TDS situations-had to intervene during the pupils' work. This was explained by too little attention or effort to create an appropriate milieu for the intended knowledge. In the following, I present how three students'through their comparison of a priori and a posteriori analyses-in each case, reflected on limitations of the milieu of the implemented situation, and suggested improvements that might possibly strengthen its adidactical functioning. In the last case, the strength of the designed situation is also accounted for.

The target knowledge for Uma's pupils (upper secondary first year) was a formula for the number of components of the n-th element of a shape pattern, . This knowledge was the solution to a problem embedded in a situation with a barbeque area of variable size with fixed shape, covered by tiles as illustrated by the shape pattern in Figure 6. Uma described how emphasising the structure of the elements of the pattern by, for example, using different colours, might have made it easier for the pupils to implement the knowledge aimed, without the teacher's guidance and help. This idea of Uma can be seen as stimulating algebraic thinking. She also commented on the importance of the devolution for the adidactical functioning of the milieu.

The target knowledge for Tracy's pupils (upper secondary second year) was the theorem expressing that the sum of two consecutive triangular numbers is equivalent to the square number of rank the same as the largest of the triangular numbers. The theorem was symbolised as

, where Tracey presented a model of the theorem, a generic example illustrating that , as shown in Figure 7. The target knowledge was the solution to a problem embedded in a situation with production of quadratic birthday cards, made by two staircase figures (representing triangular numbers) as shown in Figure 8. Tracey explained that the colours of the material milieu were an obstacle to pupils' engagement with the knowledge at stake; the pupils referred to staircase figures by their colours instead of by their structure (sum of natural numbers) and rank. Further, generalisation of the birthday card problem was supposed to be brought into effect by the task: "Formulate a method to make a birthday card of an arbitrary size, and argue why the method works". This part was problematic because the pupils understood the concept arbitrary (in Norwegian: "vilkårlig") in the meaning of any birthday card made by the figures available in the milieu, instead of any birthday card made by imaginary staircase figures of any size (as intended).

The target knowledge for Tina's pupils (grade 8) was an explicit formula for the number of components of the n-th figure of a shape pattern (shown in Figure 9). This knowledge was the solution to a problem at a four-way crossing, where a car in the middle blocked the crossing and caused one car at each of the four ways to be stuck every second. The situation after two seconds is shown in Figure 10 (this was the material milieu along with a pile of paper cut-outs of cars). For small numbers of seconds, Tina expected the pupils to use a recursive approach. Hence, she tried to motivate for an explicit formula by asking for the number of cars after 1 minute. Some pupils, however, used a strategy where they treated the situation as a direct proportionality (representing the four cars added each second) and added 1 in the end for the car in the middle. One pupil reasoned that since there were 21 cars after 5 seconds, he multiplied 20 by 2, then multiplied 40 by 6, and added 1 in the end. This gave the right answer, but it was not what Tina was after. She was after a functional relationship between the seconds that had passed and the total number of cars. Tina explained that in planning it was not anticipated that anyone would use the mentioned method, so the milieu did not have adequate feedback to handle it.

In conclusion, Tina wrote: "The results show that the work in advance of implementation is very important regarding the knowledge actually developed by the pupils […] The results show, however, that feedback from peers has been instrumental in helping pupils to correct misunderstandings in adidactical situations. This I see as a strength of the designed milieu." Another strength reported by Tina is in institutionalisation, the pupils were exposed to a new shape pattern, the relationships of which they were able to represent through algebraic symbols. Tina claimed that this was evidence that the knowledge developed through the experiment had in fact been generalised beyond the designed situation.

DISCUSSION

There were institutional constraints on the study reported here: the time available for instruction at campus was only 4 lessons (a modifiable condition); and the students had no knowledge of the pupils before the experiment (an unmodifiable condition). The fact that the students-contrary to their intention-had to intervene during the pupils' work is not unique to the model for instructional design. What the methodology of the model offers, however, is validation of an implemented teaching situation based on comparison between a priori and a posteriori analyses; this is similar to the methodology of didactical engineering [START_REF] Artigue | Perspectives on design research: The case of didactical engineering[END_REF]. In the model for instructional design, the a priori analysis is constituted by an epistemological analysis and development of an epistemological model, whereas the a posteriori analysis is an analysis of the realisation of the designed situation.

The answer to the research question posed-about what the students learned from using the model for instructional design-is that the students were able to identify relationships between the adidactical milieu and the knowledge progress in each case. The points addressed in the students' reports were about features of the milieu that constituted limitations on generalisation processes: In Uma's case it was about the devolution, and a material milieu that failed to focus on structure and algebraic thinking. In Tracey's case it was about features of the material milieu that suppressed focus on structure, and pupils' not knowing the meaning of the term "arbitrary". In Tina's case it was about a milieu without feedback for a linear generalisation method used by many of the pupils. However, Tina also emphasised a quality of the milieu: how pupils' progress in adidactical situations were enabled by feedback from peers.

Such comparisons between a priori and a posteriori analyses will identify relationships between the nature of the milieu and the knowledge progress in adidactical situations, and thereby create awareness of what enables and what prevents pupils from reaching the target knowledge. In this way, the model has potential to inform future design (aiming at the same knowledge): it might provide deeper insight into the nature of the knowledge at stake, and a better understanding of necessary conditions on the milieu concerning its adidactical functioning.

In conclusion, the impact of the study on my practice as a teacher educator is that the students' experiments are used as objects of study (with the same or other students), where the task is to revise the experimented situations and their milieu to better match them with their original didactic intention, and subsequently, to implement them with pupils to validate them. In this way, the experimented situations are considered material milieus for instructional design. This represents a meta-level perspective on instructional design in mathematics teacher education.

Figure 1 .

