Antoine Demiquel

Vassos Achilleos

Georgios Theocharis

Vincent Tournat

Modulation instability in nonlinear flexible mechanical metamaterials

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Modulation instability in nonlinear flexible mechanical metamaterials

A. Demiquel, V. Achilleos, G. Theocharis, and V. Tournat Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique -Graduate School (IA-GS), CNRS, Le Mans Université, France (Dated: November 17, 2022) In this paper, we study modulation instabilities (MI) in a one-dimensional chain configuration of a flexible mechanical metamaterial (flexMM). Using the lumped element approach, flexMMs can be modeled by a coupled system of discrete equations for the longitudinal displacements and rotations of the rigid mass units. In the long wavelength regime, and applying the multiple-scales method we derive an effective nonlinear Schrödinger equation for slowly varying envelope rotational waves. We are then able to establish a map of the occurrence of MI to the parameters of the metamaterials and the wavenumbers. We also highlight the key role of the rotation-displacement coupling between the two degrees of freedom in the manifestation of MI. All analytical findings are confirmed by numerical simulations of the full discrete and nonlinear lump problem. These results provide interesting design guidelines for nonlinear metamaterials offering either stability to high amplitude waves, or conversely being good candidates to observe instabilities.

I. INTRODUCTION

In the context of nonlinear waves, flexible mechanical metamaterials have recently emerged as a rich and versatile platform, opening the way for fundamental studies and potential applications [START_REF] Deng | Nonlinear waves in flexible mechanical metamaterials[END_REF]. Such flexible mechanical metamaterials (flexMMs) can be defined as artificial compliant structures able to support large deformations and mechanical instabilities leading to new modes of functionality [START_REF] Bertoldi | Flexible mechanical metamaterials[END_REF]. As a result, a plethora of original quasi-static behaviors and functions have already been reported, with applications to soft robotics [START_REF] Rafsanjani | Programming soft robots with flexible mechanical metamaterials[END_REF], structure reconfigurability [START_REF] Haghpanah | Multistable Shape-Reconfigurable Architected Materials[END_REF] or mechanical logic devices [START_REF] Raney | Stable propagation of mechanical signals in soft media using stored elastic energy[END_REF][START_REF] Bilal | Bistable metamaterial for switching and cascading elastic vibrations[END_REF][START_REF] Jiang | Bifurcationbased embodied logic and autonomous actuation[END_REF], as examples. In addition and more recently, the study of their dynamic properties has revealed that the nonlinearity is most often geometric in nature, resulting from large local deformations, which makes the nonlinear dynamic response governed by the architecture and therefore controllable [START_REF] Deng | Nonlinear waves in flexible mechanical metamaterials[END_REF]. This latter possibility opens the way to targeting specific dynamical properties, which have been known to be described by existing fundamental equations (such as nonlinear Klein-Gordon equations found in [START_REF] Deng | Elastic Vector Solitons in Soft Architected Materials[END_REF]) or which could illustrate and reveal new relevant dynamic equations.

Up to now, the specific behaviors of the reported flexMM designs could be accurately modeled as rigid units able to translate and rotate, connected with highly compliant springs of longitudinal, shear and bending nature. On the one hand, the derived nonlinear and discrete equations of motion for multiple degrees of freedom can be efficiently solved by numerical integration [START_REF] Deng | Nonlinear waves in flexible mechanical metamaterials[END_REF]. On the other hand, several steps towards analytical solutions can be taken, including the consideration of periodicity, long wavelength compared to the lattice period, expansions to first order nonlinear and dispersive terms, for instance. A review of the main nonlinear wave processes and corresponding equations in flexMM reported to date can be found in [START_REF] Deng | Nonlinear waves in flexible mechanical metamaterials[END_REF]. These include among others the observation of mechanical vector solitons, their interactions and tuning [START_REF] Deng | Nonlinear waves in flexible mechanical metamaterials[END_REF][START_REF] Deng | Elastic Vector Solitons in Soft Architected Materials[END_REF][START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF], the observation of cnoidal waves [START_REF] Mo | Cnoidal wave propagation in an elastic metamaterial[END_REF] and of transition waves [START_REF] Haghpanah | Multistable Shape-Reconfigurable Architected Materials[END_REF][START_REF] Raney | Stable propagation of mechanical signals in soft media using stored elastic energy[END_REF][START_REF] Bilal | Bistable metamaterial for switching and cascading elastic vibrations[END_REF][START_REF] Jiang | Bifurcationbased embodied logic and autonomous actuation[END_REF]. However, nonlinear modulated waves in flexMM is an unexplored field. Many interesting wave phenomena are expected to be revealed, including the manifestation of modulation instability (MI) and the resulting formation of localized waves such as envelope solitons or breathers [START_REF] Zakharov | Modulation instability: The beginning[END_REF][START_REF] Dudley | Rogue waves and analogies in optics and oceanography[END_REF][START_REF] Copie | The Physics of the one-dimensional nonlinear Schrödinger equation in fiber optics: Rogue waves, modulation instability and self-focusing phenomena[END_REF].

The phenomenon of MI has attracted a significant research interest in a range of different wave systems, both continuum (water surface [START_REF] Benjamin | The disintegration of wave trains on deep water Part 1. Theory[END_REF][START_REF] Chabchoub | Rogue Wave Observation in a Water Wave Tank[END_REF][START_REF] Chabchoub | Time-Reversal Generation of Rogue Waves[END_REF], plasmas [START_REF] Ghosh | Modulational instability of electron plasma waves in a cylindrical wave guide[END_REF], optical fibers [START_REF] Tai | Observation of modulational instability in optical fibers[END_REF][START_REF] Shukla | Modulational instability of short pulses in long optical fibers[END_REF], Bose-Einstein condensates [START_REF] Strecker | Formation and propagation of matter-wave soliton trains[END_REF]) and discrete (electrical transmission lines [START_REF] Kengne | Modulational instability criteria for coupled nonlinear transmission lines with dispersive elements[END_REF], granular chains [START_REF] Liu | Breathers in a locally resonant granular chain with precompression[END_REF]) described by the universal nonlinear Schrödinger equation (NLSE) [START_REF] Ablowitz | Discrete and Continuous Nonlinear Schrodinger Systems[END_REF][START_REF] Peyrard | Physics of solitons[END_REF][START_REF] Solli | Optical rogue waves[END_REF]. MI analysis conventionally describes the early (linear) stage of the exponential growth of perturbations of an unstable plane wave background [START_REF] Tai | Observation of modulational instability in optical fibers[END_REF][START_REF] Shukla | Modulational instability of short pulses in long optical fibers[END_REF][START_REF] Potasek | Modulation instability in an extended nonlinear Schrödinger equation[END_REF][START_REF] Cheng | Controllable rogue waves in coupled nonlinear Schrödinger equations with varying potentials and nonlinearities[END_REF][START_REF] Diakonos | Symmetric solitonic excitations of the Abelian-Higgs "classical vacuum[END_REF][START_REF] Kraych | Statistical Properties of the Nonlinear Stage of Modulation Instability in Fiber Optics[END_REF]. Recently, a renewed interest in MI has appeared, motivated by the search for extreme waves, and has led to the analysis of various initial conditions not limited to plane waves as well as to the study of the subsequent nonlinear stages of instability beyond the initial linear stage. [START_REF] Sarma | Modulational instability of few-cycle pulses in optical fibers[END_REF][START_REF] Sarma | Modulational instability of coupled nonlinear field equations for pulse propagation in a negative index material embedded into a Kerr medium[END_REF][START_REF] Xiang | Modulation instability in metamaterials with saturable nonlinearity[END_REF]. Along these lines, numerous theoretical and experimental works in water wave tanks and optical fibers appeared in the literature [START_REF] Copie | The Physics of the one-dimensional nonlinear Schrödinger equation in fiber optics: Rogue waves, modulation instability and self-focusing phenomena[END_REF][START_REF] Chabchoub | Rogue Wave Observation in a Water Wave Tank[END_REF][START_REF] Chabchoub | Time-Reversal Generation of Rogue Waves[END_REF][START_REF] Liu | Breathers in a locally resonant granular chain with precompression[END_REF][START_REF] Bonnefoy | From modulational instability to focusing dam breaks in water waves[END_REF][START_REF] Xu | Observation of modulation instability and rogue breathers on stationary periodic waves[END_REF][START_REF] Kibler | The Peregrine soliton in nonlinear fibre optics[END_REF][START_REF] Tikan | Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation[END_REF][START_REF] Tikan | Local Emergence of Peregrine Solitons: Experiments and Theory[END_REF][START_REF] Pierangeli | Observation of Fermi-Pasta-Ulam-Tsingou Recurrence and Its Exact Dynamics[END_REF][START_REF] Zhao | Controlled generation of high-intensity optical rogue waves by induced modulation instability[END_REF].

It is the main objective of this paper to study the phenomenon of MI in nonlinear flexMM. To do so, starting from a discrete, nonlinear lump model, which was found to describe well the dynamics of flexMM, we derive a NLS equation for the slowly varying envelope of waves of the rotational degree of freedom. Then, we analyze under which conditions, modulation instability of plane waves emerges by random perturbations. We finally compare the theoretical results with numerical simulations of the full nonlinear lump model. We show that, via an initial condition problem, the coupling between the degrees of freedom of the particles as well as the mechanical parameters of the metamaterial (see section II), can allow modulation instability to occur and under which conditions. It is composed by two rows of rigid mass units (gray squares) linked by elastic connectors (thick, blue lines) extending along x-direction with a lattice constant α. The rigid units can be of various shapes (for example crosses, spheres, cubes) and are characterized by a mass m and a moment of inertia J. The elastic bonds (for example highly flexible plastic films) are characterized by three effective stiffness. We consider symmetric movements relative to the horizontal symmetry axis of the system. The displacements of the n and n -1 particles from the equilibrium position are shown in panels (b) and (c) for the two different considered cases. In panel (b) the mass units can only rotate, case (I) while in panel (c), the mass units can both rotate and longitudinally translate, case (II).

