Abstract :

Thermomechanical behaviour of NiTi SMA is due to the complex interaction between several deformation mechanisms including lattice elasticity, phase transformations between austenite (A), martensite (M) and Rphase (R), reorientations of R and/or M variants, twinning and detwinning phenomena and the usual plasticity due to dislocation movements. In this work, thermomechanical tensile behaviour of a commercial Ti-50.8at% Ni alloy wire of 0.

F E' S ε = (1)
Tous les essais ont été réalisés avec f=100Hz et ε dyn =0.1%, choisi assez faible pour pouvoir avoir des mesures de E', en début d'essai, à faible déformation statique (ε stat >ε dyn ). Des points caractéristiques sont indicés de (1) à (8). Lors de l'étape initiale de déformation jusqu'au point 2, considérée usuellement comme la déformation élastique de la phase austénitique, on constate une forte variation du module E'. Ceci est expliqué par l'intervention dès ce stade de déformation de mécanismes autres que l'élasticité du réseau cristallin. Dans ce stade initial de déformation précédant le plateau, lors d'essai de traction sur tubes, Favier et al [START_REF] Favier | Homogeneous and heterogeneous deformation mechanisms in an austenitic polycrystalline Ti-50.8at%Ni thin tube under tension. Investigation via temperature and strain fields measurements[END_REF] ont montré par mesures couplées de champs cinématiques et thermiques qu'une partie importante de la déformation était due aux transformations de phase A→R et A→M, débutant bien avant le plateau. Ces transformations créent des interfaces dont les mouvements pendant les mesures dynamiques provoquent une diminution du module de conservation et une augmentation du module d'amortissement [START_REF] Van Humbeeck | Damping capacity of thermoelastic martensite in shape memory alloys[END_REF]. Lorsqu'une transformation A-R a lieu entre le début de la charge et le plateau, le point (1) sera utilisé pour caractériser cette transformation A→R. Il est connu que le plateau de déformation est associé à des phénomènes de localisation de déformation [START_REF] Favier | Homogeneous and heterogeneous deformation mechanisms in an austenitic polycrystalline Ti-50.8at%Ni thin tube under tension. Investigation via temperature and strain fields measurements[END_REF] ; les points (2) et (3) correspondent aux début et fin de cette localisation à la charge. Entre les points (3) et (4), le matériau est constitué de martensite orientée et d'austénite résiduelle qui continue à se transformer [START_REF] Favier | Homogeneous and heterogeneous deformation mechanisms in an austenitic polycrystalline Ti-50.8at%Ni thin tube under tension. Investigation via temperature and strain fields measurements[END_REF]. Le début de la décharge entre (4) et (5) a lieu de manière homogéne et implique immédiatement en plus de l'élasticité du réseau cristallin un mécanisme de déformation impliquant la transformation inverse (M orientée)-A. Entre les points ( 5), ( 6), (7), la déformation est localisée, due en grande partie à la transformation inverse M→R. Entre (7) et la fin de la décharge, la déformation est homogène, mais n'est pas uniquement due à de l'élasticité de l'austénite ; le point (8) est utilisé pour caractériser la transformation inverse R→A.

Description de l'essai balayage en température et DMA

Les figures 4a) et 4b), représentent les valeurs de E' pour l'ensemble des essais respectivement durant le déchargement et le déchargement auxquelles ont été superposées les valeurs des points caractéristiques identifiés sur les différentes courbes isothermes.

Résultats des essais balayage en température et DMA

Des essais ont été réalisés à différentes contraintes 50MPa, 100MPa, 200MPa, 300MPa, 390MPa, 490MPa, 590MPa, 790MPa et 990MPa. La figure 5, présente les résultats sous 300MPa. La courbe supérieure représente la déformation en fonction de la température, tandis que la courbe inférieure représente le module de conservation E' en fonction de la température. Sur cette courbe ont été placés un certain nombre de points caractéristiques, identifiés de (1') à (11'). Lorsque T diminue, E' diminue tout d'abord régulièrement entre (1') et (2'), puis beaucoup plus fortement jusqu'à atteindre un minimum en (3'). La baisse importante de E' entre (2') et (3') est associée à une déformation sous contrainte, due à une première transformation de phase de type A→R. Ensuite E' augmente et atteint un maximum en (4') puis subit une chute importante en (5'). La chute entre (5') et (6') correspond à une deuxième transformation de type R→M confirmée par la grande déformation correspondante ; des travaux supplémentaires sont à effectuer pour confirmer que cette déformation entre (5') et (6')

Conclusion

Cette étude expérimentale par une méthode originale, a permis de cartographier sur les valeurs de module de conservation en fonction de la température et de la contrainte. Après avoir identifié les domaines d'existence des différentes phases par une analyse du module de conservation, les modules de conservation et leurs évolutions ont pu être proposés pour chacune d'entre elles. Il apparait que les transformations de phases influencent les valeurs des modules de conservation de manière très importante y compris loin des lignes de transformation.