 1 Figure 1. A model for instructional design (reproduced from Strømskag, 2017, p. 912)

Figure 2 .

 2 Figure 2. A model of the target knowledgeA situation that preserves meaning of the target knowledge I invented a situation based on an imaginary company called TILEL selling a special kind of tile formations that can be used to cover squares. The tile formations have shapes as L-forms, and consist of an odd number of unit squares. The idea of this situation is derived from the model of the target knowledge, shown in Figure2. The task has two main parts: -Finding a method for building a square of side length a natural number, using Lforms from TILEL. -Explaining why the method will work for any natural number.

Figure 3 .

 3 Figure 3. The material milieu for the intended theorem (paper cut-outs)

  Figure 4.

Figure 6 .

 6 Figure 6. The material milieu for Uma's barbeque area problem (iconic representations)

Figure 7 .

 7 Figure 7. Tracey's model of the target knowledge

Figure 9 .

 9 Figure 9. Tina's model of the target knowledge


From Nonlinear Oscillations to Chaos Theory

In this work we propose to reconstruct the historical road leading from nonlinear oscillations to chaos theory by analyzing the research performed on the following three devices: the series-dynamo machine, the singing arc and the triode, over a period ranging from the end of the XIX th century till the end of the Second World War.

Thus, it will be shown that the series-dynamo machine, i.e. an electromechanical device designed in 1880 for experiments, enabled to highlight the existence of sustained oscillations caused by the presence in the circuit of a component analogous to a "negative resistance".

The singing arc, i.e. a spark-gap transmitter used in Wireless Telegraphy to produce oscillations and so to send messages, allowed to prove that, contrary to what has been stated by the historiography till recently, Poincaré made application of his mathematical concept of limit cycle in order to state the existence of sustained oscillations representing a stable regime of sustained waves necessary for radio communication.

During the First World War, the singing arc was progressively replaced by the triode and in 1919, an analogy between series-dynamo machine, singing arc and triode was highlighted. Then, in the following decade, many scientists such as André Blondel, Jean-Baptiste Pomey, Élie and Henri Cartan, Balthasar Van der Pol and Alfred Liénard provided fundamental results concerning these three devices. However, the study of these research has shown that if they made use of Poincaré's methods, they did not make any connection with his works.

In the beginning of the twenties, Van der Pol started to study the oscillations of two coupled triodes and then, the forced oscillations of a triode. This led him to highlight some oscillatory phenomena which have never been observed previously. It will be then recalled that this new kind of behavior considered as "bizarre" at the end of the Second World War by Mary Cartwright and John Littlewood was later identified as "chaotic". 1

Introduction

The aim of this work is to trace the history of the foundations of Chaos theory through the analysis of the works performed on the following three devices: the series-dynamo machine, the singing arc and the triode, over a period ranging from the end of the XIX th century till the end of the Second World War.

In 1880, by sending the current produced by a dynamoelectrical into a magnetoelectrical machine forming thus a series-dynamo machine, the French engineer Jean-Marie Gérard Anatole Lescuyer highlighted a nonlinear phenomenon that will be later considered by Paul Janet as sustained oscillations and by Balthasar Van der Pol as relaxation oscillations 1 . If the cause of this phenomenon was rapidly identified as being the presence in the circuit of a component analogous to a "negative resistance", its mathematical modeling was out of reach at that time.

A quarter of a century later, at the time of the emergence of Wireless Telegraphy, it became of tremendous need to find the condition for which the oscillations produced by a spark-gap transmitter called singing arc were sustained. Actually, this condition representing a stable regime of sustained waves necessary for radio communication was established by Henri Poincaré in 1908 during a series of "forgotten lectures" he gave at the École Supérieure des Postes et Télécommunications (today Telecom ParisTech). Contrary to what was stated by the historiography till recently, Poincaré made thus the first correspondence between the existence of sustained oscillations and the concept of limit cycle that he had introduced in his second memoir "On the curves defined by differential equations". In other words, he proved that the periodic solution of the nonlinear ordinary differential equation characterizing the oscillations of the singing arc corresponds in the phase plane to an attractive closed curve, i.e. a stable limit cycle.

During the First World War, the singing arc was progressively replaced by the triode which was also able to sustain oscillations but even more importantly to amplify the electric signal.

In 1919, the French engineer Paul Janet established an analogy between the series-dynamo machine, the singing arc and the triode and stated thus that their sustained oscillations belong to the same nonlinear phenomenon. Then, in the following decade, many scientists such as André Blondel, Jean-Baptiste Pomey, Élie and Henri Cartan, Balthasar Van der Pol and Alfred Liénard provided fundamental results concerning these three devices. However, it appears that if they made use of Poincaré's methods, they did not make any connection with his works.

In the beginning of the twenties, Van der Pol started to study the oscillations of two coupled triodes and then, the forced oscillations of a triode. This led him to highlight new oscillatory phenomena that he called oscillation hysteresis, automatic synchronization and frequency demultiplication. Nevertheless, in this case, if the oscillations are still sustained, the solution is no more periodic but exhibits a new kind of behavior that will be called "bizarre" at the end of the Second World War by Mary Cartwright and John Littlewood and that will be later identified as "chaotic".

The series-dynamo machine: the expression of nonlinearity

At the end of the nineteenth century, magneto-or dynamo-electric machines were used in order to turn mechanical work into electrical work and vice versa. With the former type of machine, the magnetic field is induced by a permanent magnet, whereas the latter uses an electromagnet. These machines produced either alternating or direct current indifferently. Thus, in 1880, a French engineer named Jean-Marie-Anatole Gérard-Lescuyer made an experiment by associating a dynamo-electric machine used as a generator with a magneto-electric machine, which in this case can be considered as the motor (Fig. 1).

