Metic Cakmak
email: metincakmak@gmail.com

Helmut Hlavacs
email: helmut.hlavacs@univie.ac.at

A Need for Interactive Music Videos

We present a system that creates new versions of a music video dynamically showing a rapper on the fly. There are also versions which allow users to interact with the content in real time. We present results from experiments asking an audience how they like these possibilities.

Introduction

Music videos [START_REF] Vernallis | Experiencing Music Video: Aesthetics and Cultural Context Columbia University Press[END_REF][START_REF] Escher | Rapper's handbook: A guide to freestyling, writing rhymes, and battling[END_REF] have been companions to modern pop and rock songs for decades now. However, all music videos share one common property: they are static. Once having been filmed, they remain as they are, never again change, though in the past more and more examples of interactive selection of content and perspectives have appeared [START_REF]Red Hot Chilli Peppers, Look Around[END_REF][START_REF]Death Grips, I've Seen Footage[END_REF][START_REF] Dylan | Like a Rolling Stone[END_REF][START_REF]When the night time comes[END_REF][START_REF]Music Video Interactive[END_REF][START_REF] Ok Go | All is not lost[END_REF]. New computing technologies on the other hand nowadays make possible real-time rendering of complex scenes with stunning, photo realistic quality. These features have mainly been used in modern high-quality computer games. In our work we transfer this technology to the realm of music videos. This makes it possible to produce dynamically produced music videos, having the capability to change, to adapt, to be created newly again and again, each time they are watched. Consumers can even interact with them just like they can interact with computer games.

Automated and Interactive Music Videos

The practical work of this study was the creation of a video clip for a rap song using the engine Unity 3D. At first we have analyzed hundreds of video clips of rap songs, and categorized their content. Then we analyzed the rap song at hand and planned the scenes according to the song text. The scenes have been realized directly in a 3D environment which we created in Unity 3D.

The rapper "Massimo Schena" is a young Austrian musician. He writes usually about actual and major issues facing teenagers in the community. The main topics are drug abuse, unemployment of teenagers, the lack of chances offered to teenagers, the lack of education and other teenager related issues.

At first we created a 3D character of Massimo using the Blender 3D character creation tool. Then the animations of moves which he likes to do during a performance were created with the same tool. We have created the animations according to a video clip capturing the moves. The rapper mostly prefers dance styles like break dancing, freestyle, dougie, hip hop and jerk and the hand gestures included mainly def wave, the slim shady chop, and the ninja star.

After the creation of the character and the animations, we selected an appropriate song from the repertoire of Massimo called "Mehr Perspektiven"1 , a song demanding more chances and options for teenagers to live a fruitful life.

Experimental Evaluation

We asked a test audience to evaluate, whether our enhancements concerning interactivity and automation indeed lead to more entertaining videos. In particular we started with two hypotheses:

1. Automation and interactivity make video clips more interesting / attractive.

2. Automated and interactive video clips are preferred to be seen in the future.

We chose a within-subjects design with 20 subjects, watching the static, the automated, and the interactive videos 8 times each. In order to check for memory effects we chose ten subjects to watch the videos sequentially, while the other ten subjects watched them in random order (in the end there were no differences between the groups). We chose individuals as much as possible from different age, gender or social groups. However, the most reachable audience for us were young students and therefore the majority of our audience is rather young. After watching the videos we asked the participants to fill out a questionnaire containing 20 questions. While 80% of respondents think that the third video clip is more preferable for the near future, 15% think the second video clip and only 5% of them think that the first video clip will be more preferred.

Also, we have implemented a paired t-tests to pair questions for testing our hypotheses. As a conclusion we can say that both automation and dynamical change, as well as interaction will be important features to add to future music videos. Above all, interaction is key to create music videos which keep the audience interested. As music videos try to bring a visual stimulus to the audience, it is important to keep this stimulus fresh and surprising throughout many subsequent presentations, and people want to be able to influence the outcome at least to some degree. This is similar to normal stories, which become boring after a while, since we think that surprise, change and new stimuli are key to keep an audience interested over some time.

Conclusion and Future Work

Our hypotheses are confirmed by our experiments, which show significant differences between the presented videos. However, more research is definitely necessary to include more viewers from different age groups, test for differences between gender groups and social background, or come up with videos with different content, and higher quality.

Fig 1 .Fig. 2 .