FIG. 1 - 3 Résultats 3 . 1

 1331 FIG. 1 -Principe de mesure lors d'un balayage statique + DMA

FIG. 3 -FIG. 4 -

 34 FIG. 3 -Résultats d'un essai balayage statique + DMA réalisé à 20°C a) b)

FIG. 5 -FIG. 6 -FIG. 7 -

 567 FIG. 5 -Résultats d'un essai de balayage en température + DMA réalisé sous une contrainte de 300MPa a) b)

1 Etude du comportement thermomécanique d'un fil fin de NiTi par spectromètre dynamique mécanique et thermique T. Alonso a , D. FAVIER a , G. Chagnon a

  Les alliages à mémoire de forme de type NiTi présentent des phénomènes de déformation complexes liés aux transformations entre les différentes phases Austénitique (A), Martensitique (M) et phase R (R). Ces phénomènes sont observés à l'aide de nombreux dispositifs expérimentaux ayant pour objectif de déterminer les caractéristiques nécessaires pour la modélisation et la simulation. Ce travail présente les résultats expérimentaux issus d'une analyse par un dispositif non conventionnel, qui

	a. Université de Grenoble Alpes/CNRS
	Résumé :

couple un spectromètre mécanique et thermique, plus connu sous le nom DMTA (Dynamical Mechanical and Thermal Analysis) à une machine de traction classique. Des mesures du module de conservation (E') ont été effectuées pendant deux types d'essais sur des fils de NiTi à grains nanométriques. Les essais du premier type sont des tractions isothermes pour des températures allant de -100°C à +150°C, en charge/décharge et en charge jusqu'à rupture. Les essais de second type sont effectués sous contrainte constante s'échelonnant de 50 MPa à 1000MPa, au chauffage et refroidissement entre 200°C et -140°C. A partir de l'évolution de E' sur l'ensemble des essais, il est possible de détecter les changements de phase par une analyse des variations de E'. Il est ainsi possible de déterminer les domaines d'existence des différentes phases, séparés par les lieux de transformations directes et inverses. De plus, il a été effectué une

analyse de l'évolution de E' en fonction de la température et de la contrainte de traction dans les domaines où l'alliage est monophasique, totalement austénitique ou martensitique. Les évolutions de E' sont importantes. Pour l'austénite, E' augmente avec la température, contrairement aux alliages métalliques usuels.

Mots clefs: Shape Memory Alloy, NiTi, Young Modulus, Austenite, Martensite, R-Phase 1 Introduction

  Habituellement les mesures DMTA sont effectuées lors de refroidissement et réchauffement sous faible contrainte[START_REF] Besseghini | DMA characterization of a Ni50.5at%Ti shape memory alloys[END_REF][START_REF] Cai | Damping behavior of TiNi based shape memory alloys[END_REF][START_REF] Chang | Damping Characteristics of Cold-Rolled and Annealed Equiatomic TiNi Shape Memory Alloy[END_REF][START_REF] Silva | Dynamic Properties of NiTi Shape Memory Alloy and Classic Structural Materials: A Comparative Analysis[END_REF][START_REF] Liu | Apparent modulus of elasticity of near-equiatomic NiTi[END_REF][START_REF] Roy | Mechanical characterization of NiTi SMA wires using a dynamic mechanical analyzer[END_REF] .Ce travail présente des résultats issus d'une étude utilisant une DMTA un peu particulière car elle est combinée à une machine de traction classique. Ce dispositif permet entre autres de réaliser deux types d'essai, d'une part des essais où l'évolution du module de conservation (E') est mesurée lors d'un essai de traction isotherme, d'autre part des essais où l'évolution de E' est mesurée lors d'un balayage en température sous contrainte constante. L'association de ces deux types d'essais permet de connaître intégralement le domaine d'existence des différentes phases en fonction de la contrainte de traction et de la température, mais aussi de déterminer l'évolution du module élastique de conservation pour chacune des phases en fonction de la température et de la contrainte.

	alors que les modules élastiques macroscopiques -des phases (A) et (M) -et les transformations de phases
	induites par la contrainte sont déterminés par des essais de tractions isothermes. Plus finement les modules
	élastiques locaux (module de conservation) peuvent être déterminés par un spectromètre dynamique
	mécanique (DMTA). 2 Dispositif expérimental et procédures
	2.1 Matériau
	These tests are analysed to study dependences of phase transformations on stress and temperature and to
	bring new information concerning the value of the elastic moduli in the different stress-temperature regions.
	Les alliages à mémoire de forme de type NiTi quasi-équiatomique font apparaître des mécanismes de
	déformation complexes incluant l'élasticité du réseau cristallin, des transformations de phases solides entre
	austénite (A), martensite (M), R-Phase (R), des phénomènes d'orientations/réorientations, de maclages et
	démaclages, ainsi que des mécanismes usuels de plasticité. La conception, la modélisation et la simulation de
	ces alliages nécessitent de connaître le comportement thermomécanique et de déterminer de nombreux
	paramètres, notamment les températures et les contraintes introduisant les transformations de phases ainsi
	que les modules élastiques des différentes phases. Classiquement, les températures de transformations de
	phases sous contrainte nulle sont déterminées à l'aide de la calorimétrie différentielle à balayage (DSC),

1 mm in diameter was investigated by means of a device which combines a conventional tensile device with a dynamic mechanical and thermal analyser (DMTA). This nonconventional tensile device permits to study storage modulus (E') as functions of tensile stress, strain and temperature history. First, E' was measured during usual isothermal tensile tests over a wide testing temperature range between -100°C and 150°C. Second, E' was measured during thermal cycling under a constant tensile stress between 50 and 1000 MPa over wide testing temperature range between -150 and 250°C.

Les essais ont été effectués sur un fil Ti-50.8at%Ni de diamètre 0.1mm (référence : FWM NiTi#1-SE) commercialisé par la société Fort Wayne Metals

[7]

. Ce fil de faible diamètre composé de grains nanométriques possède d'excellentes caractéristiques mécaniques, d'une part une très bonne super élasticité à température ambiante avec 0.5% de déformation résiduelle après un essai de traction à 10% et d'autre part une résistance à la rupture de 1600MPa. De par sa faible plasticité, ce fil est bien adapté à l'étude des modules d'élasticité des différentes phases.