Figure 1: The Gérard-Lescuyer's paradoxical experiment [START_REF] Ph | Non-linear theory of maintenance of oscillations[END_REF].

Gérard-Lescuyer [START_REF] Gérard-Lescuyer | Sur un paradoxe électrodynamique[END_REF][START_REF] Gérard-Lescuyer | On an electrodynamical paradox[END_REF] reports on the found effects in a note published in the Comptes rendus de l'Académie des Sciences de Paris and in the Philosophical Magazine in the following way: "As soon as the circuit is closed the magnetoelectrical machine begins to move; it tends to take a regulated velocity in accordance with the intensity of the current by which it is excited; but suddenly it slackens its speed, stops, and start again in the opposite direction, to stop again and rotate in the same direction as before. In a word, it receives a regular reciprocating motion which lasts as long as the current that produces it."

While observing the periodical reversal of the magneto-electric machine's circular motion, despite the direct current, he wondered about the causes of this oscillatory phenomenon that he was unfortunately unable to isolate. Gérard-Lescuyer [START_REF] Gérard-Lescuyer | Sur un paradoxe électrodynamique[END_REF][START_REF] Gérard-Lescuyer | On an electrodynamical paradox[END_REF] wrote in his conclusion:

"What are we to conclude from this? Nothing, except that we are confronted by a scientific paradox, the explanation of which will come, but which does not cease to be interesting."

It was actually proven by the count Théodose du Moncel [START_REF] Th | Réactions réciproques des machines dynamoélectriques et magnéto-électriques[END_REF] a few weeks later, then by Aimé Witz [START_REF] Witz | Des inversions de polarités dans les machines série-dynamos[END_REF][START_REF] Witz | Recherches sur les inversions de polarité des série-dynamos[END_REF], and by Paul Janet [START_REF] Janet | Sur les oscillations électriques de période moyenne[END_REF], that the gap situated between the brushes of the dynamo is the source of an electromotive force (e.m.f.), i.e. a potential difference at its terminals symbolized by a nonlinear function of the intensity that flows through there. However, the mathematical modeling of this e.m.f. was out of reach at that time. Therefore the essence of Gérard-Lescuyer's paradox is the presence of an e.m.f, which has a nonlinear current-voltage characteristic acting as a a negative resistance and leading to sustained oscillations.

Half a century later, the famous Dutch physicist Balthasar Van der Pol [START_REF] Van Der Pol | Über Relaxationsschwingungen II[END_REF] noted: "Relaxation oscillations produced by a motor powered by a D.C. series-dynamo. The fact that such a system is able to produce relaxation oscillations was already briefly discussed. In an article written by Mr. Janet (we find a reference to Gérard Lescuyer (CR 91, 226, 1880) where this phenomenon had already been described."

3 The singing arc:

Poincaré's forgotten lectures

At the end of the nineteenth century a forerunner to the incandescent light bulb called electric arc was used for lighthouses and street lights. Regardless of its weak glow it had a major drawback: the noise generated by the electrical discharge which inconvenienced the population. In London, physicist William Du Bois Duddell (1872-1917) was commissioned in 1899 by the British authorities to solve this problem. He thought up the association of an oscillating circuit made with an inductor L and a capacitor C (F on Fig. 2) with the electrical arc to stop the noise (see Fig. 2). Duddell [START_REF] Bois Duddell | On Rapid Variations in the Current through the Direct-Current Arc[END_REF][START_REF] Bois Duddell | On Rapid Variations in the Current through the Direct-Current Arc[END_REF] created a device that he named singing arc. Duddell had actually created an oscillating circuit capable of producing not only sounds (hence its name) but especially electromagnetic waves. This device would therefore be used as an emitter for wireless telegraphy until the triode replaced it. The singing arc or Duddell's arc was indeed a "spark gap" device meaning that it produced sparks which generated the propagation of electromagnetic waves shown by Hertz's experiments as pointed out by Poincaré [30, p. 79]:

"If an electric arc is powered by direct current and if we put a self-inductor and a capacitor in a parallel circuit, the result is comparable to Hertz's oscillator. . . These oscillations are sustained exactly like those of the pendulum of a clock. We have genuinely an electrical escapement."

On July 4 th 1902, Henri Poincaré became Professor of Theoretical Electricity at the École Supérieure des Postes et Télégraphes (Telecom Paris-Tech) in Paris where he taught until 1910. The director of this school, Édouard Éstaunié , then 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: "From the first words it became apparent that we were going to attend the research work of this extraordinary and awesome mathematician. . . Each obstacle encountered, a short break marked embarrassment, then a blow of shoulder, Poincaré seemed to defy the annoying function."

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. More precisely, he demonstrated the existence, in the phase plane, of a stable limit cycle.

To this aim Poincaré [START_REF] Poincaré | Sur la télégraphie sans fil[END_REF] studied Duddell's circuit that he represented by the following diagram (Fig. 3) consisting of an electromotive 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. Then, he called x the capacitor charge, x ′ the current intensity in the branch including the capacitor, ρx ′ the term corresponding to the internal resistance of the self and various damping and θ (x ′ ) the term representing the e.m.f. of the arc the mathematical modeling of which was also out of reach for Poincaré at that time. Nevertheless, Poincaré was able to establish the singing arc equation, i.e. the second order nonlinear differential equation (1) for the sustained oscillations in the singing arc:

Lx ′′ + ρx ′ + θ ( x ′ ) + Hx = 0 (1)
Then, by using the qualitative theory of differential equations that he developed in his famous memoirs [START_REF] Poincaré | Sur les courbes définies par une équation différentielle[END_REF][START_REF] Poincaré | Sur les courbes définies par une équation différentielle[END_REF], he stated that: "One can construct curves satisfying this differential equation, provided that function θ is known. Sustained oscillations correspond to closed curves, if there exist any. But every closed curve is not appropriate, it must fulfill certain conditions of stability that we will investigate." Thus, he plotted a representation of the solution of equation ( 1): 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. This representation led him to state the following stability condition: "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 [34, p. 261] has introduced it in his own famous memoir "On the curves defined by differential equations" and as Poincaré [35, p. 25] has later defined it in the notice on his own scientific works [START_REF] Poincaré | Notice sur les Travaux Scientifiques de Henri Poincaré[END_REF]. 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. So, in the next 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 the following inequality.