II. PROPERTIES AND MODELING OF THE CONSIDERED FLEXIBLE MECHANICAL METAMATERIAL

A. Problem position and modeling of the structure

The considered structure is inspired from the flexible Lego chain implemented in ref. [START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF] and it consists of rigid units (an assembly of Lego bricks), that are linked to the next neighbors by highly flexible plastic films. A periodic chain can then be constructed by connecting pairs of units along one direction as shown in Fig. 1(a). The plastic films connecting the rigid bodies are physically modeled by massless springs. Three springs are needed to represent the plastic films connections, a longitudinal spring with stiffness k l , a shear spring with a shear stiffness k s , and a bending spring with a bending stiffness k θ . Two rows of masses were originally used in ref. [START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF] because this chain configuration possesses a symmetry axis ensuring symmetry of the motion and no experimental buckling of the chain out of this axis. The motion takes place in the plane of the chain and in the general case, each mass should have 3 degrees of freedom, one rotation and two displacements. In the context of soliton propagation [START_REF] Deng | Elastic Vector Solitons in Soft Architected Materials[END_REF][START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF], it has been shown numerically and experimentally that ignoring the transversal displacement is a reasonable assumption. Indeed, the numerically and experimentally observed transversal displacement amplitude is an order of magnitude smaller than the longitudinal one. A 2degree-of-freedom (dof) model was therefore used for this system, and could be used as a starting point for obtaining relevant analytical solutions.

In the present study, we also ignore the transversal displacements and we consider two cases. Case (I), Fig. 1(b), where each rigid unit is free only to rotate (thus is described by one dof θ), and case (II), Fig. 1(c), where each rigid unit both rotates and is longitudinally displaced (thus is described by two dofs θ and u). Based on the mirror symmetry of the two lines configuration along the y-axis, we look for symmetric excitations for which the two rigid units of each column move along x with the same amount and rotate at an opposite angle.

As done in [START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF], a positive direction of rotation is from now on defined alternately for neighboring units since the natural rotation is alternated, upon static compression or long-wavelength propagation. The corresponding normalized equations of motion for the n-th column are then written [START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF],

∂ 2 U n ∂T 2 = U n+1 -2U n + U n-1 - cos θ n+1 -cos θ n-1 2 , 1 α 2 ∂ 2 θ n ∂T 2 = -K θ (θ n+1 + 4θ n + θ n-1 ) + K s cos θ n [sin θ n+1 + sin θ n-1 -2 sin θ n ] -sin θ n [2 (U n+1 -U n-1 ) + 4 -2 cos θ n -cos θ n+1 -cos θ n-1 ] , (1)
where we have introduced the following normalized variables and parameters: the longitudinal displacement of unit n, U n = u n /a, the normalized time T = t k l /m, an inertial parameter α = a/ m/(4J), and stiffness parameters K θ = 4k θ /k l a 2 and K s = k s /k l . Above, m and J are the mass and the moment of inertia of the rigid units, while a is the unit cell length (distance between the centers of the masses).

B. Discrete dispersion relations

A particularity of this system, compared to other mechanical chains with two dofs, [START_REF] Pichard | Localized transversal-rotational modes in linear chains of equal masses[END_REF][START_REF] Prodan | Dynamical Majorana edge modes in a broad class of topological mechanical systems[END_REF][START_REF] Köpfler | Topologically Protected Twist Edge States for a Resonant Mechanical Laser-Beam Scanner[END_REF][START_REF] Allein | Linear and Nonlinear Elastic Waves in Magnetogranular Chains[END_REF][START_REF] Miyazawa | Topological state transfer in Kresling origami[END_REF], is that in the linear limit, the two motion (displacements and rotations) are decoupled, i.e. each degree of freedom follows its own dynamics, independent of the other (see Appendix).

The corresponding dispersion relations are given by

ω (1) = 2 sin qa 2 , (2)
ω (2) = ± 4α 2 (K s -K θ ) sin 2 qa 2 + 6α 2 K θ . (3)
The first branch, Eq. ( 2), describes propagating longitudinal waves with the typical monoatomic dispersion relation. The second branch, Eq. ( 3), describes propagating rotational waves with a Klein-Gordon type dispersion relation and a lower cutoff frequency at ω = α √ 6K θ . From Eq. (3), it is clear that the dispersion relation of the structure can be highly tuned through the inertial parameter α (changing the mass and the shape of the rigid particles) as well as the stiffness parameters K s , K θ (changing the elastic parameters of the plastic films). Four examples of the dispersion relation for different values of the bending stiffness K θ are shown in Fig. 2 with solid lines. The rest of the parameters are chosen to be consistent with the literature [START_REF] Deng | Elastic Vector Solitons in Soft Architected Materials[END_REF][START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF][START_REF] Mo | Cnoidal wave propagation in an elastic metamaterial[END_REF][START_REF] Guo | Nonlinear architected metasurfaces for acoustic wave scattering manipulation[END_REF]. Note also that the concavity of the dispersion relation for the rotation dof is defined by the sign of δ = K s -K θ , see Fig. 2(a-c) vs (b-d). As we explain below, the sign of δ plays a key role in the stability of the plane waves in the system.

C. Continuum Limit

Considering waves with wavelengths that are sufficiently larger than the unit cell distance, i.e. λ a, one can employ the continuum limit approximation. Therefore, we define two continuous functions U (X, T ) and θ(X, T ), interpolating the displacement and rotation of the n-th pair of rigid units located at the position x n = na, where n is an integer, such that

U (X n , T ) = U n (T ) , θ(X n , T ) = θ n (T ) , X n = x n a .
(4) If we further assume weak nonlinearity, namely θ 1, keeping terms up to θ 3 , see also [START_REF] Deng | Elastic Vector Solitons in Soft Architected Materials[END_REF][START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF], Eqs. (1) yield,

∂ 2 U ∂T 2 = ∂ 2 U ∂X 2 + θ ∂θ ∂X , (5)
∂ 2 θ ∂T 2 = C 1 ∂ 2 θ ∂X 2 -C 2 θ -C 3 θ 3 -C 4 θ ∂U ∂X , (6)
where

C 1 = α 2 [K s -K θ ], C 2 = 6K θ α 2 , C 3 = 2α 2 and C 4 = 4α 2 .
The system of equations (5-6) is a simple dispersion-less wave equation for the displacement field U , Eq. ( 5), coupled through a nonlinear term, with a Klein-Gordon equation for the rotation field θ, Eq. ( 6).

Pulse soliton solutions of (5-6) were theoretically obtained and experimentally observed in [START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF], revealing the validity of the continuum coupled equations.

The linear dispersion relations of Eqs. (5-6) are given by:

ω (1) = k , (7)
ω (2) = C 1 k 2 + C 2 , (8)
and they are shown in Fig. 2 with dashed lines. For the cases we plot, one can see that as long as the wavenumber q ≤ 1, the continuum equations capture well the dispersive characteristics of the discrete model.

III. MODULATED WAVES IN FLEXMM

Although there are several recent studies on pulse nonlinear waves, the existence, stability and propagation of nonlinear modulated waves, in the form of plane waves or wavepackets in flexMM remain unexplored. Only recently, the existence and stability of discrete breathers in flexMM was explored [START_REF] Duran | Discrete breathers in a mechanical metamaterial[END_REF]. Here, we derive the theoretical framework for the description of long-wave, nonlinear modulated waves. For this purpose, we apply below the multiple-scales method [START_REF] Peyrard | Physics of solitons[END_REF][START_REF] Holmes | Introduction to Perturbation Methods[END_REF] to Eqs. [START_REF] Raney | Stable propagation of mechanical signals in soft media using stored elastic energy[END_REF][START_REF] Bilal | Bistable metamaterial for switching and cascading elastic vibrations[END_REF].

A. Multiple-Scales

We are looking for U and θ, in the form of a perturbative expansion, where T i = i T and X i = i X, with i = 0, 1, . . . N and represents a small parameter. X 0 and T 0 correspond to the original "fast" spatial and temporal scales of the carrier wave, while X i and T i with i = 0 define progressively the "slow" spatial and temporal scales of the envelope.

U = N i=1 i u i (X 0 , . . . X N , T 0 , . . . T N ) , θ = N i=1 i θ i (X 0 , . . . X N , T 0 , . . . T N ) , (9)
By inserting the expansions of Eq. ( 9) into the system of Eqs. [START_REF] Raney | Stable propagation of mechanical signals in soft media using stored elastic energy[END_REF][START_REF] Bilal | Bistable metamaterial for switching and cascading elastic vibrations[END_REF], and taking into account the derivative operators of the new spatial and temporal variables [see appendix B Eqs. (B1)], we end up with the following hierarchy of equations at successive orders of ,

O( ) L(1) 0 u 1 = 0 , L(2) 0 θ 1 = 0 , O ( 2 )
L(1)

0 u 2 = - L(1) 1 u 1 + M(1) 0 θ 2 1 , L(2) 0 θ 2 = - L(2) 1 θ 1 + θ 1 M(2) 0 u 1 , O( 3 )      L(1) 0 u 3 = - L(1) 1 u 2 - L(1) 2 u 1 + M(1) 1 θ 2 1 + 2 M(1) 0 θ 1 θ 2 , L(2) 0 θ 3 = - L(2) 1 θ 2 - L(2) 2 θ 1 + M(3) θ 3 1 + θ 1 M(2) 0 u 2 +θ 1 M(2) 1 u 1 + θ 2 M(2) 0 u 1 , (10)
where the linear operators, L(i) j and M(i) j , applied to the linear and nonlinear terms of the equations (5-6) respectively are defined in the appendix B Eqs. (B2).