2.2 Spectromètre dynamique mécanique et thermique

Tous les essais ont été réalisés avec un spectromètre dynamique mécanique et thermique (DMTA) Eplexor® 500N

[8]

. Ce dispositif non conventionnel est composé d'une unité supérieure qui permet d'appliquer une charge statique, en étant pilotée en déplacement/déformation ou en force/contrainte, et d'une unité inférieure qui permet d'appliquer les paramètres dynamiques avec une fréquence de 1Hz à 100Hz, en étant pilotée en déplacement/déformation ou en force/contrainte. Une enceinte thermique permet de réaliser les essais sur une gamme de température comprise entre -150°C et +500°C. Deux types d'essais ont été réalisés, d'une part des essais isothermes avec des mesures dynamiques lors d'un essai de traction que l'on nommera balayage statique + DMA, d'autre part des essais avec des mesures dynamiques lors d'un refroidissement puis réchauffement à contrainte constante que l'on nommera balayage

température + DMA. 2.2.1 Description de l'essai (balayage statique + DMA) Cet

  Puis un cycle dynamique est réalisé, consistant en une déformation dynamique sinusoïdale centrée sur ε stat d'amplitude ε dyn avec une fréquence f. Durant ce cycle dynamique, la force dynamique correspondante F dyn est mesurée, ce qui permet de déterminer d'une part le déphasage δ entre ε dyn et F dyn et d'autre part, le module de conservation E'. Pour une éprouvette de section S 0 , E' représentant la partie élastique du comportement est donné par

	dyn
	0 dyn

essai, représenté Figure

1

, consiste à faire des mesures de DMA tout au long d'un essai en traction isotherme. L'essai de traction est contrôlé en déformation dite statique. Après chaque incrément de déformation statique de pas ∆ε stat , la déformation statique ε stat et l'effort statique correspondant F stat sont mesurés.

  Le point (10') est considéré pour caractériser la transformation R→A. Les figures 6a) et 6b), tracées dans le plan contrainte de traction-température, représentent les lieux des changements de phases déterminées par l'analyse des courbes sous différentes contraintes. Ces lieux sont raisonnablement linéaires, comme prévus par la relation de Clausius-Clapeyron. Les droites ont alors été prolongées en traits pointillés jusqu'à contrainte nulle. La courbe noire issue de la mesure par Differential Scanning Calorimetry (DSC) a été superposée, et permet de valider la transformation A→R, R→A détectée avec les mesures issues de la DMA. La limitation à -100°C de la DSC utilisée n'a pas permis de détecter la transformation R→M.Les cartes d'isovaleurs de E' issus des essais de balayage statique (figure4) et de balayage en température (figure7) montrent que même loin des lieux des transformations de phase, le module de conservation E' évolue fortement. A titre d'exemple, pour un matériau à haute température et faible contrainte que l'on peut considérer comme totalement austénitique, E'(Austénite) est mesuré sur la Figure4augmentant de 75 à 80GPa lorsque par exemple T atteint 90°C ; il atteint 85GPa sur la figure 7 à T=200°C. Cette augmentation de E' de l'austénite avec la température, non usuel pour les matériaux métalliques, doit être investiguée plus en détail. Les mesures des modules E' des phases R et martensite M sont du fait du processus expérimental des modules de phase R ou M orientées. Des essais sont en cours pour mesurer ces mêmes modules sur des matériaux avec de la martensite auto-accomodée. Dans ce cas, seule l'anisotropie du fil initial interviendra.

	Les figures 7a et 7b représentent les valeurs de E' pour l'ensemble des essais respectivement durant le
	refroidissement et le réchauffement auxquelles ont été superposés les domaines d'existence des différentes
	phases.
	4 Détermination des modules d'élasticité des différentes phases

est localisée. E' augmente à nouveau lors du refroidissement entre (6') et (7'). Lors du réchauffement E' subit une augmentation importante entre (8') et (9') qui correspond à la transformation inverse certainement localisé M→R.


Miroslav Sulc

Pierre Pugnat

Rafik Ballou

Guy Deferne

Lionel Duvillaret

Lucie Flekova

Michael Finger

Jan Hošek

Thomas Husek

Axion Search by Laser-based Experiment OSQAR

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Emmanuel Bernuau

Denis Efimov

Wilfrid Perruquetti

Analysis of scale invariance property applying homogeneity

The problem of scalability of trajectories in homogeneous and locally homogeneous systems is considered. It is shown that the homogeneous systems have scalability property, and locally homogeneous systems possess this property approximately. The issue of closeness of trajectories for the system and its local homogeneous approximation is investigated. The notions of scale invariance and fold change detection are also studied. The results are illustrated by a cascade system analysis.

INTRODUCTION

Many biological systems demonstrate an adaptation ability, i.e. they are capable to maintain the levels of essential variables in the admissible bounds in the presence of wide deviations of the range of inputs [START_REF] Alon | An Introduction to Systems Biology: Design Principles of Biological Circuits[END_REF]. In some cases they even demonstrate a more strong property called (approximate) scale invariance, when the output of the system becomes (almost) the same after the input scaling [START_REF] Shoval | Symmetry invariance for adapting biological systems[END_REF], [START_REF] Skataric | A characterization of scale invariant responses in enzymatic networks[END_REF], [START_REF] Hamadeh | Transient dynamic phenotypes as criteria for model discrimination: fold-change detection in rhodobacter sphaeroides chemotaxis[END_REF]. Constructive conditions of presence of this property are demanded in systems biology.