 12 Fig 1. Suicide Fig. 2. Drunk Based on the scenes, we created three different versions of a video. The static version never changes, while the automatic version can change camera angles, type and number of characters seen, and the sequence of the scenes, which are dynamically adapted on the fly. Finally, the interactive version makes it possible to actually change scenes. For instance, in scene 5, the viewer can prevent the teenager form committing suicide by clicking the mouse.

https://www.youtube.com/watch?v=veyDvOcGUBM


Tim Marsh
email: t.marsh@griffith.edu.au

Sim Joo Jin

Chia Ching Hong

Entertainment and Language Learning: Voice Activated Digital Game and Interactive Storytelling Trials in Singapore Schools

Keywords: assessment, studies, trials, serious games, games for learning

We describe the Ministry of Education's (MOE) English Language Oracy Portal project that aims to make learning English engaging and effective through the introduction of game-based learning and interactive storytelling/storybooks incorporating automated speech-assessment-feedback mechanisms in Singapore schools. In particular, we describe pilot studies and trials with 720 students and their teachers from twelve schools, and report the most important findings to inform development to make improvements and recommend strategies for their integration in the curriculum and classroom for the final rollout in primary and secondary schools across Singapore.

Introduction

Game-based learning provides engaging, fun, exciting and entertaining gameplay that motivates students to learn [START_REF] Gee | Learning by design: games as learning machines[END_REF][START_REF] Prensky | Fun, play and games: what makes games engaging, In: Digital Game-Based Learning[END_REF][START_REF] Marsh | Fun and Learning: Blending Design and Development Dimensions in Serious Games Through Narrative and Characters[END_REF]. Research in the field of applied linguistics also suggests that motivation is a key factor in language learning [START_REF] Dorneyi | Motivation in action: a process model of L2 motivation[END_REF] and digital games have been found to possess the potential for motivating the development of speaking skills [START_REF] Johnson | Serious Use of a Serious Game for Language Learning[END_REF]. Despite English being officially designated in 1965 as the main language of instruction within the Singapore education system, Singapore's English Language Curriculum and Pedagogy Review Committee (ELCPRC) and the Ministry of Education (MOE), identified a wide range of language abilities and language use among Singaporean students and focus groups with employers revealed "a decline in oral fluency and writing skills" among Singaporean employees. Therefore, ELCPRC and MOE identified a need "to ensure that every student is equipped with the English language competency and skills needed for learning, for work and for life in a global economy". This paper describes one effort to address this need by introducing a gaming environment and an interactive storytelling/storybook in primary and secondary schools to motivate Singaporean students to improve their oral English Language (EL) skills. We describe pilot study trials and report on findings before rollout in schools across Singapore.

2

Studies: Pilot and Trials 720 (M, F), 480 primary school students (9-10 years) and 240 secondary school students (12-13 years) from 12 schools across a range of academic abilities took part in studies with the two interactive digital English learning applications: V.A.S.T. (Voice Activated Spy Tech) and Reading Champs, a gaming environment and interactive storybook, respectively (see figure 1). Both applications have rich graphics, animations, audio and interactive features that are comparable in many respects to commercial off-the-shelf interactive games and applications. Both applications incorporate automated speech-assessment-feedback mechanisms in four key areas: accuracy, fluency, word stress, and tone (AFWT), providing personalized guidance to help with students' individual learning needs. The rationale and aims for the EL Oracy Portal applications study trials is to:

• investigate the effectiveness of the applications, in supporting students in the development of their EL oral reading skills • identify the factors that affect students' ability to make productive use of the feedback provided by the applications • identify the extent that applications may be incorporated into the formal curriculum and classroom that yields the most teaching and learning value Three data collection approaches were identified to shed light on the study's rationale and aims: (i) observation of students in classroom sessions (ii) focus group sessions with students and teachers (iii) student and teacher survey/questionnaire.

Findings

Students enjoyed interacting with V.A.S.T. and Reading Champs judging by observed intonations and excitement when speaking/reading, facial expressions, and laughter. Students moved between in-game and out-of-game/storybook to discuss, collaborate and share with fellow students about in-game features, reading/speaking approaches and strategies for advancing. During this out-of-game/storybook behavior, students appeared to continue to be engaged in game/storybook-related activities. In focus groups, students verified this in-game and out-of-game collaborative behavior, and emphasized that it heightened enjoyment and contributed to learning. Teachers agreed that students enjoyed playing with applications, looked forward to lessons and were positive about collaboration between students outside of the game/storybook. In ques-tionnaires, all students rated applications high (~4, scale 1 to 5) for fun, enjoyable, focused, and for wanting to play/read again. Male students' ratings were slightly higher than female students for mixed classes (M, F). Students' self-report ratings for learning show mid-to-high ratings. Table 1

Discussion and Conclusion

Studies show V.A.S.T. and Reading Champs provide appealing automated reading and speaking practice environments that are highly effective for motivating students to improve their oral EL skills in an engaging and entertaining manner. A number of recommendations have been made for their effective implementation and use towards rollout in primary and secondary schools across Singapore.

Fig. 1 .