∫ θ ( x ′ ) x ′ dt < 0 (2)
It has been proved by Ginoux [12,[START_REF] Ginoux | Analyse mathématiques des phénomènes oscillatoires non linéaires[END_REF][START_REF] Ginoux | Self-excited oscillations: from Poincar to Andronov[END_REF][START_REF] Ginoux | Histoire de la théorie des oscillations non linéaires[END_REF][START_REF] Ginoux | History of nonlinear oscillations theory, Archimede[END_REF] that this stability condition (2) flows from a fundamental result introduced by Poincaré in the chapter titled "Exposants caractristiques" ("Characteristics exponents") of his "New Methods of Celestial Mechanics" [38, Vol. I, p. 180].

Until recently the historiography considered that Poincaré did not make any connection between sustained oscillations and the concept of limit cycle he had introduced and credited the Russian mathematician Aleksandr' Andronov [START_REF] Andronov | Predel nye cikly Puankare i teori kolebani i[END_REF][START_REF] Andronov | Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues[END_REF] for having been the "first" to establish this correspondence between periodic solution and limit cycle.

Concerning the singing arc, Van der Pol [START_REF] Van Der Pol | Oscillations sinusoïdales et de relaxation[END_REF] also noted in the beginning of the thirties: "In the electric field we have some very nice examples of relaxation oscillations, some are very old, such as spark discharge of a plate machine, the oscillation of the electric arc studied by Mr. Blondel in a famous memoir [START_REF] Andronov | Predel nye cikly Puankare i teori kolebani i[END_REF] The original of this valve called "Abraham lamp" is still in the Arts et Métiers museum to this day (Fig. 5). It has a cylindrical structure, which appears to have been designed by Abraham. In November 1917, Abraham consequently invented with his colleague Eugene Bloch (1878-1944) a device able to measure wireless telegraphy emitter frequencies: the so-called multivibrator (see Ginoux [START_REF] Ginoux | Analyse mathématiques des phénomènes oscillatoires non linéaires[END_REF][START_REF] Ginoux | Self-excited oscillations: from Poincar to Andronov[END_REF][START_REF] Ginoux | Histoire de la théorie des oscillations non linéaires[END_REF][START_REF] Ginoux | History of nonlinear oscillations theory, Archimede[END_REF]).

Wireless telegraphy development, spurred by war effort, went from craft to full industrialization. The triode valves were then marketed on a larger scale. More reliable and stable than the singing arc, the consistency of the various components used in the triode allowed for exact reproduction of experiments, which facilitated research on sustained oscillations.

Janet's analogy

In April 1919, the French scientist Paul Janet (1863-1937) published an article entitled "Sur une analogie électrotechnique des oscillations entretenues" [START_REF] Janet | Sur une analogie électrotechnique des oscillations entretenues[END_REF] which was of considerable importance on several levels. Firstly, it underscored the technology transfer taking place, consisting in replacing an electromechanical component (singing arc) with what would later be called an electronic tube. This represented a true revolution since the singing arc, because of its structure it made experiments complex and tricky, making it almost impossible to recreate. Secondly, it revealed "technological analogy" between sustained oscillations produced by a series dynamo machine like the one used by Gérard-Lescuyer [START_REF] Gérard-Lescuyer | Sur un paradoxe électrodynamique[END_REF][START_REF] Gérard-Lescuyer | On an electrodynamical paradox[END_REF] and the oscillations of the singing arc or a three-electrode valve (triode). Janet [24, p. 764] wrote: "It seemed to me interesting to mention the unexpected analogies of this experiment with the sustained oscillations so widely used to-day in wireless telegraphy, for example, those produced in Duddell's arc or in the lamp with three-electrodes lamps used as oscillators. . . Producing and sustaining oscillations in these systems mostly depends on the presence, in the oscillating circuit, of something comparable to a negative resistance. The dynamo-series acts as a negative resistance, and the engine with separated excitation acts as a capacity." Thus, Janet considered that in order to have analogies in the effects, i.e. in order to see the same type of oscillations in the series-dynamo machine, the triode and the singing arc, there must be an analogy in the causes. Therefore, since the series-dynamo machine acts as a negative resistance, responsible for the oscillations, there is indeed an analogy. Consequently, only one equation must correspond to these devices. In this article, Janet provided the nonlinear differential equation characterizing the oscillations noted during Gérard-Lescuyer's experiment:

L d 2 i dt 2 + [ R -f ′ (i) ] di dt + k 2 K i = 0 (3)
where R corresponds to the resistance of the series dynamo machine, L is the self-induction of the circuit and K/k 2 is analogous to a capacitor and f (i) is the electromotive force of the series-dynamo machine. However, as recalled by Janet [24, p. 765], its mathematical modeling was also out of reach at that time.

"But the phenomenon is limited by the characteristic's curvature, and regular, non-sinusoidal equations actually occur. They are governed by the equation ( 3), which could only be integrated if we knew the explicit for of the function f (i)."

By replacing in Eq. ( 3) i with x, R with ρ, f ′ (i) with θ(x), and k 2 /K with H, one find again Poincaré's singing arc equation (2). Thus, both ordinary differential equations are analogous but are not of the same order. Nevertheless, it appeared that Janet did make no connection with Poincaré's works.