The conditions of scale invariance obtained in [START_REF] Shoval | Symmetry invariance for adapting biological systems[END_REF], [START_REF] Skataric | A characterization of scale invariant responses in enzymatic networks[END_REF], [START_REF] Hamadeh | Transient dynamic phenotypes as criteria for model discrimination: fold-change detection in rhodobacter sphaeroides chemotaxis[END_REF] are based on some kind of symmetry verification and linearization. It is shown that if a system is symmetric, then under some additional minor restrictions it has the scale invariance property. Roughly speaking a system admits the scale invariance property if its transient behavior stays approximately the same when the input signal level is scaled. Such a type of dynamical behavior is a kind of symmetry. Homogeneity is a concept, which has been introduced in order to study the influence of symmetry on behavior (stability, robustness etc.) of solutions of nonlinear systems [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]. Homogeneity is also a kind of symmetry. Verification of the presence of this type of symmetry is rather simple since it is based on arithmetic operation and/or algebraic tests.

The objective of this note is to investigate the scale invariance property using the homogeneity framework. First, it is shown that the results on the presence of scale invariance under a generic symmetry of the differential equations describing the system can be easily re-casted in the homogeneity framework. Second, it is shown how local homogeneity implies approximate scale invariance in the system. Several examples demonstrating efficiency of the homogeneity application for scaling analysis are given.

The outline of the presentation is as follows. In Section 2 some preliminary results and definitions of homogeneity and scale invariance are presented. In Section 3 the link between homogeneity and scalability is established. The relation between approximate scalability and local homogeneity is investigated in Section 4.

PRELIMINARIES

Notation

In this paper, the following notation will be used.

• We denote R + = {x ∈ R : x ≥ 0} and R * + = R + \ {0}. If E ⊂ R we set E + = E ∩ R + and E * + = E ∩ R * + . We denote N = Z + and N * = Z + ∩ R * + . • If n ∈ N * , we denote Lip(R n ) the set of locally Lipschitz continuous vector fields on R n . • Denote Ψ the flow of a F ∈ Lip(R n ), then Ψ t (x)
represents the state of the trajectory of F at time t with the initial condition x, i.e. Ψ 0 (x) = x. We denote

d dt Ψ t (x) the time derivative of this curve ( d dt Ψ t (x) = F (Ψ t (x)) by definition) and d x Ψ t the differential at x of the diffeomorphism x → Ψ t (x), t being fixed. • If f is a C 1 vector field on R n , we denote Jacf (x) the Jacobian matrix of f taken at the point x ∈ R n . • For two C 1 vector fields f, g : R n → R n , denote their Lie brackets [f, g] = ∂g ∂x f (x)-∂f ∂x g(x). For a C 1 vector field f : R n → R n and a C 1 function V : R n → R the Lie derivative of V along the vector field f is defined as L f V = ∂V ∂x f (x). • Given matrices A k ∈ R n×n , 1 ≤ k ≤ K, K ∈ N * ,
define the convex hull of these matrices

Co(A 1 , . . . , A K ) ={A ∈ R n×n : A = K k=1 α k A k , α k ∈ [0, 1], K k=1 α k = 1}.
• For a symmetric matrix A ∈ R n×n denote λ max (A) and λ min (A) the biggest and smallest eigenvalues respectively.

• If A and B are two subsets of R n , we denote A + B = {a + b, a ∈ A, b ∈ B}. • For a measurable function u : R + → R m define the norms u L 1 [a,b] = b a |u(t)|dt and u L ∞ [a,b] = sup t∈[a,b] |u(t)| for any 0 ≤ a ≤ b ≤ +∞.

Homogeneity concepts

In order to use the symmetry that some systems present, the weighted homogeneity has been introduced [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF].

Definition 1. Set r = (r 1 , . . . , r n ) with r i ∈ R * + .
A dilation associated to the generalized weight r is a group action:

λ ∈ R * + → Λ r (λ) with Λ r (λ) : x → (λ r1 x 1 , . . . , λ rn x n ).
The mapping Λ r (λ) is often simply denoted by Λ r (the λ is omitted). Definition 2. A function V : R n → R (resp. a vector field F : R n → R n ) is said to be r-homogeneous of degree d iff for all λ > 0 and for all x ∈ R n we have λ

-d V (Λ r x) = V (x) (resp. λ -d Λ -1 r F (Λ r x) = F (x)
). Note that a vector field F = (F 1 , . . . , F n ) is r-homogeneous of degree d iff for each i ∈ {1, . . . , n}, the function F i is r-homogeneous of degree r i + d. The homogeneous norm can be defined as follows

N (x) = n i=1 |x i | ρ/ri 1/ρ , ρ ≥ max 1≤i≤n r i , this norm is r-homogeneous.
Being a kind of symmetry, the homogeneity property should be invariant under a change of coordinates. But this is not true for the weighted homogeneity. This motivates for a geometric, coordinate-free definition of homogeneity [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF]. Definition 3. A vector field ν : R n → R n is said to be Euler if ν is C 1 and the origin is Globally Asymptotically Stable (GAS) for -ν. In the sequel, we will restrict ourselves to complete Euler vector fields. We will always write Φ the flow of ν. Definition 4. Let ν be an Euler vector field. A function V is said to be ν-homogeneous of degree d iff for all s ∈ R we have: e -ds V (Φ s (x)) = V (x). A vector field F is said to be ν-homogeneous of degree d iff for all s ∈ R we have:

e -ds (d x Φ s ) -1 F (Φ s (x)) = F (x).
It is straightforward to check that considering the generalized weight r = (r 1 , . . . , r n ), with r i > 0 and setting ν

(x) = r 1 x 1 ∂ ∂x1 + . . . + r n x n ∂
∂xn in a fixed basis, a function or a vector field will be r-homogeneous iff it is νhomogeneous with the same degree.

Proposition 5. A C 1 function V is ν-homogeneous with degree d iff L ν V = dV . A C 1 vector field F is ν- homogeneous with degree d iff [ν, F ] = dF .
The main feature of homogeneous systems (i.e. systems defined by a homogeneous vector field) is the following proposition about a kind of symmetry of trajectories, or a commutation property of flows [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF]. Proposition 6. Let F ∈ Lip(R n ) and denote by Ψ its flow. Then F is ν-homogeneous of degree d iff

Φ s • Ψ e ds t = Ψ t • Φ s ∀s, t ∈ R. (1)
Due to this result, if a homogeneous system is locally asymptotically stable (attractive), then it is GAS.