The first set of equations [START_REF] Mo | Cnoidal wave propagation in an elastic metamaterial[END_REF] of order O( ), corre-sponds to the linearized system of Eqs. [START_REF] Raney | Stable propagation of mechanical signals in soft media using stored elastic energy[END_REF][START_REF] Bilal | Bistable metamaterial for switching and cascading elastic vibrations[END_REF]. Using the fact that in the linear regime the two fields are decoupled, we will focus on the particular case which, at the leading order, there is only rotational motion, i.e.,

u 1 = 0 , θ 1 = B(X 1 , T 1 , X 2 , T 2 , ...)e i(kX0-ωT0) + c.c , (11)
with ω and k satisfying the dispersion relation Eq. ( 8) and c.c stands for the complex conjugate.

Let us proceed to the next order of the perturbation scheme, O( 2), and substitute the solutions [START_REF] Zakharov | Modulation instability: The beginning[END_REF] into the second set of equations [START_REF] Mo | Cnoidal wave propagation in an elastic metamaterial[END_REF] to obtain,

L(1) 0 u 2 = M(1) 0 θ 2 1 , L(2) 0 θ 2 = - L(2) 1 θ 1 . ( 12
)
The right-hand-side of the last equation is a secular term, as it acts as a source term proportional to e iσ (σ = kX 0 -ωT 0 ) with which the linear operator L(2) 0 on the left is in resonance. This implies that the solution θ 2 would blow up as t → ∞ and thus the perturbation scheme will fail. The only way for the expansion to be bounded is to set the secular term to zero, which translates to the following relation for the envelope function B,

D 1 B + v g D 1X B = 0 . (13)
Here we have introduced the group velocity given by

v g = C 1 k √ C 1 k 2 + C 2 = C 1 k ω . ( 14
)
Once the secular term is removed, the system of equations of the second order in in Eq. ( 10) is now reduced to,

L(1) 0 u 2 = ikB 2 e 2iσ + c.c , L(2) 0 θ 2 = 0 . (15)
The first equation has the following solution,

u 2 = ikB 2 4(k 2 -ω 2 ) e 2iσ + c.c , (16)
where the homogeneous part of the solutions is omitted due to our choice of initial conditions U (0, X) = U (0, X) = 0. For θ 2 we choose the trivial solution, i.e. θ 2 = 0, since any other solution can be incorporated in B.

B. Nonlinear Schrödinger Equation (NLSE)

We now proceed with the O( 3 ) order of the perturbation scheme. By using u 1 = 0 and θ 2 = 0, as discussed above, the last equation of Eq. ( 10) is reduced to

L(2) 0 θ 3 = - L(2) 2 θ 1 + M(3) θ 3 1 + θ 1 M(2) 0 u 2 . ( 17
)
Similar to the previous order, there are secular terms in the right-hand side of Eq. [START_REF] Ghosh | Modulational instability of electron plasma waves in a cylindrical wave guide[END_REF] proportional to e iσ : the L(2) 2 θ 1 , and parts of the M(3) θ 3 1 and θ 1 M(2) 0 u 2 terms. To find their secular contributions, we develop the operators as well as the functions on which they are applied. For the first of them,

M(3) θ 3 1 = -C 3 B 3 e 3iσ -3C 3 |B| 2 Be iσ + c.c , (18)
the secular contribution is -3C 3 |B| 2 Be iσ . For the next one,

θ 1 M(2) 0 u 2 = C 4 k 2 B 3 2(k 2 -ω 2 ) e 3iσ + C 4 k 2 |B| 2 B 2(k 2 -ω 2 ) e iσ + c.c , (19)
the secular contribution is

C4k 2 |B| 2 B 2(k 2 -ω 2 ) e iσ .
To avoid the resonant driving we set all the secular terms equal to zero (17-18-19),

L(2) 2 θ 1 + 3C 3 - C 4 k 2 2 (k 2 -ω 2 ) |B| 2 Be iσ = 0 . ( 20
)
It is possible to simplify this expression Eq. ( 20) by using the variables ξ i = X i -v g T i , τ i = T i , i.e. a reference frame moving with the group velocity. Within this frame Eq. ( 13), becomes ∂B/∂τ 1 = 0 and Eq. ( 20) leads to the following nonlinear Schrödinger (NLS) equation,

i ∂B ∂τ 2 + g 1 2 ∂ 2 B ∂ξ 2 1 + g 2 |B| 2 B = 0 . (21)
Eq. ( 21) describes the evolution of the envelope B of the modulated rotational waves, in the co-moving space variable and the second order slow time. The coefficients g 1 and g 2 are given by the following expressions,

g 1 = d 2 ω dk 2 = C 1 -v 2 g √ C 1 k 2 + C 2 , g 2 = - 1 2 √ C 1 k 2 + C 2 3C 3 + C 4 k 2 2k 2 (C 1 -1) + 2C 2 . (22)
Furthermore, Eq. ( 21) can be rewritten as a function of a single nonlinear parameter g = g 2 /g 1 by applying the following change of variable τ2 = g 1 τ 2 ,

i ∂B ∂ τ2 + 1 2 ∂ 2 B ∂ξ 2 1 + g|B| 2 B = 0 . (23)
In its current form, the NLS equation has two distinct behaviors depending on the sign of the nonlinearity coefficient: it is known as focusing when g > 0 and defocusing for g < 0. Among other different properties between these two cases, an important one is the stability of plane wave solutions. More precisely, for the focusing case, it is known that plane waves are subject to modulational instabilities [START_REF] Copie | The Physics of the one-dimensional nonlinear Schrödinger equation in fiber optics: Rogue waves, modulation instability and self-focusing phenomena[END_REF][START_REF] Cheng | Controllable rogue waves in coupled nonlinear Schrödinger equations with varying potentials and nonlinearities[END_REF][START_REF] Diakonos | Symmetric solitonic excitations of the Abelian-Higgs "classical vacuum[END_REF][START_REF] Akhmediev | Rogue waves and rational solutions of the nonlinear Schrödinger equation[END_REF][START_REF] Akhmediev | Modulation instability and periodic solutions of the nonlinear Schrödinger equation[END_REF][START_REF] Akhmediev | Exact first-order solutions of the nonlinear Schödinger equation[END_REF][START_REF] Mora | Modulational instability and rogue waves in one-dimensional nonlinear acoustic metamaterials: case of diatomic model[END_REF], which is the main interest of the present work. Therefore, below we establish the conditions under which MI appears in the proposed flexMM.

C. Modulation instability (MI)

We seek solutions of Eq. ( 23) in the form of a perturbed plane wave [START_REF] Zakharov | Modulation instability: The beginning[END_REF],

B(ξ 1 , τ2 ) = (A 0 + b(ξ 1 , τ2
))e i(k0ξ1-ω0 τ2+ θ(ξ1,τ2)) , [START_REF] Peyrard | Physics of solitons[END_REF] with b the amplitude and θ the phase of small perturbations. The unperturbed plane wave satisfies the dispersion relation,

ω 0 = k 2 0 2 -gA 2 0 . (25)
Inserting Eq. ( 24) into Eq. ( 23), we find at first order a set of linear equations for the perturbations b and θ. We thus assume harmonic solutions of the form,

b = f 1 e i(Kξ1-Ω τ2) , θ = f 2 e i(Kξ1-Ω τ2) , (26)
where the perturbation frequency Ω and wavenumber K follow the dispersion relation,

Ω = Kk 0 ± |K| K 2 4 -gA 2 0 . (27)
We can now identify two different regions of stability of the plane waves. On the one hand, where g < 0 the perturbations are oscillating functions and remain bounded. Thus we call this region modulational stable. On the other hand, for g > 0 there exists a band of unstable wavenumbers satisfying K < K c where,

|K c | = 2A 0 √ g , (28)
resulting in a complex frequency Ω = Ω R ± iΩ I with

Ω R = Kk 0 , Ω I = |K|A 0 g - K 2 4A 2 0 . (29)
We call this region modulational unstable. The small unstable wavenumbers lead to an exponential growth of the perturbations, with a growth rate Ω I . Thus any perturbation with wavenumbers within the instability band should lead to MI. Another important parameter for studying MI is the wavenumber with the maximum growth rate,

|K m | = A 0 2g . ( 30
)
We notice that both the critical wavenumber K c and the wavenumber corresponding to the fastest growth rate of the perturbations K m , depend on the parameter g and the initial amplitude A 0 .

Parametric study of the coefficient g

It is now clear that the stability of modulated waves in the flexMM depends on the sign g. As already discussed in section II, we study two distinct cases: (I) allowing only rotations and (II) with 2 dofs per unit, i.e. including both rotation and longitudinal displacement [Fig. 1(b-c)]. The corresponding nonlinear coefficient g(δ, α, K θ , k) for the two cases is given by,

g = -3α 2 δα 2 -v 2 g , (31)
for case (I), and

g = -3α 2 δα 2 -v 2 g 1 + k 2 3k 2 (α 2 δ -1) + 18K θ α 2 , ( 32
)
for case (II).