Blondel's triode equation

According to the historiography, it is common knowledge the Dutch physicist Balathasar Van der Pol is credited for having stated the differential equation of the triode in his famous publication entitled "On relaxation oscillations" published in 1926 [START_REF] Van Der Pol | On "relaxation-oscillations[END_REF]. However, it was proved by Ginoux [START_REF] Ginoux | Analyse mathématiques des phénomènes oscillatoires non linéaires[END_REF][START_REF] Ginoux | Self-excited oscillations: from Poincar to Andronov[END_REF][START_REF] Ginoux | Blondel et les oscillations auto-entretenues[END_REF] on the one hand that the triode equation was actually stated by Van der Pol in 1920 in a publication entitled: "A theory of the amplitude of free and forced triode vibrations," [START_REF] Van Der Pol | A theory of the amplitude of free and forced triode vibrations[END_REF] and on the other that the French engineer André Blondel sated the triode equation one year before him.

As previously pointed out, the main problem of these three devices was the mathematical modeling of their oscillation characteristics, i.e., the e.m.f. of the series-dynamo machine, of the singing arc and of the triode.

Thus, in a note published in the Comptes Rendus of the Académie des Sciences on the 17 th of November 1919, Blondel proposed to model the oscillation characteristic of the triode as follows [START_REF] Blondel | Amplitude du courant oscillant produit par les audions générateurs[END_REF]:

i = b 1 (u + kν) -b 3 (u + kν) 3 -b 5 (u + kν) 5 ... (4)
Then, substituting i by its expression in the triode equation, neglecting the internal resistors and integrating once with respect to time, he obtained

C d 2 u dt 2 - ( b 1 h -3b 3 h 3 u 2 -... ) du dt + u L = 0 (5)
Let's notice that this equation is perfectly equivalent to those obtained by Poincaré and Janet. Nevertheless, if Blondel solved the problem of the mathematical modeling of the oscillation characteristic of the triode he did make no connection with Poincaré's works despite of the fact that he knew him personally.

Pomey's contribution

Less than on year later, the French engineer Jean-Baptiste Pomey (1861-1943) proposed a mathematical modeling of the e.m.f. of the singing arc in his entitled: " Introduction à la théorie des courants téléphoniques et de la radiotélégraphie " and published on June 28 th 1920 (this detail would be of great importance in the following). Pomey [39, p. 375] wrote: "For the oscillations to be sustained it is not enough to have a periodic motion, it is necessary to have a stable motion." Then, he proposed the following "law" for the e.m.f. of the singing arc:

E = E 0 + ai -bi 3 (6)
and posing i = x ′ (like Poincaré) he provided the nonlinear differential equation of the singing arc:

Lx ′′ + Rx ′ + 1 C x = E 0 + ax ′ -bx ′3 (7)
By posing H = 1/C, ρ = R and θ (x ′ ) = -E 0 -ax ′ + bx ′3 it is obvious that Eq. ( 1) and Eq. ( 7) are completely identical 2 . 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.

At the same time, Van der Pol [START_REF] Van Der Pol | A theory of the amplitude of free and forced triode vibrations[END_REF] proposed the following mathematical modeling of the oscillation characteristic of the triode in an article published on July 17, 1920:

i = ψ (kv) = αv + βv 2 + γv 3 (8)
Van der Pol [40, p. 704] precised that, by symmetry consideration, one can choose β = 0 and provided the triode equation:

C d 2 v dt 2 - ( α -3γv 2 ) dv dt + 1 L v = 0 (9)
Taking into account that β can be chosen as equal to zero, one finds no difference between the Eq. ( 6) and the Eq. ( 8). Nevertheless, nothing proves that Van der Pol had read Pomey's book.

Five years later, 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 (3) are sustained. Within ten days, Élie Cartan and his son Henri sent an article entitled: "Note sur la génération des oscillations entretenues" [START_REF] Cartan | Note sur la génération des oscillations entretenues[END_REF] in which they proved the existence of a periodic solution for Janet's equation [START_REF] Blondel | Amplitude du courant oscillant produit par les audions générateurs[END_REF]. In fact, their proof was based on a diagram which corresponds exactly to a "first return map" diagram introduced by Poincaré in his memoir "Sur les Courbes définies par une équation différentielle" [34, p. 251].

Van der Pol's relaxation oscillations

Van der Pol's most famous publication is probably that entitled "On relaxation oscillations" [START_REF] Van Der Pol | On "relaxation-oscillations[END_REF]. However, what is least well-known is that he published four different versions of this paper in 1926 in the following order:

1. Over Relaxatietrillingen [START_REF] Van Der Pol | Over Relaxatietrillingen[END_REF] (in Dutch); 2. Over Relaxatie-trillingen [START_REF] Van Der Pol | Relaxatie-trillingen[END_REF] (in Dutch); 3. Über Relaxationsschwingungen [START_REF] Van Der Pol | Über ,Relaxationsschwingungen[END_REF] (in German); 4. On relaxation-oscillations [START_REF] Van Der Pol | On "relaxation-oscillations[END_REF] (in English).

In these four articles, Van der Pol presents the following generic dimensionless nonlinear differential equation for relaxation oscillations which is neither attached to the triode, nor to any other device (series-dynamo machine or singing arc):

v -ε(1 -v 2 ) v + v = 0. ( 10
)
Early on, Van der Pol [40, p. 179] realized that the equation ( 10) was not analytically integrable: "It has been found to be impossible to obtain an approximate analytical solution for [START_REF] Bois Duddell | On Rapid Variations in the Current through the Direct-Current Arc[END_REF] with the supplementing condition (ε ≪ 1), but a graphical solution may be found in the following way." So, he used the isoclynes method to graphically integrate the nonlinear differential equation [START_REF] Bois Duddell | On Rapid Variations in the Current through the Direct-Current Arc[END_REF] for the relaxation oscillations. Moreover, let's notice that he didn't make any reference to Poincaré's works but to Andronov's article [START_REF] Andronov | Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues[END_REF].