Example 1. Consider the following scalar system:

ẋ = - ax b + |x| ,
where a > 0 and b > 0 are parameters. Obviously, this system is not homogeneous with respect to conventional Definition 2. However, it is straightforward to verify using the last proposition that it is homogeneous in the sense of Definition 4 with

d = 1 for ν(x) = b -1 x(b + |x|).

Scalability and scale invariance

In this section, n, m ∈ N * . Consider now a system ẋ = f (x, u),

y = h(x), (2)
where x ∈ R n is the state, u ∈ U ⊂ R m is the input and y ∈ R is the output, f : R n+m → R n is locally Lipschitz continuous. The set U is the set of admissible inputs, namely a subset of the measurable inputs from R to R m . We assume that U is invariant with respect to a positive scaling. We assume that for any ū ∈ R m , there exists a unique σ(ū) ∈ R n such that f (σ(ū), ū) = 0. Let us denote by x(t, x 0 , u) the current state of the system under input u and with initial condition x 0 . Finally, we denote y(t, x 0 , u) = h(x(t, x 0 , u)).

Following [START_REF] Shoval | Symmetry invariance for adapting biological systems[END_REF], [START_REF] Skataric | A characterization of scale invariant responses in enzymatic networks[END_REF] let us introduce the main properties under consideration in this note. Definition 7. The system (2) is said to be positive Scalable (pS) with respect to the inputs from U if there exists a function β : R * + → R * + such that for any ū ∈ R m , any input u ∈ U and any λ > 0:

y(t, σ(λū), λu) = β(λ)y(t, σ(ū), u) ∀t ≥ 0.
It is worth to stress that comparing with Shoval et al.

[2011], [START_REF] Skataric | A characterization of scale invariant responses in enzymatic networks[END_REF], [START_REF] Hamadeh | Transient dynamic phenotypes as criteria for model discrimination: fold-change detection in rhodobacter sphaeroides chemotaxis[END_REF], in this work only positive scaling λ is considered, that is why the property is called pS. In [START_REF] Shoval | Symmetry invariance for adapting biological systems[END_REF] much more general P-invariance property has been considered, where P is a set of C 0 input transformations, while in the present work only multiplication by a constant λ is analyzed.

In this definition the constant ū ∈ R m represents the influence of inputs applied before the instant t = 0, then the system starts its movement from the equilibrium σ(ū) corresponding to that input. The input u ∈ U is applied for t ≥ 0 and the scalability property means that if all inputs have been scaled by a λ > 0 (i.e. ū and u), then the output of the system is scaled by β(λ).

If this property is satisfied and the system is excited by two different inputs u ∈ U and u ′ ∈ U, then y(t, σ(ū), u) y(t, σ(ū ′ ), u ′ ) = y(t, σ(λū), λu) y(t, σ(λū ′ ), λu ′ ) for any λ > 0, therefore the scaling factor λ is unimportant and only the "ratio" u u ′ has to be taken into account. Definition 8. A pS system (2) with β = 1 is said to admit a positive Fold Change Detection (pFCD) property.

A pFCD system (2) has exactly the same response for an input u applied at the initial condition σ(ū) as for an input λu applied at the initial condition σ(λū).

APPLICATION OF HOMOGENEITY

In the work [START_REF] Shoval | Symmetry invariance for adapting biological systems[END_REF] the generic conditions, for a dynamical systems to have invariant response with respect to symmetries in P, have been proposed (for the case β = 1). Homogeneity is a particular kind of symmetry, thus in this section we reformulate and partially repeat the results of [START_REF] Shoval | Symmetry invariance for adapting biological systems[END_REF] using homogeneity theory. This development will be used in the next section for investigation of approximate scale invariance property based on the homogeneity framework.

We assume the following homogeneity properties for the system (2). Assumption 1. There exists an Euler vector field ν 1 : R n → R n , whose flow is denoted by Φ 1 , such that f (Φ s 1 (x), e s u) = d x Φ s 1 f (x, u). The function h is ν 1 -homogeneous with degree p. Remark 9. Note that under these conditions, if for u = 0 the system (2) is GAS, then (2) is Input-to-State Stable (ISS) Bernuau et al. [2013aBernuau et al. [ ,b, 2014] ] (in the case of coordinate-free homogeneity the ISS property depends on characteristics of ν).

In this section, we will study the pS property of the system (2) under the class of locally bounded inputs, denoted by Ū, but we will first look at this property for a smaller class of inputs, denoted by U. Let us define this class.

Let q ∈ N * + , for all g ∈ Lip(R q ), we denote Ξ g the flow of g. A triple (φ, g, ξ 0 ), with a continuous φ : R q → R m , g ∈ Lip(R q ) and ξ 0 ∈ R q , is said to be an admissible qtriple if there exists an Euler vector field ν 2 : R q → R q with flow Φ 2 such that g is ν 2 -homogeneous with degree 0 and φ•Φ s 2 = e s φ for all s ∈ R (if m = 1, then this property means that φ is ν 2 -homogeneous with degree 1).