In practice, the sign of g is determined by the choice of the carrier wavenumber k and the geometrical characteristics of the flexMM. This shows the great flexibility that the proposed system offers in order to manipulate weakly nonlinear waves. In Fig. 3 we plot a map of the sign of g as a function of the wavenumber k and δ. In all cases, white (resp. black) regions correspond to g > 0 (resp. g < 0). From the left panel, it is clear that for case (I) with only rotations, the sign of g solely depends on the sign of delta. However for case (II), things are different and the coupling between the rotation and the longitudinal motion creates intermediate regions of focusing and defocusing behavior depending also on the wavenumber k. The different panels of Fig. 3 also show how these regions "move" towards larger k by changing the value of the inertia parameter α.

Another interpretation of the results plotted in Fig. 3 is that the coupling between the rotations and longitudinal displacements creates stripes of stability (black shaded regions) in the otherwise unstable single dof lattice with only rotations [panel (a)]. At the same time this coupling forms regions of instability (white) where solely rotational motion would have been stable. Once again, this result shows the great tunability and richness of the system regarding nonlinear wave propagation.

IV. NUMERICAL SIMULATIONS OF THE FLEXMM

In this section we use direct numerical simulations of the system's discrete equations [START_REF] Deng | Nonlinear waves in flexible mechanical metamaterials[END_REF], in order to verify our analytical predictions. In particular we first want to check the stability of plane waves as this is predicted by the sign of g (defocusing vs focusing) of the effective NLS. In addition, in the case of modulational instability, we want to compare the unstable generated wavenumber, according to the ones that the MI analysis predicts. Furthermore, we use the numerical simulations to uncover as well the dynamics of the system long after the emergence of the MI. We thus solve the Eqs. (1) using a 4th order Runge-Kutta iterative integration scheme for a total of N = 500 sites, using periodic boundary conditions. We focus on the case with α = 2.5 (Fig. 3 panels (a) and(c)) although any other choice of α could have been done in principle.

g > 0 g < 0 (a) (b) (c) (d)
As initial conditions, we apply plane waves on the rotations only, with wave-number k, whose amplitude is perturbed by a random noise

θ(n, 0) = 2 (1 + b 0 ) cos(kn) , θ(n, 0) = 2 ω(k)(1 + b 0 ) sin(kn) , (33)
with = 0.01 and b 0 ∈ [-10 -3 , 10 -3 ] is a random number taken from a uniform distribution. As mentioned above, in all the cases we use U (n, 0) = U (n, 0) = 0 for the longitudinal displacements. Here random noise was chosen as a perturbation, not only because it is relevant to realistic experimental conditions but also since it is an efficient way to excite all the wave-numbers including the unstable ones. Moreover, we can confirm in this way our analytical results by identifying the two characteristic wave-numbers K c and K m using Eqs.( 28) and ( 30) during the lattice dynamics simulation.

Here we note the following technical point. Due to the periodic boundary conditions, the spectrum is wrapped between [0; π]. During the manifestation of the MI, we expect to identify at least the following wavenumbers: the carrier k, and the most unstable wavenumber K m . However we know that we always excite at least the third harmonics 3k. In order for all these frequencies to be well identified, we thus choose parameters such that the k + K m is smaller than 3k. To do so, we use an alternative representation of Fig. 3, using as colormap the values of K m . The two points denoted by squares and circles in left and right panel respectively, are the two examples that we will study in details below.

Case (I)

Case (II)

A. Inducing MI by coupling the rotations with displacements

We first focus on a point, in the parameter space spanned by δ and k, indicated by the square in Fig. 4. This corresponds to the plane wave wavenumber k = 0.81681 and δ = 0.003. We fix from now the value of α = 2.5. As a reminder, the values of δ fixes the difference between shear and bending stiffness (δ = K s -K θ ), while the value of α the ratio of mass to moment of inertia of the particles. According to the theory, this point is described by a defocusing NLSE (g < 0) for the case (I). Thus, the plane wave is supposed to be stable. In contrary, when both the dof are considered, i.e., case (II), and for exactly the same parameters, the plane wave become modulationally unstable, since the system is described by a focusing NLSE (g > 0). To confirm our theoretical prediction, we solve the discrete system of Eqs. (1) using the initial conditions [START_REF] Bonnefoy | From modulational instability to focusing dam breaks in water waves[END_REF]. In Fig. 5, we show the results for the case (I).

Here both the evolution of the rotation [panel (a)] and its space Fourier transform [panel (b)] indeed show that a random perturbation on an initial plane wave remains bounded, thus the plane wave is stable.

The rotations θ n show small amplitude oscillations in time with a frequency ω, following the dispersion relation Eq. ( 8) at the given k. Even after more than 500 oscillations, only the wavenumber of the carrier wave is present in the spectrum, indicating the stability. For the exact same flexMM and the same initial condition, if we allow the coupling between the two dofs, namely if we consider the case (II), the dynamics is radically different. This scenario is shown in Fig. 6. As predicted by the theory, the wavenumbers of the perturbation that belong to the instability band, start growing. This is clear by the two sidebands that are developed symmetrically around the excited wavenumber k = 0.81681 in panel (b) of Fig. 6. More precisely, the center of these side bands corresponds to the point k ± K m since the most unstable wavenumber rises first. The generation of these wavenumbers is directly revealed on the rotations as large amplitude localized structures appear [see Fig. 6 (a)]. For later times, after the instability kicks in, and when the amplitude of the rotations becomes large enough, we observe a spectrum with many excited wavenumbers.

In this case, since rotations are coupled to the longitudinal displacements U n , we expect to see some dynamics in the displacements too. Indeed, as expected from our analysis in Eq. ( 16), U starts oscillating with a wavenumber 2k as shown in Fig. 6(d), and at later times following the evolution of θ, larger amplitude modulated waves are also emerging in the displacements U n .

B. Stabilizing plane waves using the coupling of dofs

The second configuration which we focus on is the "complementary" one. It corresponds to the green circles in Fig. 4, where the uncoupled system (case I) is described by a focusing NLS, thus we expect the plane waves to be modulationally unstable, while by allowing the coupling between the two dofs (case II), the effective NLS is focusing and thus, the plane waves are stable. To confirm these theoretical predictions, we use the same initial conditions as in Eq. ( 33) but with k = 0.92991 and δ = -0.003 and we solve again numerically the system of Eqs. [START_REF] Deng | Nonlinear waves in flexible mechanical metamaterials[END_REF]. The result of the case (I) is shown in Fig. 7. Following our analysis, the numerical simulations confirm that an initially perturbed plane wave develops initially the expected side branches at k ± K m . At the final steps of the simulation, all the wavenumbers are excited. On the other hand, when both dofs are present (case (II)) and for exactly the same parameter values, the corresponding numerical result, shown in Fig. 8(a-b), verifies the stability of the plane wave solution. We see that for the same total time of propagation as in the decoupled case, θ shows stable oscillations with a wavenumber k while U oscillates at 2k, as per the theory.

C. MI growth rate: theory vs numerics

To further support our theoretical findings we perform numerical simulations, for both cases, by varying the amplitude of the initial excitation A 0 , and we compare the predictions of the MI linear stability analysis [see Eqs. (28-30)], with the early stage of the MI manifestation in simulations. In particular, in Fig. 9(a-b), we plot the MI growth rate for both cases and for two different amplitudes A 0 . The dotted lines correspond to the analytical values of k + K m , while the dashed ones to k + K c . In Fig. 9(cd), we plot the Fourier spectrum of the rotation field θ at times that correspond to the early stage of MI. In both cases a stronger initial excitation results in a larger unstable band showing a maximum shifted to larger k values. Note also how well the theoretically predicted bandwidth, Fig. 9(a-b), matches the numerically obtained bandwidth. This observation constitutes another more quantitative validation of the derived effective NLS description.

V. CONCLUSIONS

FlexMM are mechanical structures with some unique features, including the geometrical nonlinearity coming from the large rotations of the building blocks, the presence of several dofs that are nonlinearly coupled, and the great tunability of the dispersion relation. Therefore, flexMMs offer a perfect experimental platform to explore a plethora of nonlinear wave phenomena.

In this paper, we focused on the archetypal nonlinear phenomenon of MI. To that end, starting from a discrete, nonlinear lump model, that has been proved to accurately describe their dynamics, we first derived a NLS equation for slowly varying rotational envelope waves. We then studied the stability of the rotational plane waves to small perturbations via the MI analysis for the derived NLS. Analytical and numerical results revealed that, under proper values of the physical parameters of the flexMM, namely under some particular values of the inertia and stiffness parameters, it is possible to observe MI in these flexMMs. More importantly, we have analyzed the role of the coexistence of two dofs. In particular, the interplay between the two dofs can lead to regions of stability, in an otherwise unstable flexMM which supports only rotations, i.e., only one of the two dofs, and vice versa.

This work constitutes an attempt to understand more generally the dynamics of modulated waves in nonlinear flexMMs. Several natural extensions of this work include the initial excitation of both rotational and longitudinal modulated waves, leading to a coupled NLS with a much richer MI dynamics, as well as the study of the discreteness when shorter modulated waves are considered, along the lines of [START_REF] Remoissenet | Low-amplitude breather and envelope solitons in quasi-one-dimensional physical models[END_REF][START_REF] Daumont | Modulational instability: first step towards energy localization in nonlinear lattices[END_REF]. Both aspects are currently under investigation and results will be presented in future publications. Other interesting perspectives are the generation and dynamics of coherent structures like Peregrine breathers and extreme wave effects in nonlinear flexMMs. We believe that the present work reveals the great potential that nonlinear flexMMs have, for the observation and control of both typical and novel nonlinear phenomena related to modulated waves.