 1 Fig. 1. Classroom trails in schools (left) with V.A.S.T. and Reading Champs (right)

  below shows examples of the issues, concerns and recommendations identified in studies:

	(i) Set-up, Integration & Tutorial: relating to support, guidance and tutorials
	1	students should be made aware and guided in the use of features through handout/tutorial provided by teachers before gameplay
	2 minimize technical issues associated with sound-levels and microphone input
	3	encourage acceptance/sustained use by student collaboration in-game and off-game in classroom; implement blended learning in-game / off-game
	(ii) Re-Design / Re-development: generally more critical recommendations:
	4	deter students from continuously skipping through the reading/speaking tasks in V.A.S.T. and encourage more reading/speaking challenges
	5	ensure students understand speech assessment feedback to make improvements to their speaking e.g. "awesome", "try again" corresponding to AFWT
	6 introduce arrows, breadcrumbs, waypoints to aid navigation
	7 incorporate short in-game tutorial on V.A.S.T's complex recording function
	8	shift students engaged in wandering around/exploring the gaming environment to engage in missions associated with reading/speaking and learning
	9	V.

A.S.T.'s in-game analytics should provide clear, accurate and reliable assessments and feedback on students' speech and performance


Mohamed Ben Ellefi

Zohra Bellahsene

Konstantin Todorov

Datavore: A Vocabulary Recommender Tool Assisting Linked Data Modeling

Keywords: Linked Data, Vocabulary Recommendation, Modeling Web Data

In this paper, we introduce the vocabulary recommendation system Datavore (Data vocabulary recommender). The tool is oriented towards metadata designers providing ranked lists of vocabulary terms to reuse in the web of data modeling process, together with additional metadata and cross-terms relations. Datavore relies on the Linked Open Vocabulary ecosystem for acquiring vocabularies and metadata. The system is able to deal with noisy and multilingual input data.

Introduction and Motivation

Following the fourth principle of linked data, a large number of datasets from different domains and using different vocabularies have been published and interlinked on the web. To guide data discovery and reuse, catalogs of linked data, such as the Data Hub1 , have been created.

With the increasing use of Linked Open Data (LOD), it becomes more and more important for data providers not only to publish their data but also to model and describe them following the LOD best practices2 . Here, we draw the reader's attention to the recommendation of building on, instead of replicating, existing RDF schema and vocabularies, in an effort to improve interoperability [START_REF] Hyland | Cookbook for open government linked data[END_REF]. Thus, an important step towards the web data modeling task is the discovery of all relevant vocabularies to reuse.

In this paper, we introduce Datavore, a vocabulary recommender system, which uses the Linked Open Vocabularies3 (LOV) as a vocabulary search engine. In addition to a list of ranked recommended concepts/properties, the tool provides important additional metadata, as well as cross-terms relations in the form of a set of triples combinations, handling noisy and multilingual input. To our knowledge, there is only one comparable tool Karma [START_REF] Knoblock | Semi-automatically mapping structured sources into the semantic web[END_REF] -which is a semiautomatic tool that proposes a mapping between semantic types (an OWL class or the range of a data property) from the introduced ontology to the data source columns-the main contribution of Datavore is the fact of handling the multilingual data as well as the benefit of the whole LOV up-to-date vocabularies in service of the linked data modeling task.

Overview of Datavore

We proceed to describe the workflow of the system, shown in Figure 1. Cleaning and Translating [START_REF] Knoblock | Semi-automatically mapping structured sources into the semantic web[END_REF]. In most cases, the extracted string of characters needs to be cleaned-up by removing or modifying the unwanted characters. We use the Microsoft Translator java API5 in order to clean up the initial string and render it in a linguistically correct form. For example, translate(Cat à c gorie, fr, fr) returns Catégorie. In case there are no sufficient or satisfactory results by using the source language, the system uses the same service to translate the source item into English, the most common language on the LOD.

Terms Search (3). We opted for the LOV as a vocabulary search engine, which, to the best of our knowledge, is the only purpose-built vocabulary search engine available on the Web with an up-to-date index. As a design decision, Datavore -Modeling Linked Data generally requires an ontology engineer and a domain expert [START_REF] Schaible | Lover: support for modeling data using linked open vocabularies[END_REF].

-Datavore uses the LOV search service, for which, as for any search engine, the choice of input keywords is crucial. For that reason, Datavore enables the user to update the source terms from the interface.

-The complexity of the inter-concept relations step is of order O(N 2 M 1 M 2 ), where, N is the cardinality of the source terms, M 1 and M 2 are respectively the cardinalities of the two compared lists of recommended concepts.

-Datavore has been implemented in Java and it is available as a GU I desktop application6 together with a demonstration video7 .