We denote T q the set of admissible q-triples. The set of admissible inputs u we consider is given by: U ={u : R → R m : ∃q > 0, ∃(φ, g, ξ 0 ) ∈ T q : u(t) = φ(Ξ t g (ξ 0 ))}. Any triple (φ, g, ξ 0 ) ∈ T q such that u(t) = φ(Ξ t g (ξ 0 )) ∈ U, we will call an associated q-triple for the corresponding u ∈ U. Therefore, we will consider the class of inputs U generated by a homogeneous differential equation:

ξ(t) = g(ξ(t)), u(t) = φ(ξ(t))
with the initial condition ξ 0 ∈ R q . Thus the scaling property under consideration in this note can be interpreted as the output scalability for a consequent connection of two dynamical systems, when positive rescaling of initial conditions for the first system leads to a corresponding scaling of the output of the second system. Proposition 10. The set U is a vector space.

All proofs are excluded due to the space limitations.

Homogeneity conditions for pS property

For an initial condition x 0 ∈ R n and an input u ∈ U, consider an associated q-triple (φ, g, ξ 0 ) and define:

F x ξ = f (x, φ(ξ)) g(ξ) . ( 3
)
We denote Ψ the flow of F . Denote p 1 : R n × R q → R n and p 2 : R n × R q → R q the first and second canonical projections respectively, then y(t, x 0 , u

) = h(p 1 • Ψ t (x 0 , ξ 0 )). Let us denote ν x ξ = ν 1 (x) ν 2 (ξ)
and Φ the flow of ν.

Lemma 11. If Assumption 1 is verified, the following properties hold:

1. The vector field F is ν-homogeneous with degree 0.

Denote Φ

s 1 = p 1 • Φ s , then Φ s 1 (σ(ū)) = σ(e s ū). 3. Denote Ψ t 2 = p 2 • Ψ t , then the mapping Ψ 2 is the flow of g and Ψ t 2 • Φ s 2 = Φ s 2 • Ψ t 2 .
Theorem 12. Under Assumption 1, the system (2) is pS with respect to inputs from U and β(λ) = λ p . Moreover, if p = 0, then the system admits pFCD property. Corollary 13. Under Assumption 1, the system (2) is pS with respect to polynomials inputs. Theorem 14. Under Assumption 1, the system (2) is pS with respect to locally bounded inputs. Lemma 15. Let u be a bounded input defined on [0, T ]. Suppose that there exists a sequence (u n ) n≥1 of bounded inputs converging to

u in L 1 [0, T ] (i.e. u n (t)-u(t) L 1 [0,T ] → 0 when n → +∞) such that u n L ∞ [0,T ] ≤ 2 u L ∞ [0,T ] .
Let x be the trajectory of the system (2) with input u and initial condition x(0) = x 0 and x n be the trajectory of the system (2) with input u n and initial condition x n (0) = x 0 . Then x n converges to x uniformly on [0, T ].

The case of constant inputs

Let U be the class of constant inputs, in this case g(ξ) = 0, φ(ξ) = ξ, then ν 2 (ξ) = ξ. Following the result of Lemma 11, to verify Assumption 1 it is equivalent to check that the vector field

F (x, u) = [f (x, u) T 0] T is homogeneous for ν(x, u) = [ν 1 (x) T u] T with degree d = 0.
Consider examples of such a case study. Example 2. Consider a linear system with f (x, u) = Ax + bu, y = c T x, where A is an invertible matrix of dimension n × n, b and c are some vectors in R n . This system is (weighted) homogeneous with d = 0 and p = 1 and, hence, it admits the pS property with respect to constant inputs. For y = sign(c T x) the system has pFCD property. Example 3. Take

f (x, u) = - ax 1 + |x| + e -|x| 1 + |x| u, h(x) = e p|x| x p ,
where a > 0, p > 0, then F and h are homogeneous for ν 1 (x) = x 1+|x| with the degrees d = 0 and p respectively. Thus this nonlinear system is pS with respect to constant inputs.

LOCAL HOMOGENEITY

Previously we assumed that the vector field F is homogeneous, which is a rather restrictive hypothesis. This assumption may be relaxed using local homogeneity [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF]. Definition 16. Let F 0 be a vector field such that e -ds (d x Φ s ) -1 F (Φ s (x)) converges to F 0 (x) uniformly on compact sets as s → -∞ (s → +∞). Then F 0 is called the local ν-homogeneous approximation of degree d at 0 (at +∞) of F .

The local ν-homogeneous approximation F 0 of degree d is always homogeneous with degree d. Only one degree can give us an interesting approximation. An easy verification shows that, if for a given d, F 0 is non-vanishing, then all others approximations are either vanishing or not defined. When such a non-vanishing ν-homogeneous approximation F 0 exists, we call it the ν-homogenization of F . It is possible to show that if F 0 is the local ν-homogeneous approximation of degree d at 0 (at +∞) of F , then for any ǫ > 0 there is s ǫ < 0 (s ǫ > 0) such that sup y∈Sν ,sign(sǫ)s≥|sǫ| |w(s, y)| ≤ ǫ, w(s, y) = e -ds (d y Φ s ) -1 F (Φ s (y)) -F 0 (y), where S ν = {x ∈ R n : N (x) = 1} and N is a homogeneous norm. Further in this section the case of weighted homogeneity with ν(x) = diag (r 1 x 1 , . . . , r n x n ) for some r i ∈ R * + , 1 ≤ i ≤ n is considered (it is the most useful case for a local approximation calculation). In this case for any x ∈ R n there exist s ∈ R and y ∈ S ν such that x = Φ s (y) and

|F (x) -F 0 (x)| = |e ds d y Φ s w(s, y)| = |e ds diag(e r1s , . . . , e rns )w(s, y)| ≤ γ(s)|w(s, y)|, γ(s) = e (rmin+d)s s ≤ 0, e (rmax+d)s s > 0,
where r min = min 1≤i≤n r i and r max = max 1≤i≤n r i . The following stability properties are satisfied [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF], [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]: Theorem 17. Let F be a continuous vector field and assume that there exists a non-zero vector field F 0 being its local approximation of degree d at 0 (at +∞). If F 0 is GAS, then F is locally asymptotically stable (then F has bounded trajectories).