ACKNOWLEDGEMENT

Figure 1 .

 1 Figure1. (a) Sketch of the chain configuration periodic flexMM under consideration. It is composed by two rows of rigid mass units (gray squares) linked by elastic connectors (thick, blue lines) extending along x-direction with a lattice constant α. The rigid units can be of various shapes (for example crosses, spheres, cubes) and are characterized by a mass m and a moment of inertia J. The elastic bonds (for example highly flexible plastic films) are characterized by three effective stiffness. We consider symmetric movements relative to the horizontal symmetry axis of the system. The displacements of the n and n -1 particles from the equilibrium position are shown in panels (b) and (c) for the two different considered cases. In panel (b) the mass units can only rotate, case (I) while in panel (c), the mass units can both rotate and longitudinally translate, case (II).

Figure 2 .

 2 Figure 2. Dispersion relations of Eqs. (2-3) (solid lines) and of the continuum approximation (dashed lines) following Eqs. (7-8). In all the examples, we fix the coefficients α = 2.5, Ks = 0.01851 and we vary K θ . (a) K θ = 1.534.10 -4 , (b) K θ = 0.1, (c) K θ = 0.01551, and (d) K θ = 0.02151. Left (right) panels correspond to δ > 0 (δ < 0).

Figure 3 .

 3 Figure 3. Sign of the nonlinear coefficient g as a function of k and δ, for α = 1.5, 2.5, 3.5. Panel (a) corresponds to the case (I) while panels (b,c,d) to case (II).

Figure 4 .

 4 Figure 4. Most unstable wave number Km (colormap) as a function of δ and k for α = 2.5. In both panels, two particular points are indicated: a blue square point for k = 0.81681 and δ = 0.003 and a green circle point at the position k = 0.92991 and δ = -0.003.

Figure 5 .

 5 Figure 5. Panel (a) represents the evolution in time (T ) of the absolute value of the rotation amplitude along the chain (N ). Panel (b) represents the evolution in time of its k-spectrum. The results correspond to the case (I), blue square point (k = 0.81681, δ = 0.003)

Figure 6 .

 6 Figure 6. . Panels (a-c) represent the evolution in time (T ) of the absolute value of the rotation respectively displacement amplitudes along the chain (N ). Panels (b-d) represents the evolution in time of the k-spectrum for the rotation and longitudinal displacement. The results correspond to case (II)blue square point (k = 0.81681, δ = 0.003).

Figure 7 .

 7 Figure 7. Panel (a) represents the evolution in time (T ) of the absolute value of the rotation amplitude along the chain (N ). Panel (b) represents the evolution in time of its k-spectrum. The result correspond to case (I), green circle point (k = 0.92991, δ = -0.003)

Figure 8 .

 8 Figure 8. Panels (a-c) show the evolution in time (T ) of the absolute value of the rotation and displacement amplitudes along the chain (N ). Panels (b-d) show the evolution in time of the k-spectrum for the rotation and longitudinal displacement. The results correspond to case (II), green circle point (k = 0.92991, δ = -0.003).

Figure 9 .

 9 Figure 9. Panels (a) and (b) show the theoretical modulational instability band for the blue point in case (II), the green point in case (I) and for two different amplitudes A0. In panels (c) and (d), we plot the Fourier transform of the θ, corresponding to the parameters of the blue square (respectively green circle) at two different instances. For case (II), we choose t = 2500 for A0 = 0.8 and t = 1100 for A0 = 1.4. For case (II), t = 5000 for A0 = 0.8 and t = 1700 for A0 = 1.4. The dotted lines correspond to the analytical values of k b/g + Km, while the dashed ones to k b/g + Kc. k b/g are the wave numbers of the initial plane waves and K the wave number of the perturbation.

The authors acknowledge the support from the project ExFLEM ANR-21-CE30-0003-01.

Appendix A: Dispersion relation

The dispersion relation of the metastructure is obtained by linearizing the motion equations (sin θ ≈ θ) and assuming that the chain is excited by a harmonic source of ω pulsation, propagating along increasing x. The harmonic solutions of the linear system are represented by these three vectors, when one poses x i = ia, x i±1 = (i ± 1)a:

(A1) By substituting these harmonic solutions in the linearized equations, we obtain the eigenvalue problem,

with λ = ω 2 the eigenvalue and φ = U 0 θ 0 the eigenvector, and

(A3) The coupling between modes comes from the antidiagonal terms of the K matrix. Since all mass units are aligned at the initial time, these anti-diagonal coefficients are zero, so the modes are decoupled.

Appendix B: Multiple scales

The different scales imply that the differentials of X and T must be redefined according to the different scales X i and T i used. By defining the notation D i = ∂ ∂Ti and in an analogous way D iX = ∂ ∂Xi , we can write,

The operators L(i) j and M(i) j are given by the following expressions:


Rébecca Deneckère

Charlotte Hug

Ali Jaffal

Manuele Kirsch Pinheiro

Bénédicte Le Grand

Raúl Mazo

Irina Rychkova

Context Management and Intention Mining for Adaptive Systems in Mobile Environments: from Business Process Management to Video Games?

Keywords: context management, digital traces analysis, intention mining, adaptive processes

This article describes some recent advances in the Information Systems research domain to support users of mobiles devices such as tablets or smartphones. Indeed, we show how information about the evolving context of users can be organized and analyzed through context management. We also describe an original approach called intention mining, which analyzes traces of users' actions on the system and extracts the rational behind these actions (i.e., the user goals/intentions). Finally, we explain how adequate process modeling formalisms allow for personalized recommendations (and therefore adaptive scenarios) in order to optimize users experience with the information system. So far, these approaches have mostly been applied to business process models, and we discuss their interest for video games, in particular urban games which combine physical and real worlds.

Introduction

This article describes some recent advances in the Information Systems research domain to support users of mobiles devices such as tablets or smartphones. Our goal is to provide these users with a better guidance through their daily use of these interfaces. The objective is to generate recommendations that are adapted to a given user's context and that take into account his/her objectives. These recommendations systems also take into account the experience of the other system users. In order to make this possible, we show how information about the evolving context of users can be organized and analyzed through context management. We also describe an original approach called intention mining, which analyzes traces of users' actions on the system and extracts their goals (intentions). Finally, we explain how an adequate process modeling formalism allows for personalized recommendations (and therefore adaptive scenarios) in order to optimize users experience with the information system.

So far, these approaches have mostly been applied to business process models, and the second objective of this work is to study their interest for video games, in particular urban games which combine physical and real worlds.

Context Management

Context management in information systems: state of the art

The "one size fits all" way of using Information Systems (ISs) does not work anymore. Users now expect these systems to be context-aware, i.e., to adapt to users' specific contexts [START_REF] Baldauf | A survey on context-aware systems[END_REF]. The context represents any information that can be used to characterize the situation of an entity (a person, place, or object) that is considered relevant to the interaction between a user and an application [START_REF] Dey | Understanding and using context[END_REF].

More generally, we can consider the context as everything that surrounds the center of interest of the individual and that provides additional information that can help understand his/her focus [Mostefaoui et al., 2004].

Gamers playing to an urban game are very sensitive to their contexts. When playing their game, they evolve outdoors in the real word and, as a result, can modify their behavior to match this context. For instance, if the gamer is heading to a particular place but that rain starts pouring, then he will certainly delay his walk to do something else until the rain stops (exchange with another gamer, try to solve a clue, or any other activity that can be done inside instead of outside a building).

The management of context information has become strategic in the context of pervasive information systems [START_REF] Kourouthanassis | Enhancing the user experience with pervasive information systems[END_REF] [ [START_REF] Najar | Semantic representation of context models: a framework for analyzing and understanding[END_REF] [ [START_REF] Villalonga | Modeling of sensor data and context for the real world Internet[END_REF]. Data generated from these systems by observing user interactions and the environment in which they take place is extremely valuable for systems designers and service providers. It serves as a basis to enhance their systems and services according to user's context (e.g., location, date, time, past and current activities). In the case of urban games, context can be composed of the geolocation of the gamer, the time of day (or night), the proximity of other gamers, the weather forecast, etc.).

Analyzing the impact of context information on the user's actions becomes a key aspect for successful pervasive information systems. Data mining techniques can be used to this end, in order to extract relevant information to support decision and prediction [START_REF] Fayyad | From data mining to knowledge discovery in databases[END_REF] [ [START_REF] Chen | Data mining: an overview from database perspective[END_REF]. Several works such as [Ramakrishnan et al., 2014] are considering data mining techniques for analyzing context data. These techniques can play an important role in understanding and discovering the intrinsic relationships between data. Among them, Formal Concept Analysis (FCA) [START_REF] Priss | Formal Concept Analysis in Information Science[END_REF][START_REF] Wille | Formal Concept Analysis as Mathematical Theory of Concepts and Concept Hierarchies[END_REF] represents an interesting method of conceptual clustering. It helps extracting implicit knowledge and finding a natural data structure, while associating user actions to observed context elements. We describe in the following section how FCA can be used to manage context information.