Conclusion and Future Work

We introduced Datavore, a vocabulary recommender system based on LOV assisting linked data modeling. Among the original features of the tool is the fact that it provides metadata of the recommended predicates and that it automatically retrieves existent relations between the predicates to guide metadata design. LOV is a trustworthy search engine but the actual version contains less than 480 vocabularies. In the future, we intend to extend Datavore via the Swoogle search engine, which contains over 10, 000 ontologies.

Fig. 1 .

 1 Fig. 1. Datavore Workflow

Table 1 .

 1 LIRMM Open Data Team Example.

	id Person Name Profession Lab City Name Postal Adress Country
	1 M. Ben Ellefi PhD Student LIRMM Montpellier	34090	France
	2	K. Todorov	Assoc. Pr. LIRMM Montpellier	34000	France
	3 Z. Bellahsene	Pr.	LIRMM Montpellier	34000	France

http://datahub.io/

http://www.w3.org/2011/gld/wiki/Linked_Data_Cookbook

http://lov.okfn.org/dataset/lov/

We note that considering the actual values from the CSV misled the recommendation in most cases that is why we limited ourselves to the column names.

https://code.google.com/p/microsoft-translator-java-api/

Download and unzip this file: http://www.lirmm.fr/benellefi/Datavore_ExeFile

http://www.lirmm.fr/benellefi/Datavore_VideoDemo

http://www.datalyse.fr/ -FSN-AAP Big Data n3.

Acknowledgements. This research is funded under the Datalyse project 8 .

Authors Suppressed Due to Excessive Length queries the LOV search service with the extracted cleaned or/and translated terms. The result is a list of concepts for each source term ranked by the LOV metric, which is based on the vocabulary terms popularity in the LOD datasets and in the LOV ecosystem.

Metadata Extraction [START_REF] Schaible | Lover: support for modeling data using linked open vocabularies[END_REF]. Metadata designers are recommended to select popular vocabularies found in the search phase but it is not straightforward to judge which vocabulary is better suited to the application. Datavore queries the LOV endpoint (/dump file) to extract the needed metadata to help designers to choose the appropriate vocabularies. As a result, for each concept c, extract: (i) the set of object properties having c as domain that includes labels and hierarchical relations, (ii) the set of datatype properties that can be used with c as domain, and (iii) a link to the vocabulary web site. In addition, we provide a "test-yourluck " option, which recommends to the user only one, the top ranked, datatype property. This "lucky" property has the highest Levenshtein string similarity [START_REF] Levenshtein | Binary codes capable of correcting deletions, insertions, and reversals[END_REF] with the source term.

Inter-Concept Relations (5). From the extracted lists of recommended concepts, Datavore queries the LOV endpoint (/dump file) to retrieve cross-lists relations, i.e., relations between concepts from different lists. These metadata are crucial for selecting the best combination of predicate names to reuse for the input dataset.

Example Scenario

Imagine a designer who wants to model the data in Table 1 using an ontology editor. Datavore will guide him/her to find vocabularies to reuse, returning a sorted list of concepts for each column name. For the column "City Name" Datavore queries the LOV using the keyword "City" and returns the sorted list of concepts {"akt:City", "place:City", "lgdo:City", etc.}. When the designer selects the concept "akt:City", Datavore presents to him/her the following metadata: (i) literals (like rdfs:label, rdfs:comment, etc.) and the hierarchical relations of "akt:City", (ii) a set of datatype properties like "foaf:name" that have "akt:City" as rdf:domain to represent the column "City Name". After the concepts extraction, Datavore queries the LOV again to extract inter-columns triples and recommends to the user a set of relations between column names. In our example, the recommended relation between the two columns "Person Name" and "PostalAdress" is the triple: < f oaf : P erson >< akt : hasAddress >< akt : P ostalAddress >.

Technical Notes

For a proper use of Datavore, we take note of the following.

-Datavore is meant to be used in complementarity with ontology development tools for dataset modeling.


Gabriel Lagasquie

A note on simple randomly switched linear systems

Keywords:

published or not. The documents may come

A note on simple randomly switched linear systems

Introduction

Piecewise Deterministic Markov Processes (PDMPs) is a huge class of stochastic processes introduced by Davis [START_REF] Davis | Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models[END_REF]. In this note we are studying a particular subclass of these processes. we consider (X t , I t ) t 0 evolving on the space R d × {0, 1} solving the equation:

Ẋt = F (X t , I t )
where F is a smooth function and (I t ) is a jump process on {0, 1}. We call λ i the jump rate of the process (I t ) from state i to state 1 -i. Such processes play a role in modeling problems in various fields such as molecular biology [START_REF] Radulescu | Théorèmes limites pour des processus de Markov à sauts. Synthèse des résultats et applications en biologie moleculaire[END_REF], population dynamics [START_REF] Benaïm | Lotka volterra in fluctuating environment or how switching between beneficial environments can make survival harder[END_REF], Internet traffic [10]. Even if these processes are easy to define, their stability is never clear.