The local approximation gives us a local information on the behavior of the system, it can be defined for any finite value s 0 ∈ (-∞, +∞) [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF].

It can also be used to investigate the scale invariance property but locally. In the paper [START_REF] Skataric | A characterization of scale invariant responses in enzymatic networks[END_REF] it has been observed that the class of systems for which the scale invariance property holds exactly is very limited, and the notion of approximate scale invariance has been introduced. Definition 18. For an ε ∈ R + , the system ( 2) is said to be ε-positive Approximately Scalable (ε-pAS) with respect to the inputs from U if there exists a function β : R * + → R * + such that for any ū ∈ R m , any input u ∈ U and any λ > 0: sup

t≥0 |y(t, σ(λū), λu) -β(λ)y(t, σ(ū), u)| ≤ ε.
The system ( 2) is said to be logarithmic ε-pAS with respect to the inputs from U if for any ū ∈ R m , any input u ∈ U and any λ > 0 there exists β : R *

+ → R * + such that sup t≥0 y(t, σ(λū), λu) β(λ)y(t, σ(ū), u) -1 ≤ ε.
The difference between ε-pAS and logarithmic ε-pAS is that the former one is better suited for analysis of approximate scalability at the origin (for small values of |y|), while the latter property is better adapted for the scalability investigations for big values of y.

Obviously, if the system (2) is ISS with respect to the input u, then always there is an ε > 0 (dependent on u) such that it is ε-pAS. The question is when this property is satisfied for a relatively small ε. An answer on this question has been given in [START_REF] Skataric | A characterization of scale invariant responses in enzymatic networks[END_REF] applying the time-separation and the linearization techniques. In this work we will use the local homogeneity framework for the same purpose. An advantage is that the local homogeneity allows an approximation at infinity (not only at the origin as with the linearization) to be defined.

Relation of trajectories of F and F 0

First, let us try to establish the conditions under which the trajectories of a locally Lipschitz system ẋ = F (x), (4) and its ν-approximation ẋ = F 0 (x), (5) starting from the same initial conditions stay close. More precisely, denote Ψ and Ψ 0 the flows of F and F 0 respectively, select an ε > 0, then we are looking for conditions that

|Ψ t (x) -Ψ t 0 (x)| ≤ ε for given x ∈ R n for some time interval t ∈ [0, T ]. Define T r x = {t ≥ 0 : Ψ t (x) ∈ N r }, where N r = {x ∈ R n : sign(r)N (x) ≥ sign(r)e r }, note that the set T r
x can be empty if Ψ t (x) / ∈ N r for all t ≥ 0. Lemma 19. Let ε > 0 be given and F 0 be the local νhomogeneous approximation of degree d at 0 (at +∞) of F . Let F 0 be a C 1 vector field and there exist matrices

A k ∈ R n×n , 1 ≤ k ≤ K for some K ∈ N * such that for all x ∈ R n JacF 0 (x) ∈ Co(A 1 , . . . , A K ), (6)
A T k P + P A k = -Q ∀1 ≤ k ≤ K for some matrices P = P T > 0, Q = Q T > 0. Then for all x ∈ N s ε/ρ , |Ψ t (x) -Ψ t 0 (x)| ≤ εκ(x), κ(x) = max{ξ • N (x) d+rmin , ξ • N (x) d+rmax } while t ∈ T s ε/ρ x
, where ρ = 2

λ 3 max (P ) λ 2 min (Q)λmin(P ) and ξ is a function from K ∞ .
The condition that the Jacobian matrix of F 0 belongs to a convex hull of matrices ( 6) is widely used in the theory of convergent systems [START_REF] Pavlov | Convergent dynamics, a tribute to boris pavlovich demidovich[END_REF] (together with the stability condition for all matrices in the hull) or for the incremental stability analysis [START_REF] Ingalls | A small-gain lemma with applications to input/output systems, incremental stability, detectability, and interconnections[END_REF]. If F 0 is the local ν-homogeneous approximation at 0 of F , then it is possible to select ε > 0 sufficiently small in such a way that ξ • N (x) ≤ 1, then the system has ε-pAS property while trajectories stay into N s ε/ρ if d = 0. For the approximation at +∞ the situation is more complicated, in this case decreasing the value of ε implies increasing the norm of initial condition N (x) for x ∈ N s ε/ρ . However, if we would consider the coordinate ratio

Ψ t i (x) Ψ t 0,i (x)
for any 1 ≤ i ≤ n under assumption that both variables Ψ t i (x) and Ψ t 0,i (x) are separated from zero, then

Ψ t i (x) Ψ t 0,i (x) -1 = Ψ t i (x) -Ψ t 0,i (x) Ψ t 0,i (x)
.

Next, let d = 0, ξ(s) = ξ 0 s for some ξ 0 > 0, r max = r min = 1 (these parameters correspond to a linear approximation dynamics F 0 ) and let Ψ t 0,i (x) ≥ ηN (x) for some η > 0, then

Ψ t i (x) Ψ t 0,i (x) ≤ ε ξ 0 η for all t ∈ T s ε/ρ x
. In this case for a sufficiently small value of ε the solutions of (4) and (5) stay close in "logarithmic" sense.

Application of local homogeneity for pAS analysis

The main idea of this section is to combine the results of Theorem 12 and Lemma 19.