Use of Formal Concept Analysis for Context Management

The ways mobile interfaces are used depend a lot on user context: geographical location, time of the day, day of the weeks, meteorological conditions, quality of network connection, etc. understanding the influence of these context elements makes it possible to propose better personalization and recommendation features and to improve adaptation mechanisms. However, identifying relevant context elements (for a given user) from the highly heterogeneous collected context data is a challenge not only because of data heterogeneity, but also because this relevance is personal and may vary significantly from user to user. Context analysis methods are then necessary in order to establish context relevance for a given user. In earlier work [START_REF] Jaffal | Unified and Conceptual Context Analysis in Ubiquitous Environments[END_REF], we have used Formal Concept Analysis (FCA) to this end and shown its interest to cluster context elements with regard to associated user activities (and vice versa) into overlapping classes. Telephone

0 0 0 1 (3) 1 (2) 1 (5) 1 0 0 1 (2) 1 VDM 0 0 0 1 (4) 1 (2) 1 (5) 1 (4) 0 0 0 1 (2)
Flappy Bird

1 0 0 1 1 1 1 1 0 1 1 Youtube 0 0 0 1 0 1 1 0 0 0 0
The input of the FCA algorithms is a binary relation -called formal context -between a set of objects and a set of attributes that describes these objects. Table 1 shows such a formal context in which objects are the applications executed by a user on a tablet, and attributes describe the context in which these applications have been used 1 . For example, Youtube has been used at home, but also on a 3G network and in the morning. Note that the only possible values in the formal context are 0 and 1, which does not reflect the real numbers of time various applications have been used (indicated in parentheses): SMS are associated to university and restaurant in a similar way in the formal context, although the number of SMS sent in each location is very different.

Fig. 1. Galois lattice generated from the formal context of Table 1 Figure 1 represents the Galois lattice generated from the formal context of Table 1.

Each cluster of the lattice is called a (formal) concept which groups objects that have attributes in common. For example, this lattice shows that Flappy Bird, SMS and telephone applications have all been used in the morning, in the afternoon, in the evening, during transport, from a 3G network and at home. These context elements are common to these applications. One interest in terms of recommendation is that if the user happens to be in a similar context in the future, the information system can automatically personalize the interface according to the user preferences, e.g., display a game in the foreground of the screen when the user enters the metro station.

Note that the concepts become more and more specific when going from the top to the bottom of the lattice, from concepts with many applications sharing few context elements to concepts with few applications sharing many context elements. Indeed, the concept containing Flappy Bird, SMS and telephone applications can be specialized into a concept that only contains Flappy Bird and SMS applications, with two additional context elements: coffee break and university.

The lattice provides a clustering of applications according to the context elements they have been associated to. Conversely, context elements are described with regard to the applications associated to them. We can notice from the lattice that 3G connection and morning context elements appear in all concepts, which means that they are not significant here and should not be taken into account for recommendations to this specific user. Conversely, the SMS application has been used with all context elements, and is therefore context-independent.

We have proposed a methodology to reflect in FCA results the relative importance of context elements on user activities, by refining the input data using a semantic frequency measure.

In order to build a formal context from collected data, Formal Concept Analysis requires to replace the numeric values (Table 1) by binary values in order to represent the relationships between the objects and the formal context attributes in concept lattice. In table 1, we have followed a simplistic approach, by directly replacing null values indicated by users by a zero and any positive value by one. More elaborate methods exist: Pensa et al. [START_REF] Pensa | Assessment of discretization techniques for relevant pattern discovery from gene expression data[END_REF]] use a cut-off threshold to replace values greater than or equal to the threshold values by 1 and the rest by 0. The method in [START_REF] Ma | Variable threshold concept lattice and dependence space[END_REF]] is to set a threshold δ, and then turn the context into a binary one by replacing values less than δ with 0 and 1 the others. However, these thresholds are calculated based on numerical values in the formal context regardless of the semantic link between the attributes.

The originality of our frequency measure described in [START_REF] Jaffal | Refinement Strategies for Correlating Context and User Behavior in Pervasive Information Systems[END_REF] is that it is a semantic measure that relies on a context ontology, illustrated in Figure 2.

Fig. 2. Context Ontology

The lattice represented on Figure 3 is obtained after the refinement process we propose. It is more interesting than the original lattice of Figure 1 for recommendation and personalization purposes, as more clusters of applications and context elements are generated, leading to a finer-grained classification. In Figure 1 for example, Flappy bird, SMS, telephone, VDM and Youtube applications have in common the context elements 3G, home and morning, whereas they are separated into 2 concepts in Figure 3: Gmail, VDM and Youtube are associated to 3G and morning, and Flappy bird, telephone, VDM and Youtube are associated to 3G and home. It means that after the refinement strategy, Flappy bird and telephone are no longer associated to the morning (and will therefore not be considered as very likely to be used in the morning). We also notice that some context elements have disappeared from the lattice, as they have no sufficient impact on the user applications: coffee break, restaurant, parents' home and lunch break. It is interesting to remark that the number of concepts in the lattice increases despite the disappearance of 4 context elements. This shows that the precision of the obtained results has indeed increased significantly with regard to the initial lattice.

Fig. 3. Galois lattice obtained after the semantic refinement strategy of the formal context

Interest for Video Games

We have shown previously how the semantic refinement strategy we proposed leads to a better understanding of context impact. Such understanding represents an important step towards personalization and recommendation features. Urban games have many common features with pervasive information systems: users interact with their system in a transparent and mobile way. Examples of context elements that could be relevant in urban games are shown on Figure 4.

Fig. 4. Example of context elements in urban games

The application of the approach we have proposed in the previous section is illustrated on Figure 5.

Fig. 5. Context management applied to urban games

This methodology could be could be interesting in order to make the following actions possible:

1) Analysing gamers' behaviour in similar contexts (Figure 6) For instance, we can imagine that when gamers are located in a specific place and use a specific device, they systematically choose to perform a precise activity, whereas if their context changes (for instance same place but other device), they choose to do something else. Information about gamers' behavior according to their context can be very useful for game designers in order to propose different kinds of services accordingly. We can imagine that the expected behavior is the one obtained with the use of a smartphone, and propose a new clue to tablet users to help them perform the expected activity.

Fig. 6. Analysing gamers' behaviour in similar contexts

2) Analysing the contexts generating a similar behavior (Figure 7)

In a similar way, game designers could be very interested in knowing the contexts leading to the same behavior. Let us consider a key activity in a game, which a set of players has a lot of difficulty to perform. The analysis of the players' contexts may highlight combinations of context elements that confuse these gamers and prevent them from completing the task correctly. As with the previous example, the game designer could then decide to offer a clue to the players who are within these contexts to help them go further in the game.

Fig. 7. Analyzing the contexts generating a similar behavior

3) Recommending specific actions according to the gamer's context (Figure 8)

Context analysis, performed at run time, can also be a way to recommend some activities to the gamers, according to their current context. For instance, taking into account weather modifications, the gamer can be proposed in-door or out-door quests, to keep his interest in the game by playing in comfortable conditions.

Digital Traces Analysis and Intention Mining

Although context management is essential to understand the interactions of users with an information system (and, similarly, the actions performed by players of video games), we believe that it is also necessary to capture the intentions of the users when they perform actions. Indeed, we claim that the actions executed by information system users follow a strategy that aims at satisfying an intention within a specific context. Context management has been addressed in the previous section; we know focus on the identification of users intention through a process we call intention mining, described in Section 3.1.

Process and intention mining: state of the art

Analyzing activities in the Information Technologies (IT) context is a very lucrative business. While surfing on websites, all our activities-products and services searches, on-line game activities, comments and posts on blogs or social networks, among others-are traced and analyzed to propose us the most adapted advertisements.

Games, as other kind of software, leave digital traces of the users that can be observed, stored and analyzed, as illustrated on Figure 9. Video and computer gaming have become increasingly popular branches of the entertainment industry and lots of traces are generated. Due to the large variety of available games, players have a plethora of choices. User selection can be influenced by the content of games, the quality of the services, the popularity of games, the choices of in-game friends, etc. Retaining players with longer player lifetime can yield more revenue for game com-panies. The ways a player interacts with in-game friends, as well as his/her in-game play strategy are also some interesting facts that can be analyzed in the traces. The player behavior, as any user behavior, can also be described as a process, and process can be defined with process models. The approach of tracing and analyzing activities has been used with great success in the field of process mining, in particular for enhancing software development. Processes can also be seen as teleological [START_REF] Ralph | A Teleological Process Theory of Software Development[END_REF]. A teleological process is a process that takes into account the teleological behaviors of process enactment (behavior attached to the notion of goal). It describes the intentions (goals, objectives) associated with a result that an individual wants to obtain. Many research works in intentional process modelling demonstrate that the fundamental nature of processes is mostly intentional and the processes should be modelled from an intentional point of view [START_REF] Davis | User acceptance of computer technology: a comparison of two theoretical models[END_REF]] [ [START_REF] Plihon | Un environnement pour l'ingénierie des méthodes[END_REF] [ [START_REF] Rolland | A multi-model view of process modelling[END_REF]. The notion of context also plays a key role for the intention, since a given intention emerges in a given context, which not only promotes its appearance, but also influences the realization of this intention [START_REF] Rolland | L'ingénierie des méthodes: une visite guidée[END_REF]. Process mining applied to intentional process models is called intention mining, which is at the heart of our solution. This original concept of "Intention Mining" aims at automatically discovering intentions from traces of user activities [START_REF] Hug | Map-TBS: Map process enactment traces and analysis[END_REF] using pattern mining [START_REF] Zaki | SPADE: An Efficient Algorithm for Mining Frequent Sequences[END_REF]. Hidden Markov models technique [START_REF] Baum | Statistical Inference for Probabilistic Functions of Finite State Markov Chains[END_REF]

Potential interest of intention mining in the area of video games

In this section, we describe how the intention mining techniques presented previously could be applied in the context of video games. Player behaviors traces, analyzed with intention mining techniques, could in particular make the following activities possible:

1) Discovery of the underlying player's intentions and strategies hidden in the traces. Intention mining allows identifying whether there exists a generic behavior of the players, through the identification of players' usual activities and the usual ordering of these activities, as shown in Figure 10. For instance, if gamers usually change their direction after finding a clue, this information could be useful for the game designer in order to offer new services adapted to this generic behavior (for instance to automatically display a map when the user enters the clue in its device). Moreover, from a marketing point of view, knowing the gamer's general behavior can add some highlighting information when designing ads for the game. Once the actual behavior of players is identified, it is possible to compare it with the one expected by the game designer, as shown on Figure 11. Some unexpected issues may appear, such as a gamer intention never achieved, a strategy never used, a new intention that was not expected by the game designer, etc.