More precisely, we are here interested in the long time qualitative behaviour of the planar randomly switched process (X t , I t ) t 0 ∈ R d × {0, 1} solving the equation

Ẋt = A It X t X 0 = x 0 , (1)
where A 0 , A 1 are d × d real matrices, and (I t ) is a Markov process on E = {0, 1} with constant jump rates:

λ 0 = βu, λ 1 = β(1 -u)
where β > 0 and u ∈ (0, 1). The process ((X t , I t )) t≥0 is a piecewise deterministic Markov process as defined in [START_REF] Davis | Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models[END_REF] and several examples have been studied in [START_REF] Lawley | Sensitivity to switching rates in stochastically switched ODEs[END_REF] or [5]. It appears that the following quantity dictates the large time behaviour of this process according to the parameters A 0 , A 1 , β and u: χ := χ(A 0 , A 1 , β, u) = lim t→+∞ 1 t log ( X t ) a.s. Indeed if χ is positive then the system explodes almost surely whereas if χ is negative the system is stable and goes to zero almost surely.

The asymptotic behaviour of such processes when (I t ) is any deterministic process and d = 2 is now understood. Given two matrices in M 2 (R), [START_REF] Balde | A note on stability conditions for planar switched systems[END_REF] gives a necessary and sufficient condition for the existence of a deterministic function (I t ) that produces a blow up. But the articles [START_REF] Lawley | Sensitivity to switching rates in stochastically switched ODEs[END_REF] and [6] have shown that the behaviour of planar randomly switched systems may be surprising: according to the value of the switching rate β, the process can either blow up or go to zero even if each system is stable. In [START_REF] Lawley | Sensitivity to switching rates in stochastically switched ODEs[END_REF], the authors build a randomly switched system living in the space R 2n which behaviour comes down to alternating blow up periods and stable periods a finite number of times as the jump rate grows.

A question raised from these works is the existence of a planar randomly switched system which behaviour is also alternating between blow up periods and stable periods. The goal of this note is to construct such a process. Our idea is the following: put a stochastic control in a deterministic planar switched system which satisfies the qualitative long time behaviour condition (using the criteria in [START_REF] Balde | A note on stability conditions for planar switched systems[END_REF]). For a > 0 and b > 1, we define two real matrices as follow:

A 0 = -a b -1 b -a and A 1 = -a 1 b -b -a .
Thanks to these two matrices, we define the continuous process ((X t , I t )) t≥0 ∈ R 2 ×{0, 1} solution of (1). The path of X t follows a spiral clockwisely during a random time with exponential law of parameter λ i before switching on an other spiral as described in Figure 1.

We also define:

χ := χ(a, b, β, u) = lim t→+∞ 1 t log ( X t ) a.s
which sign gives the long time qualitative behaviour of our process as said before.

The main result of this note is the following which is similar to the main result in [START_REF] Lawley | Sensitivity to switching rates in stochastically switched ODEs[END_REF]: (ii) For every β, u > 0 there exist a > 0 and b > 0 so that χ(a, b, β, u) > 0.

In Section 2, we establish an explicit expression of the function χ. In Section 3 we use it to prove Theorem 1.1.

The explicit expression of χ obtained in Proposition 2.2 allows us to calculate numerically the function χ at u constant (see Figure 2). It is also interesting to show the sign of the function χ according to β and u as we can see in Figure 3. In this figure, we can see that for (β, u) in a compact, our system explodes when t goes to +∞. We can see the same behaviour in the process studied in [START_REF] Lawley | Sensitivity to switching rates in stochastically switched ODEs[END_REF]. But this is different from what we can observe in [START_REF] Benaïm | Lotka volterra in fluctuating environment or how switching between beneficial environments can make survival harder[END_REF] and [6] where the explosion happens for β large enough.

This graphics combined to Theorem 1.1 shows us that this process does not meet our expectations. There is a unique area of blow up whereas the deterministic periodic process has several blow up periods (see Section 4). It seems that, in the case of the random process, the variance of the waiting time in each state is too high to select precisely the good jump times that would allow it to have several blow up periods.

We will see in Section 5 a simple way to lower the variance of the waiting time without complicate too much the process, but first, in Section 4 we will give an interesting result about the deterministic process associated to our system.

In the last section we focus on the decentered switched system defined as follow:

Ẋt = A It (X t -b It ) X 0 = x 0 , ( 2
)
where b 0 , b 1 are two different vectors of R d .
There is a connection between the existence of an explosive trajectory (with a deterministic control) and the properties of the invariant measure, if it exists, in the random case (exponential switch times).