Take the system in (2) and an input u ∈ U, which has an associated triple (φ, g, ξ 0 ). For the closed loop system (4) with F defined in (3), let F 0 be C 1 local νhomogeneous approximation of degree 0 at 0 (at +∞) of F for ν = (ν T 1 ν T 2 ) T for some Euler vector field ν 1 (the vector field g is ν 2 -homogeneous). In this case

F 0 (x, ξ) = (f T 0 (x, φ(ξ)) g T (ξ)) T , where f 0 is a kind of local approximation of f , i.e. [d x Φ s 1 ] -1 f (Φ s 1 (x), φ(Φ s 2 (ξ)))
s→±∞ -→ f 0 (x, φ(ξ)) uniformly on compact sets. Since all conditions of Theorem 12 are satisfied for F 0 , it has the pS property. Intuitively, in the domain of R n where F and F 0 are sufficiently close, the original system (2) has pAS property for a properly selected ε > 0. Indeed, as before denote Ψ and Ψ 0 the flows of F and F 0 respectively, then As in Lemma 19, let ε > 0 be given and there exist matrices

p 2 • Ψ t = p 2 • Ψ t 0 . Define e(t) = p 1 (Ψ t (x, ξ 0 ) -Ψ t 0 (x, ξ 0 )), then ė(t) =f 0 (p 1 • Ψ t , u(t)) -f 0 (p 1 • Ψ t 0 , u(t)) + f (p 1 • Ψ t , u(t)) -f 0 (p 1 • Ψ t , u(t)).
A k ∈ R n×n , 1 ≤ k ≤ K for some K ∈ N * such that for all x ∈ R n and u ∈ U Jacf 0 (x, u(t)) ∈ Co(A 1 , . . . , A K ), A T k P + P A k = -Q ∀1 ≤ k ≤ K for some matrices P = P T > 0, Q = Q T > 0.
Then for all x ∈ N s ε/ρ , where ρ is defined as in Lemma 19,

|Ψ t (x, ξ 0 ) -Ψ t 0 (x, ξ 0 )| ≤ εκ(x, ξ 0 ), κ(x, ξ 0 ) = max{ξ • N (x, ξ 0 ) m+rmin , ξ • N (x, ξ 0 ) m+rmax } while t ∈ T s ε/ρ x
. Therefore the system (2) has 2εκ(x, ξ 0 )-pAS property for h(x) = x i if approximation at the origin is considered or the logarithmic 2εκ(x, ξ 0 ) + ε 2 κ 2 (x, ξ 0 )-pAS property for the case of ν-homogeneous approximation at +∞. Let us consider examples of application. Example 4. Consider a nonlinear cascade for some n ∈ N

* : ẋ0 = u -a 0 x 0 , (7) ẋi
= c i x i-1 b i + |x i-1 | -a i x i , 1 ≤ i ≤ n, y = x n .
Let U be the class of piecewise constant inputs, then g(ξ) = 0, φ(ξ) = ξ and ν 2 (ξ) = ξ. Denote x = (x 0 , x 1 , . . . , x n ) T and take ν 1 (x) = x, then the system has the local approximation of degree 0 at 0

ẋ0 = ξ -a 0 x 0 , ẋi = c i b i x i-1 -a i x i , 1 ≤ i ≤ n, ξ = 0, and at +∞ ẋ0 = ξ -a 0 x 0 , ẋi = -a i x i , 1 ≤ i ≤ n, ξ = 0,
which are stable linear systems (their stability follows the lower triangular structure of both). Thus the system (7) should demonstrate pAS property in almost all state space R n+1 for p = 1 (linear scaling). The results of the system simulation for

n = 2, a 0 = a 1 = a 2 = 1, c 1 = c 2 = 1, b 1 = 1, b 2 = 2,
ẋ2 = κ 2 x 1γ 2 x 2 , y = x 2 , where x 1 ∈ R + and x 2 ∈ R + are mRNA and protein concentrations respectively, κ i > 0, γ i > 0 are production and degradation rate constants respectively, θ 1 > 0, n > 0 are constant parameters, u(t) ∈ R + is external input modeled by a biased sinusoid of the frequency ω. Take ν 1 (x) = x and ν 2 (ξ) = ξ, then the system has the same local approximation of degree 0 at 0 and at +∞

ẋ1 = ξ 1 + ξ 2 -γ 1 x 1 , ẋi2 = κ 2 x 1 -γ 2 x 2 , ξ1 = 0, ξ2 = ξ 3 , ξ3 = -ω 2 ξ 2 ,
which is a stable linear system (again, due to the lower triangular structure). Thus the system (8) should demonstrate pAS property in almost all state space R n+1 for p = 1 (i.e. the trajectories of the system are scaled proportionally the input amplitude). The results of the system simulation for n = 2, κ 1 = κ 2 = θ 1 = 1, γ 1 = γ 2 = 1.5, ω = 2 are shown in Fig. 2. The piecewise continuous input u(t) is presented in Fig. 2,b in the logarithmic scale. The output trajectories 2y(t) and y(t) corresponding to the inputs u(t) and 2u(t) respectively are plotted in the logarithmic scale in Fig. 2,a.

As we can conclude, these results of simulation confirm the theoretical development of the paper.

CONCLUSION

Local or global homogeneity of a vector field can be checked doing a simple calculus, thus verification of this property in practice is rather straightforward. In this work the relations between (approximate) scalability and (local) homogeneity are established. Some conditions on closeness of solutions of the original system and its local approximation are given. The theoretical results are confirmed by numerical experiments for models of biological cascades.

In the theory of homogeneity, it is well known that a system inherits stability/instability properties after its homogeneous approximation. The main result of this work claims that under mild additional conditions the solutions of the system and its homogeneous approximation stay close in the corresponding domains.

Fig. 1 .

 1 Fig. 1. The results of simulation for the system (7)

  Fig. 2. The results of simulation for the system (8) ẋ1 = u(t) + κ 1 x n 2 θ 1 + x n 2

⋆ This work was supported in part by the Government of Russian Federation (Grant 074-U01) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).