Fig. 11. Comparison between actual and expected players behaviors

3) Enhancement of the gaming process model

If the observed player's behavior is not consistent with the game designer's goal, e.g., if some paths are never followed, the game designer can add features in the game to guide players towards these paths (see Figure 12). Conversely, if some strategies used by the players are more efficient than the ones expected by the game designer, the expected model can be enriched by adding the identified new behaviors in the model. It is possible to recommend the next activity to a gamer, following the percentage of players that executed this activity when exhibiting the same behavior earlier in the game (see Figure 13). For instance, if the generic behavior consists in exchanging information with other gamers just after a specific step, the device used by the gamer can propose this action to him automatically as the next step option. The game can then become more adaptable to the gamer behavior, at run time. Sections 2 and 3 of this paper are dedicated to the analysis of user context and intentions. These elements allow for personalizing user interaction with their mobile interface and for supporting them with relevant recommendations based on their context intentions and previous experience. We now focus on the adequate way to model the underlying scenarios followed by users during their interaction with their mobile interfaces. We first discuss the challenges of game scenario modeling, then we briefly overview the modeling formalisms used in the domain of software engineering and Business Process Management. Finally, we discuss the interest of statecharts and life sequence charts (LSC) modeling formalisms for the domain of urban games.

Challenges of game scenario modeling

The objective of a multi-player game is to create a virtual or semi-virtual (in the case of urban games which mix virtual and real worlds) environment where players are placed. During the game, players follow some scenarios in order to achieve one or multiple goals through communicating and collaborating with each other. A game scenario needs to specify the objects, their positions in a virtual world (e.g., a maze) and some puzzle to be solved or some task to be carried out. It induces a partially ordered set of actions that the player must perform [START_REF] Gal | Writing for video games[END_REF]. In general, the game scenario is a description of the main game phases and the navigation between them.

The main challenge in game scenario is to remain "unpredictable" for the player while driving this player towards some (probably implicit) goal. Unpredictability is important: first of all, if similar situations occur too often and if the reaction of the game environment and its objects can be predicted, the player may get bored very rapidly. The scenario of the game needs to evolve together with the player.

Unpredictability is extremely important in urban games which mix real and virtual worlds, as the artificial life needs to bear a resemblance to real life. As real life is rich, unpredictable and evolving, the player needs to feel it in the simulated game environment too. Not only the situations encountered by the player should look "unique" but also the opportunities (what the player can and cannot do) in each particular situation should be defined dynamically and not embedded into the scenario. The game scenario therefore needs to unfold progressively, at run-time, and all players can be also considered as scenario designers.

The game scenario thus can be compared to a process, for which the requirements for flexibility and "unpredictability" support are also relevant. Game designers can therefore reuse process modeling languages to model the game scenarios. In the following section, we review existing modeling paradigms used for business process management.

Process modeling: state of the art

Conventional activity-oriented formalisms for process modeling represent a process as a sequence of activities. These formalisms have been mentioned in Section 3. The most popular examples of these formalisms include BPMN and UML activity diagrams. These formalisms are mostly prescriptive and provide only limited support for process adaptability at run-time.

Goal-oriented is another group of modeling formalisms that, compared to activityoriented formalisms, represent a process as a sequence of goals. The MAP modeling notation mentioned earlier in this paper is an example of goal-oriented languages.

Context-driven goal-oriented process models such as [START_REF] Rolland | An approach for de ning ways-of-working[END_REF], [START_REF] Soffer | A state-based context-aware declarative process model[END_REF] and [START_REF] Pohl | A contextual approach for process-integrated tools[END_REF]] support automated recommendations and user guidance, providing that for each goal all the situations (states) in which this goal is achievable are known.

In urban games, we might face unpredictable sequences of events and non-repeatable execution scenarios; it would therefore be hard, if at all possible to model relations between various process situations, goals and activities that must/can be executed in order to achieve these goals.

In the literature, several major perspectives of the process models are specified [Jablonski and Bussler, 1996]: the control flow perspective that captures the temporal ordering of process tasks, events, and decision points; the data perspective that captures the lifecycle of data objects (creation, usage, modification, deletion) within the process; the resource perspective that describes how the process is carried out within the organization and deals with roles and resource assignments; the operational perspective that addresses the technical aspects of process execution and specifies the elementary process tasks and their assignment to concrete applications or application components of the organizations; the context perspective that describes the attributes related to the process execution context; the performance perspective, addressing the process cost effectiveness.

Flexibility along control flow, data and resource perspectives is widely addressed: modeling languages (e.g., BPMN, EPS, C-iEPC, Declare) enable the specification of processes where the activities can be skipped, repeated, or their ordering can be chosen depending on the context (at run time). Some languages (e.g., configurable models in C-iEPC [La [START_REF] Rosa | Configurable multiperspective business process models[END_REF]) allow for dynamic resource and data management. Flexibility along the context and the operational perspectives, however, are still challenging.

Flexibility along the operational perspective would allow the player, for example, to not only select an activity to execute from the predefined list, but also to propose an alternative that was probably not even thought of at design.

To search for a formalism that would support such flexibility, we appeal to the domain of complex reactive systems.

According to [START_REF] Harel | Crafting game-models using reactive system design[END_REF], the design of games and complex reactive systems share a common set of challenges: 1) The need to construct a system from a set of many interacting small units; 2) The importance of emergent properties, unique characteristics of the global system behavior that emerge from the interaction between the small units but are not always evident from the rules of behavior of each of the components by its own; 3) The crucial role of visualization and graphics; and 4) the need to provide strong tool support that frees the mind and encourages creativity of the designers, allowing them to focus on the big picture and the important parts rather than on implementation details.

In the games where multiple players are placed in virtual reality (i.e., an "artificial life"), each "object" or "actor" of the game is specified as a random automaton sending and receiving stimuli from other objects. The global behavior is the results of the interactions between local ones [START_REF] Gal | Writing for video games[END_REF]. This allows for simulating complex behaviors and avoiding repetitive scenarios.

For the specification of unpredictable and flexible game scenarios, the state-oriented modeling paradigm and the formalisms of statecharts [Harrel, 1987] can be beneficial.

Applying the statecharts modeling formalism to game processes

Within the state-oriented paradigm, a process (a game in our case) can be described by a set of states and transitions between these states. A game starts at an initial state and terminates at a final state. A state transition is triggered when some condition is fulfilled. The sequence of states and transitions that leads from the initial state to the final state can be seen as a game scenario.

Activities carried out by a player depend on the current state of the system (the game environment in our case) and the game scenario is adapted at run time, according to the evolution of the game.

Examples of state-oriented modeling formalisms include state machines in UML [START_REF] Rumbaugh | Unifed Modeling Language Reference Manual, The (2Nd Edition)[END_REF], generic state-transition systems or state machines, such as FSM [START_REF] Plotkin | A structural approach to operational semantics[END_REF] or Petri Nets [START_REF] Murata | Petri nets: Properties, analysis and applications[END_REF], and statecharts by D. Harel [START_REF] Harel | Statecharts: A visual formalism for complex systems[END_REF] created for the specification and analysis of complex discrete-event systems.

The core difference between state-oriented (and goal-oriented) formalism is the way they specify the transition between states (or goals) within a process: whereas some formalisms (e.g. Petri Net) state that a transition is a result of an execution of an activity, the others (e.g., statecharts) state that a transition is a result of a triggering event.

In the first case, all activities need to be explicitly modeled and thus these formalisms do not provide much flexibility compared to activity-oriented formalisms mentioned above. Whereas in the second case, only the (desired or undesired) triggering events have to be specified. The activities that "produce" these events may remain unknown and can be defined at run time. This provides a supplementary degree of flexibility along the operational perspective required by urban games.

Following this paradigm, a game scenario specification can be divided into two parts: the state-transition part, defined with a set of states and transitions between states and their triggering events, and the activity part, defined by a list of activities specified by their preconditions and outcomes. The process enactment can be seen as a dynamic selection of activities to produce some outcomes (events) that make the process progress towards its (desired) final state.

In [START_REF] Kushnareva | Modeling Business Processes for Automated Crisis Management Support: Lessons Learned[END_REF], we proposed the basis for implementing the state-oriented formalism of statecharts for modeling case management processes, where the case management scenario, similarly to a game scenario, depends a lot on the context and requires flexibility. In [START_REF] Kushnareva | Modeling Business Processes for Automated Crisis Management Support: Lessons Learned[END_REF], we elaborated on this basis, proposing an approach for interactive modeling and simulation-based analysis of process specifications for crisis management. Again, crisis handling can be easily considered as a subject of a game. This approach is based on statecharts formalism and YAKINDU statecharts tools 4 .

The state-oriented paradigm allows us to exclude activities from the process design: we can state that "any activity is good as soon as it produces a desired outcome". In particular, it enables deferred activity planning, that gives more freedom to the process participant in choosing an activity that is adapted for a concrete situation.