Let us call T 0 = 0, T 1 , ..., T n the jumping times of the process (X t ) and τ 0 , τ 1 , ..., τ n , the interjump times following an exponential law of parameters λ 0 and λ 1 alternatively. Let us define Y n = X T 2n . By simply solving the equation 2 between each jumping times we obtain:

Y n = e τ n+1 A 1 e τnA 0 Y n-1 -e τ n+1 b 0 + b 1 .
This kind of stochastic equation have been the object of numerous research. Applying Kesten's renewal theorem to the discrete process (Y n ) will give some informations about the tail of the invariant measure of (X t ) (see [11] for the main article, see [1], [START_REF] De Saporta | On the multidimensional stochastic equation Y(n+1)=a(n)Y(n)+b(n)[END_REF] for different formulations of the same theorem). Theorem 1.2. If there is no deterministic control that makes the deterministic process associated to (Y t ) explodes, it means that there exists an invariant measure µ for our process and the support of µ is necessarily bounded.

In the other hand, we assume that χ < 0 and that there exists a deterministic control such as:

∃k ∈ 2N + 1 such as ∀x ∈ S d-1 , ∃t 0 , t 1 , ..., t k > 0 such as e t k A I k ...e t 1 A I 1 e t 0 A I 0 x > x . (3)
We call B = e τ k A I k ...e τ 1 A I 1 e τ 0 A I 0 and B 0 , B 1 , ..., B n n random variables i.i.d following the law of B. We also assume that:

max n 0 P B n ...B 0 x B n ...B 0 x ∈ U > 0 for all x ∈ S d-1 and any open ∅ = U ⊂ S d-1 . (4)
P(B n 0 ...B 0 ∈ .) γ 0 1 Bc(Γ 0 ) λ for some Γ 0 ∈ GL d (R), n 0 ∈ N and c, γ 0 > 0.

(

) 5
where λ is the Lebesgue measure on R n 2 . Under these assumptions, there exists a unique invariant measure µ for the process (Y t ) and it has an heavy tail:

∃p 1 > 0 such as ∀0 p < p 1 E µ [ Y p ] < +∞ and ∀p > p 1 E µ [ Y p ] = +∞.

Study of the angular process

In order to study our process ((X t , I t )) t≥0 we use the polar coordinates for X t . As a start, and as in [START_REF] Lawley | Sensitivity to switching rates in stochastically switched ODEs[END_REF], let us look at the case of the deterministic process. Let A be a 2-sized squared matrix and x ∈ R 2 \{0}. We set (x t ) t≥0 the solution of the ODE: ẋt = Ax t for all t > 0,

x 0 = x.
If x is nonzero, then it is also true for any x t with t positive. So we can define the polar coordinates (r t , θ t ) of x t . We call e θ the unitary vector (cos θ, sin θ) and u t = e θt . Then x t = r t u t . As r 2 t = x t , x t , we obtain the relations:

r t ṙt = x t , Ax t A(r t u t ) = ẋt = ṙt u t + r t ut .
And then we have:

ṙt = r t u t , Au t (6) ut = Au t -u t , Au t u t . ( 7
)
Let us write the equation ( 2) with the angle θ t . As ut = θt e θt+π/2 , by making the scalar product of ( 2) with e θt+π/2 , we obtain:

θt = Ae θt , e θt+π/2 . ( 8
)
We use now the polar coordinates in order to study the process ((X t , I t )) t≥0 . Between the jumps, the process follows the flow determined by A ∈ {A 0 , A 1 }. From Equation (3), the development of θ is deterministic and does not depend on r. As a consequence, the processes ((Θ t , I t )) t≥0 and ((U t , I t )) t≥0 are piecewise deterministic Markov processes on R × {0, 1} and S 1 × {0, 1} respectively. The principal interest of the study of these processes lies in the fact that the development of (R t ) t≥0 is determined by that of the process ((Θ t , I t )) t≥0 as shown by Equation (1). Indeed, by solving (1) between the jumps and by calling A(θ, i) = A i e θ , e θ , we obtain:

R t = R 0 exp t 0 A(Θ s , I s )ds . ( 9
)
So, as suggested by Equation ( 4), in order to study the behaviour of (R t ) t≥0 we are going to study the process ((Θ t , I t )) t≥0 and particularly its invariant measure.

Lemma 2.1. The invariant measure µ of the process ((Θ t , I t )) t≥0 is given by,

µ(dθ, i) = ρ i (θ)dθ
where:

ρ 0 = Φ d 0 and ρ 1 = C -ρ 0 d 0 d 1 ; d 0 (θ) = -b sin 2 (θ) - 1 b cos 2 (θ) and d 1 (θ) = - 1 b sin 2 (θ) -b cos 2 (θ); Φ(θ) = K + θ 0 βC(1 -u) 1 d 1 (α)
e -βv(α) dα e βv(θ) ; v is the primitive null in zero of the function -(

u d 0 + 1 -u d 1 );
K and C are two explicit constants.