Liénard's riddle

On May 1928, the French engineer Alfred Liénard (1869Liénard ( -1958) ) published an article entitled " Étude des oscillations entretenues" in which he studied the solution of the following nonlinear differential equation:

d 2 x dt 2 + ωf (x) dx dt + ω 2 x = 0 (11)
Such an equation is a generalization of the well-known Van der Pol's equation and of course of Janet's equation ( 4). 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. [START_REF] Bois Duddell | On Rapid Variations in the Current through the Direct-Current Arc[END_REF]. Then, Liénard [26, p. 906] plotted this solution (Fig. 7) and wrote: "All integral curves, interior or exterior, traveled in the direction of increasing time, tend asymptotically to the curve D, we say that the corresponding periodic motion is a stable motion." Then, Liénard [26, p. 906] explained that the condition for which the "periodic motion" is stable is given by the following inequality:

∫ Γ F (x) dy > 0 ( 12
)
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 (2) and ( 12) are completely identical 3 and represents an analogue of what is now called "orbital stability". Again, one Figure 7: Closed curve solution of Eq. ( 10), Liénard [START_REF] Liénard | Étude des oscillations entretenues[END_REF]. can find no reference to Poincaré's works 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. All these facts constitutes the Liénard's riddle.

Andronov's note at the Comptes Rendus

On Monday 14 October 1929, the French mathematician Jacques Hadamard (1865-1963) presented to the Académie des Sciences de Paris a note which was sent to him by Aleksandr Andronov and entitled "Poincaré's limit cycles and the theory of self-sustained oscillation". In this work, Andronov [START_REF] Andronov | Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues[END_REF] proposed to transform the second order nonlinear differential equation modeling the sustained oscillations by the series-dynamo machine, the singing arc or the triode into the following set of two first order differential equations:

dx dt = P (x, y) ; dy dt = Q (x, y) (13)
Then, he explained that the periodic solution of this system ( 13) is expressed in terms of Poincaré's limit cycles: "This results in self-oscillations which emerge in the systems characterized by the equation of type [START_REF] Ginoux | Analyse mathématiques des phénomènes oscillatoires non linéaires[END_REF] corresponding mathematically to Poincaré's stable limit cycles."

It is important to notice that due to the imposed format of the Comptes Rendus (limited to four pages), Andronov did not provide any demonstration. He just claimed that the periodic solution of a non-linear second order differential equation defined by ( 13) "corresponds" to Poincaré's stable limit cycles. Then, Andronov provided a stability condition for the stability of the limit cycle:

∫ 2π 0 [f x (R cos ξ, -R sin ξ; 0) cos ξ + g y (R cos ξ, -R sin ξ; 0) sin ξ] dξ < 0 (14)
In fact, this condition is based on the use of characteristic exponents introduced by Poincaré in his so-called New Methods on Celestial Mechanics [38, Vol. I, p. 161] and after by Lyapounov in his famous textbook General Problem of Stability of the Motion [START_REF] Lyapounov | Problème général de la stabilité du mouvement, Annales de la faculté des sciences de Toulouse[END_REF]. That's the reason why Andronov will call later the stability condition [START_REF] Ginoux | The First "Lost" International Conference on Nonlinear Oscillations[END_REF]: stability in the sense of Lyapounov or Lyapounov stability. It has been stated by Ginoux [12,[START_REF] Ginoux | Analyse mathématiques des phénomènes oscillatoires non linéaires[END_REF][START_REF] Ginoux | Self-excited oscillations: from Poincar to Andronov[END_REF][START_REF] Ginoux | Histoire de la théorie des oscillations non linéaires[END_REF][START_REF] Ginoux | History of nonlinear oscillations theory, Archimede[END_REF] that both stability condition of Poincaré (2) and of Andronov ( 14) are totally identical. Thus by comparing Andronov's previous sentence with that of Poincaré (see above), it clearly appears that Andronov has stated the same correspondence as Poincaré twenty years after him. Nevertheless, it seems that Andronov may not have read Poincaré's article since at that time even if the first volume of his complete works had been already published it didn't contained Poincaré's lectures on Wireless Telegraphy.

The first "lost" International Conference on Nonlinear Oscillations

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. This event, of which virtually no trace remains, was reported in an article written in Russian by Papaleksi at his return in USSR. This document, recently rediscovered by Ginoux [START_REF] Ginoux | The First "Lost" International Conference on Nonlinear Oscillations[END_REF], 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 oscillations4 . According to Papaleksi [29, 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.

The triode: from limit cycle to "bizarre" solutions

At the end of the First World War, the development of wireless telegraphy led the engineers and scientists to turn to the study of self-sustained oscillations in a three-electrode lamp subjected to a periodic "forcing" or a "coupling'". According to Mrs. Mary Lucy Cartwright [START_REF] Cartwright | Non-Linear Vibrations: A Chapter in Mathematical History[END_REF]:

"The non-linearity [in the Van der Pol equation] may be said to control the amplitude in the sense that it allows it to increase when it is small but prevents it becoming too large. The general solution cannot be obtained by the combination of two linearly independent solutions and similar difficulties arise when we add a forcing term to this equation. This was brought out very clearly by the work of van der Pol and Appleton, partly in collaboration, and partly independently, in a series of papers on radio oscillations published between 1920 and 1927. To me the work of the radio engineers is much more interesting and suggestive than that of the mechanical engineers. The radio engineers want their systems to oscillate, and to oscillate in a very orderly way, and therefore they want to know not only whether the system has a periodic solution, but whether it is stable, what its period and amplitude and harmonic content are, and how these vary with the parameters of the equation, and they sometimes want the period to be determined with a very small error. In the early days they wanted to explain why the amplitude was limited in a certain way and why in some cases the period lengthened as the harmonic content increased and not in others. The desire to know why and the insistence on how the various quantities such as amplitude and frequency vary with the parameters of the equation over fairly wide ranges meant that numerical and graphical solutions either failed to provide the answer or were far too cumbersome. Further, unless one knows something about the general behavior of the solutions, the numerical work, which is only approximate, may be misleading."