The system based on statecharts model cannot be considered as a "management" system that automatically defines and executes the scenarios, but rather a "recommendation" system that only advises the process participant on scenario planning but not drives him/her through a sequence of predefined steps. That is why this formalism is also beneficial for game design. In our vision, the "a priory" statecharts specification of a game can be analyzed using graph theory algorithms. The objective of the analysis is to search and optimize a path from some current state of the statecharts model to its target state, representing the goal of the game. As a result, the "best next state to visit", "best next transition to fire" and, consequently, "possible activities to execute" are recommended to the player.

The state-oriented model will address the internal behavior of objects involved in the multi-player game in a simulated virtual world. The internal behavior of an object describes how this object can react on any external stimuli in any context (i.e. situation).

This model can be complemented by a model of inter-object behavior [START_REF] Harel | Crafting game-models using reactive system design[END_REF]] that describes how this object interacts with other objects in particular situations. In [START_REF] Harel | Come, let's play: Scenario-based programming using LSCs and the Play-Engine[END_REF]], the inter-object behavior is modeled with scenarios that specify the interactions between objects.

In [START_REF] Harel | Crafting game-models using reactive system design[END_REF], the authors state that "the approach makes it possible to describe behaviors of different parts of the system using different reactive system design languages and tools. Among the advantages of this approach is the ability to use existing analysis tools to understand the game behavior at design time and run time, the ability to easily modify the behavior, and the use of visual languages to allow various stakeholders to be involved in early stages of building the game."

Let us illustrate our approach on a simple example inspired from urban games. Figure 14 represents a statechart that specifies a (gamer's view on a) game with a state machine defined at three hierarchical levels. On the higher level, only two states are defined: Disconnected and Connected. The transitions between these states are triggered by the events produced by a gamer: e1when the gamer connects to the game, and e2when he/she disconnects.

Fig. 14. Example statechart inspired by urban games

When connected (the second hierarchical level), the gamer can be in one of the three (exclusive) states: Navigation, when the player moves in a physical environment (e.g., walking on the street or visiting buildings, as shown on the third hierarchical level) with an objective to pursue another player, to reach some specific place defined by a mission or to search for a clue; Communication, when the player meets an actor or another player with whom he/she can communicate in order to search for a clue; Exploration, when the gamer explores the environment around him/her (e.g., a building) in order to search for a clue.

The transitions between these states are defined by logical expressions over events (e1, e2, …) . The list of events is presented in Fig. 15. Events can occur as a result of a player's actions (these events are prefixed with Player in the diagram) or can be produced by the game environment (these events have the Game prefix).

A black circle in the diagram identifies a substate that will be entered "by default" when its parent state is visited: for example, when the Navigation state is entered for the first time, its substate "On_the_street" is activated.

A black circle with a H* symbol means "entering the state by history": for example, once the gamer leaves the Communication or the Exploration state, he/she returns to his/her previous position in the Navigation state.

Fig. 15. Events list

The main idea is that the activities that the gamer can execute are not specified at all by this model. It only specifies the events that can trigger state transitions. The player is free to perform any activities, assuming that they will lead to the generation of a desired event. For example, in the Navigation state, the gamer can walk, run, stop, turn, take a metro, ride a bicycle etc. Along these lines, while in the Exploration state, the gamer can use any means at his/her disposal in order to search for a clue. This means that concrete activities can be defined by the player based on particular situation. This is especially important for urban games, where the game is taking place in a real environment.

Obviously, this example is very simple. When the scenario becomes more complex, more specific events associated with the game mission will be defined and the player behavior will be specified with more details using new hierarchical levels.

The statechart represented on Figure 14 has modeled with the YAKINDU open source environment for modeling and simulating statecharts. This model can be simulated and refined, providing more details about the game: new hierarchical levels, states, events, and transitions can be specified.

Conclusion and Perspectives

We have presented how context management, intention mining and state-oriented adaptive process modeling could be beneficial to game process analysis and modeling, in order to better understand the behavior of players during their gaming experience and provide them with relevant recommendations. This section first summarizes the contributions presented in this article. Finally, as the recent advances presented in this article have been developed independently, we present one of our perspectives for future work, which consists in integrating them into a single solution.

Conclusion: overall approach

The overall objective of the various contributions presented in this article is to provide users of an information system with relevant recommendations, which take into account their context and their intentions, as illustrated on Figure 16. We propose to build a recommendation system based on the model of user behavior using a stateoriented modeling formalism that integrates context information and intentions of users. User intentions are extracted from traces with intention mining algorithms, whereas context management relies on Formal Concept Analysis. The obtained statecharts can then be used in order to identify the next best state for the process according to the current situation of the user and recommend the most efficient path towards this next best state. [START_REF] Russel | Artificial Intelligence: A Modern Approach[END_REF], in which an intelligent agent perceives its environment through sensors, and analyzes the obtained information to determine actions to execute on the environment. In the MAPE-K loop, the adaptive system represents any software or hardware resource that can be adapted to better responds to the changing context. Thus, the adap-tive system can for example be an operating system, a specific software component in an application (e.g., the query optimizer in a database), a cluster of machines in a grid environment, a wired or wireless network, a middleware, but also an adaptive game.

Fig. 17. MAPE-K loop architecture

The elements of the MAPE-K loop and their interactions are presented as follows:

-The monitor function collects the details from the internal and external managed resources. For instance, the managed resources in an urban game can be metrics (e.g., offered capacities or powers, throughput of the network for distributed games, and score), configuration property settings and so on. In addition, the monitor function periodically aggregates, correlates, filters this information and sends it to the analyzer. With regard to the contributions presented in this article, the monitor is responsible for context management and the monitored resources are those that appear in the context ontology. -In our case, the analyzer function performs the intention mining. Moreover, it evaluates whether changes are required in the adaptive system (i.e., whether recommendations are needed in the game) according to gamer intentions and to the information provided by the monitor function. Usually, changes are required when the current system does not satisfy the constraints provided by the monitor or when a better solution exists that satisfies these new constraints. In our case, the analyzer is in charge of finding the next best state in the statecharts process model, relying on context information and on gamer intentions (which are part of the knowledge). If changes are required in the system (i.e., in the game), a change request is sent to the planner. -The planner function structures the actions needed to achieve goals and objectives. The planner creates or selects a procedure to enact a desired alteration in the managed resources. The planner's role is to identify the path that should be taken in order to reach the next best state. Thereafter, the plan is passed to the executor. -The executor function changes the structure or the behavior of the adaptive system (the game), and sometimes also of the adaptive context [START_REF] Sawyer | Constraint Programming as a Means to Manage Configurations in Self-Adaptive Systems[END_REF],

according to the information provided by the planner. In our case, it consists in actually providing users with recommendations. -The knowledge base contains the data shared by the monitor, analyzer, planner and executor functions. This shared knowledge, is created/updated by the monitor, analyzer and executor. In our case it gathers in particular the context ontology, and the behavior model.

Fig. 8 .

 8 Fig. 8. Recommending specific actions according to the gamer context

  Tracing the gaming activities of a user can then become the source of a lot of knowledge. Analyzing game traces allows understanding the characteristics of games such as player behaviour [Suznjevix et al, 2011][(Ducheneaut et al, 2007], traffic analysis [Chen et al, 2006][Kinicki et al 2008], resource management [Chambers et al, 2010][Denault et al, 2011], etc. Traces can be stored in a specific repository as the Game Trace Archive 2 which allows analyzing large and public game traces datasets [Guo et al, 2012].

Fig. 9 .

 9 Fig. 9. Digital traces capture

  [Weijters and van der Aalst, 2003] have proposed methods and tools based on Petri nets and α type algorithm to analyze events logs (traces) in order to discover process models. Process mining approaches aim at modelling users' behaviors in terms of sequences of tasks and branching in an automatic way [Van der Aalst et al., 2004] [Agrawal et al., 1998] [Cook and Wolf, 1998] [Datta, 1998] [van Dongen, 2004] [Herbst, 2000]. The resulting (i.e., mined) process models are activity-oriented. However, activity-oriented models lack flexibility 3 .

Fig. 10 .

 10 Fig. 10. Extraction of a general player behavior

Fig. 12 .

 12 Fig. 12. Refinement of the player process model

Fig. 13 .

 13 Fig. 13. Recommendation to players based on context and intention

Fig. 16 .

 16 Fig. 16. Overall approach

Table 1

 1

	Applications	University	Restaurant	Parents' home	Home	Transportation	3G	Morning	Coffee break	Lunch break	Afternoon	Evening
	Gmail	1 (2)	0	0	0	1	1 (2)	1	0	0	1	0
	SMS	1 (80)	1	1 (5)	1 (30)	50	1 (170)	1 (40)	1 (20)	1 (40)	1 (60)	1 (10)

. Example Formal Context (the values in parentheses correspond to the number of times the application was used in each context).

  ], a specific type of Dynamic Bayesian Networks, was used in the Map Miner Method[START_REF] Khodabandelou | Mining Intentional Process Models[END_REF] [START_REF] Khodabandelou | Mining Intentional Process Models[END_REF]. Moreover,[START_REF] Epure | What Shall I Do Next? Intention Mining for Flexible Process Enactment[END_REF] proposed an approach using an unsupervised technique and a knowledge tree to build intentional process models at a low level of abstraction. Information about user intentions within a specific context, together with knowledge of other user behaviors with the same intentions in similar contexts is the key to provide them with adapted recommendations.

This data comes from a dataset built from a questionnaire filled by

master students of our university. The goal of this survey was to know in which context they had been using applications (such as social networks, games, emails, etc.) on their tablets during a week.

http://gta.st.ewi.tudelft.nl

We will further discuss process models in Section

Yakindu Statechart Tools version 2.4. for Eclipse Luna, 2014, http://statecharts.org/