Proof. We define, for i ∈ {0, 1},

d i (θ) = A i e θ , e θ+π/2 .
We call L the infinitesimal generator associated to the semi-group P t of the process ((Θ t , I t )) t≥0 .

Remind that

P t = E[f (Θ t , I t )|Θ 0 = θ, I 0 = i].
L is given by:

Lf (θ, i) = d i (θ) ∂f ∂θ (θ, i) + λ i (f (θ, 1 -i) -f (θ, i)) ( 10
)
for any smooth function f . In our study case, the functions d i are given by,

d 0 (θ) = -b sin 2 θ - 1 b cos 2 θ and d 1 (θ) = - 1 b sin 2 θ -b cos 2 θ.
We notice that the functions d i are negative. This obviously implies that the process ((U t , I t )) t≥0 is recurrent, irreducible and it has a unique invariant measure. We want to write it as follows:

µ(dθ, i) = ρ i (θ)dθ. ( 11
)
where ρ 0 and ρ 1 are two 2π-periodical continuous functions. In this case, for every good functions f defined on S 1 × {0, 1}, we will have:

S 1 ×{0,1} Lf (θ, i)dµ(θ, i) = 0.
Let f be a C 1 function defined on S 1 × {0, 1}. Injected in the previous formula, the expression (6) allows us to obtain:

2π 0 Lf (θ, 0)ρ 0 (θ)dθ + 2π 0 L β f (θ, 1)ρ 1 (θ)dθ = 0. ( 12
)
Assume initially that ∀i, f (θ, i) = f (θ). So, by injecting in (6), and after integrating by parts, we obtain:

d 0 ρ 0 + d 1 ρ 1 = C ( 13
)
where C is a negative constant depending only of the parameters of the problem a, b, u and β. Assume now that f (θ, 0) = f (θ) and f (θ, 1) = 0. The same way, we obtain the relation:

(d 0 ρ 0 ) = (1 -u)βρ 1 -uβρ 0 .
Set Φ = d 0 ρ 0 . By using the relation [START_REF] Benaïm | Lotka volterra in fluctuating environment or how switching between beneficial environments can make survival harder[END_REF], we can write the last equation as a linear differential equation:

Φ = -βΦ( u d 0 + 1 -u d 1 ) + βC(1 -u) d 1 . ( 14
)
This differential equation is easily solved, we obtain:

Φ(θ) = K + θ 0 βC(1 -u) 1 d 1 (α)
e -βv(α) dα e βv (θ) where K is an integration constant and v is the primitive null in zero of the function -(

u d 0 + 1-u d 1 ). We can calculate v explicitly on the interval (-π 2 , π 2 ): v(θ) = u arctan(b tan(θ)) + (1 -u) arctan( 1 b tan(θ)).
We extend v to R + by continuity. The constants C and K are left to calculate in such a way that µ is a probability measure. So:

(i) ρ 0 and ρ 1 are non negative, (ii)

2π 0 ρ 0 (θ)dθ + 2π 0 ρ 1 (θ)dθ = 1.
The condition (i) gives us the following condition on Φ: C < Φ < 0. By developping this inequality, we obtain:

∀θ ∈ R + , e -βv(θ) - θ 0 β(1 -u) 1 d 1 (α) e -βv(α) dα > K C > - θ 0 β(1 -u) 1 d 1 (α)
e -βv(α) dα.

As θ -→ +∞, we obtain:

1 κ = K C = -β(1 -u) ∞ 0 1 d 1 (α)
e -βv(α) dα.

In order to find K, we use the condition (ii) and we get easily:

1 K = 2π 0 e βv(θ) 1 + κβ(1 -u) θ 0 1 d 1 (α) e -βv(α) dα 1 d 0 (θ) - 1 d 1 (θ) + κ 1 d 1 (θ) dθ.
We finally get expressions of ρ 0 and ρ 1 . Reciprocally, we check that the functions we obtained are solutions of our problem (one has to check the 2π-periodicity of these functions).

We want now to be able to describe the stability of our switched process according to the jump parameters u and β, and to the parameters of our matrices a and b. It appears thanks to Formula (4) that the following quantity:

χ(β) = lim t→+∞ 1 t log R t R 0 = lim t→+∞ 1 t t 0 A(Θ s , I s )ds
is worth studying because not only its sign gives the behaviour in large time of our process but also, thanks to the Ergodic Theorem and the previous Lemma, we will obtain an explicit expression of χ.