Thus, in the beginning of the 1920s, Van der Pol [START_REF] Van Der Pol | A theory of the amplitude of free and forced triode vibrations[END_REF] studied the oscillations of a forced triode, i.e. a triode powered by a voltage generator with an f.e.m. of type v (t) = E s in (ω 1 t) the equation of which reads then:

v -α ( 1 -v 2 ) v + ω 2 0 v = ω 2 1 E s in (nω 1 t) with ε = α ω 0 ≪ 1 (15)
Four years later, while using the method of "slowly-varying amplitude" that he had developed, Van der Pol [START_REF] Van Der Pol | Gedwongen trillingen in een systeem met nietlineairen weerstand (Ontvangst met teruggekoppelde triode)[END_REF] was thus able on the one hand to obtain more directly the various approximations of the amplitude of this forced system, and on the other hand, to construct a solution to the equation more easily than by using the classical Poincar-Lindstedt or Fourier methods 5 . In this paper, Van der Pol [START_REF] Van Der Pol | Forced oscillations in a circuit with non-linear resistance (reception with reactive triode)[END_REF] highlights the fact that when the difference in frequency of the two signals is inferior to this value an automatic synchronization phenomenon occurs and the two circuits oscillate with the same frequency. This led him to evidence the phenomenon of frequency entrainment, which he defined thus: "Hence the free frequency undergoes a correction in the direction of the forced frequency, giving the impression as if the free frequency were being attracted by the forced frequency."

In 1927, Van der Pol and his colleague Jan Van der Mark [START_REF] Van Der Pol | van der Mark Frequency Demultiplication[END_REF] published an article titled "Frequency Demultiplication," in which they again studied the forced oscillations of a triode, but in the field of relaxation oscillations. Then, they explained that the automatic synchronization phenomenon, observed in the case of the forced oscillations of a triode, can also occur for a range of the parameter corresponding to the relaxation oscillations, i.e. for ε ≫ 1, but in a much wider frequency field. They also reported that the resonance phenomenon is almost non-existent in forced relaxation oscillations, and that consequently, the sinusoidal e.m.f. inducing the forcing influences the period (or frequency) of the oscillations more than it does their amplitude, and added: "It is found that the system is only capable of oscillating with discrete frequencies, these being determined by whole sub-multiples of the applied frequency."

In their article, Van der Pol and Van der Mark [START_REF] Van Der Pol | van der Mark Frequency Demultiplication[END_REF] proposed, in order to evidence the frequency demultiplication phenomenon, the following construction (see Fig. 8) on which we can see a "jump" of the period for each increase in the value of the capacitor's capacitance. In order to evidence this frequency demultiplication phenomenon, Van der Pol and Van der Mark used a phone. They then described the phenomenon what they heard in the receiver: "Often an irregular noise is heard in the telephone receivers before the frequency jumps to the next lower value. However, this is a subsidiary phenomenon, the main effect being the regular frequency multiplication."

This irregular noise they heard was actually the sound manifestation of the transition which was taking place. Indeed, as the frequency varied, the solution to the differential equation [START_REF] Ginoux | Van der Pol and the history of relaxation oscillations: Toward the emergence of a concepts[END_REF], which had been until now represented by a limit cycle, i.e. by a periodic attractor, would draw a "strange attractor" transcribing the chaotic behavior of the solution. Van der Pol seemed to have reached the limits of deterministic physics with how far he went in the exploration of nonlinear and non-autonomous systems. He "flirted", as Mary Lucy Cartwright and John Edensor Littlewood [START_REF] Cartwright | On non-linear differential equations of the second order[END_REF][START_REF] Cartwright | On nonlinear differential equations of the second order, II: The Equation ÿ -kf (y) ẏ + g (y, k) = p (t) = p 1 (t) + kp 2 (t); k > 0, f (y) ≥ 1[END_REF]7,[START_REF] Cartwright | Addendum[END_REF] did twenty years later with the first signs of chaos, when they called "bizarre" the behavior of the solution to the differential equation [START_REF] Ginoux | Van der Pol and the history of relaxation oscillations: Toward the emergence of a concepts[END_REF] for specific values of the parameters. Indeed, according to Guckenheimer et al. [START_REF] Guckenheimer | The Forced van der Pol Equation I: The Slow Flow and Its Bifurcations[END_REF]: "Van der Pol's work on nonlinear oscillations and circuit theory provided motivation for the seminal work of Cartwright and Littlewood. In 1938, just prior to World War II, the British Radio Research Board issued a request for mathematicians to consider the differential equations that arise in radio engineering. Responding to this request, Cartwright and Littlewood began studying the forced van der Pol equation and showed that it does indeed have bistable parameter regimes. In addition, they showed that there does not exist a smooth boundary between the basins of attraction of the stable periodic orbits. They discovered what is now called chaotic dynamics by detailed investigation of this system."

Conclusion

Thus, the analysis of the research performed on the following three devices: the series-dynamo machine, the singing arc and the triode, over a period ranging from the end of the XIX th century till the end of the Second World War, has enabled to reconstruct the historical road leading from nonlinear oscillations to chaos theory. The series-dynamo machine has highlighted a new kind of oscillations generated by the presence of a nonlinear component in the circuit, i.e. a negative resistance. Poincaré's work on the singing arc has provided an analytical condition for the sustaining of these oscillations, i.e. for the existence of a stable limit cycle. Moreover, this has proved that Poincaré has established twenty years before Andronov the correspondence between periodic solution and stable limit cycle. In his research on the triode, Blondel has solved the question of the mathematical modeling of its oscillation characteristic, i.e. of its negative resistance and stated thus, one year before Van der Pol, the triode's equation. Then, Janet highlighted an analogy between the oscillations sustained by the series-dynamo machine, the singing arc and the triode and Van der Pol deduced that they were belonging to the same oscillatory phenomenon that he called relaxation oscillations. Though he plotted the solution of the equation that now bears his name, he didn't recognize that it was obviously a Poincaré's limit cycle. Thereafter, Cartan and then Liénard proved the existence and uniqueness of this periodic solution but did not make either a connection with Poincaré's works. Immediately after Andronov established this connection, Van der Pol and Papaleksi organized the first International Conference on Nonlinear Oscillations in Paris. Nevertheless, this meeting did not lead to any development or research in this field. At the same time, Van der Pol and Van der Mark highlighted that the forced triode was the source of a strange phenomenon that they called frequency demultiplication. At the end of the Second World War, Cartwright and Littlewood investigated this system and considered its oscillations as "bizarre". Many years later, it appeared that they had actually observed the first chaotic behavior.

Figure 2 :

 2 Figure 2: Diagram of the singing arc's circuit, from Duddell [10, 11].

Figure 3 :

 3 Figure 3: Circuit diagram of the singing arc, from Poincaré [31, p. 390].

Figure 4 :

 4 Figure 4: Closed curve solution of the sing arc equation, from Poincaré [31, p. 390].

  or the experience of Mr. JANET, and other more recent. . . " 4 The triode: from periodic solution to limit cycle In 1907, the American electrical engineer Lee de Forest (1873-1961) invented the audion. It was actually the first triode developed as a radio receiver detector. Curiously, it found little use until its amplifying ability was recognized around 1912 by several researchers. Then, it progressively replaced the singing arc in the wireless telegraphy devices and underwent a considerable development during the First World War. Thus, in October 1914, a few months after the beginning of the conflict, the French General Gustave Ferrié (1868-1932), director of the Radiotélégraphie Militaire department, gathered a team of specialists whose mission was to develop a French audion, which should be sturdy, have regular characteristics, and be easy to produce industrially. Ferrié asked to the French physicist Henri Abraham (1868-1943) to recreate Lee de Forests' audions. However, their fragile structure and lack of stability made them unsuitable for military use. After several unsuccessful attempts, Abraham created a fourth structure in December 1914, which was put in operation from February to October 1915 (Fig. 5).

Figure 5 :

 5 Figure 5: Picture of the original lamp T.M. made by Abraham (1915).

Figure 6 :

 6 Figure6: Graphical integration of equation[START_REF] Bois Duddell | On Rapid Variations in the Current through the Direct-Current Arc[END_REF]

Figure 8 :

 8 Figure 8: Representation of the phenomenon of frequency demultiplication, from Van der Pol et Van der Mark [48, p. 364].

A brief history of relaxation oscillations can be found in Ginoux and Letellier[START_REF] Ginoux | Van der Pol and the history of relaxation oscillations: Toward the emergence of a concepts[END_REF]. However, let's notice that this article has been entirely republished by M. Letellier in the chapter

of his last book while omitting to make correct reference to this work. For a detailed history of relaxation oscillations, see Ginoux[START_REF] Ginoux | Analyse mathématiques des phénomènes oscillatoires non linéaires[END_REF][START_REF] Ginoux | Histoire de la théorie des oscillations non linéaires[END_REF][START_REF] Ginoux | History of nonlinear oscillations theory, Archimede[END_REF].

(1) BLONDEL, Eclair. Elec.,[START_REF] Van Der Pol | Über ,Relaxationsschwingungen[END_REF][START_REF] Van Der Pol | Gedwongen trillingen in een systeem met nietlineairen weerstand (Ontvangst met teruggekoppelde triode)[END_REF] 81, 1905. See also J. de Phys.,8, 153, 1919.

For more details see Ginoux[START_REF] Ginoux | Self-excited oscillations: from Poincar to Andronov[END_REF][START_REF] Ginoux | Blondel et les oscillations auto-entretenues[END_REF][START_REF] Ginoux | Histoire de la théorie des oscillations non linéaires[END_REF][START_REF] Ginoux | History of nonlinear oscillations theory, Archimede[END_REF].

For more details see Ginoux[START_REF] Ginoux | Analyse mathématiques des phénomènes oscillatoires non linéaires[END_REF][START_REF] Ginoux | Self-excited oscillations: from Poincar to Andronov[END_REF][START_REF] Ginoux | Blondel et les oscillations auto-entretenues[END_REF][START_REF] Ginoux | Histoire de la théorie des oscillations non linéaires[END_REF][START_REF] Ginoux | History of nonlinear oscillations theory, Archimede[END_REF].

For more details see Ginoux[START_REF] Ginoux | Analyse mathématiques des phénomènes oscillatoires non linéaires[END_REF][START_REF] Ginoux | Self-excited oscillations: from Poincar to Andronov[END_REF][START_REF] Ginoux | The First "Lost" International Conference on Nonlinear Oscillations[END_REF][START_REF] Ginoux | Histoire de la théorie des oscillations non linéaires[END_REF][START_REF] Ginoux | History of nonlinear oscillations theory, Archimede[END_REF].

The English version of this article was published in 1927. See Van der Pol[START_REF] Van Der Pol | Forced oscillations in a circuit with non-linear resistance (reception with reactive triode)[END_REF].