Proposition 2.2. The function χ can be calculated explicitly according to the data of the problem:

χ = -a -(1 -u)βκK (b -1 b ) 2 2π 0 sin(2θ) 1 d 0 (θ) + 1 d 1 (θ) e βv(θ) ∞ θ e -βv(α) d 1 (α) dα dθ.
Proof. Since we have:

1 t log( R t R 0 ) = 1 t t 0 A(Θ s , I s )ds .
The Ergodic Theorem tells us that (we identify θ with e θ ):

1 t t 0 A(Θ s , I s )ds -→ t-→∞ 2π 0 A(θ, i)dµ(θ, i) = χ.
Lemma 2.1 gives us the explicit formulation of the invariant measure of the process ((Θ t , I t )) t≥0 which allows us to obtain the claimed expression for χ.

Stability of the switched process

In the last section, we obtained the explicit expression of χ. We are now going to give some results on some of the asymptotic values of χ.

Theorem 3.1. For any u ∈ (0, 1),

(i) ∀a, b > 0 χ(a, b, β, u) -→ -a when β → 0 or β → +∞.
(ii) For every β, u > 0 we can choose a > 0 and b > 0 so that χ(a, b, β, u) > 0.

Proof. (i) One can find an other way to prove this result in [START_REF] Lawley | Sensitivity to switching rates in stochastically switched ODEs[END_REF] in a more general way. In order to study the limit of χ in 0, we study the limit of -

1 κ = β(1 -u) ∞ 0 e -βv(α)
d 1 (α) dα in 0. We notice that:

+∞ 0 e -βv(α) d 1 (α) dα = +∞ i=0 (i+1)π iπ e -βv(α) d 1 (α) dα = +∞ i=0 e -βiπ π 0 e -βv(α) d 1 (α) dα because v(α + π) = v(α) + π = 1 1 -e -βπ π 0 e -βv(α) d 1 (α)
dα.

Multiplicating by β(1 -u) and observing the presence of a rate of increase we get:

β(1 -u) ∞ 0 e -βv(α) d 1 (α) dα -→ 1 -u π π 0 1 d 1 (α) dα = l.
We finally can say that:

∀θ, β(1 -u) ∞ θ e -βv(α) d 1 (α) dα -→ β-→0 l.
Moreover, as:

2π 0 sin(2θ) 1 d 0 (θ) + 1 d 1 (θ) dθ = 0
and the product κK is bounded as β → 0, we deduce that χ(β) -→ 0 when β goes to 0. The limit of χ at +∞ is obtained using the Laplace method:

∀θ, β(1 -u) ∞ θ e -βv(α) d 1 (α) dα ∼ +∞ (1 -u)e -βv(θ) d 1 (θ)v (θ) .
From this relation, we obtain that κ and K converge when β goes to +∞ and, by dominated convergence,

2π 0 sin(2θ) 1 d 0 (θ) + 1 d 1 (θ) e βv(θ) β(1 -u) ∞ θ e -βv(α) d 1 (α) dα dθ -→ β→+∞ 0.
So we finally conclude that χ goes to -a when β goes to +∞.

(ii) We will now show that for any parameter β, we can choose b large enough and a small enough so that the process goes to +∞ almost surely (ie χ(β) > 0). We can write the function χ as follows:

χ(β) = -a + K π 0 f (θ)g(θ)dθ
because the function under the integral is π-periodic and where:

f (θ) = (b - 1 b ) sin(2θ) 1 d 0 (θ) + 1 d 1 (θ) , g(θ) = e βv(θ)
+∞ θ e -βv(α)

d 1 (α) dα +∞ 0 e -βv(α) d 1 (α) dα .
Finally we write χ as follows:

χ(β) = -a + K π 2 0 f (θ) (g(θ) -g(π -θ)) dθ
A straightforward computation leads to:

π 2 0 f (θ)dθ = 4 γ b 2 -1 b 2 (b -1 b ) 2 log γ -1 γ + 1 where γ = 1 + 4 (b-1 b ) 2 .
We then conclude that:

π 2 0 f (θ)dθ -→ -∞ when b goes to + ∞.
In order to determine the limit of g when b goes to +∞ we rewrite g as follows using the same method as before:

g(θ) = e βv(θ)   1 -(1 -e -βπ ) θ 0 1 d 1 (α) e -βv(α) dα π 0 1 d 1 (α) e -βv(α) dα   = e βv(θ)   1 -(1 -e -βπ ) θ 0 1 cos 2 (α)+ 1 b 2 sin 2 (α) e -βv(α) dα π 0 1 cos 2 (α)+ 1 b 2 sin 2 (α) e -βv(α) dα    .
Using the fact that:

v(θ) -→ b→+∞ π 4 if θ ∈ (0, π 2 ) v(θ) -→ b→+∞ 3π 4 if θ ∈ (0, π 2 ),
it is now easy to see that:

g(θ) -→ b→+∞ e β π 4 if θ ∈ (0, π 2 ) g(π -θ) -→ b→+∞ e -β π 4 if θ ∈ (0, π 2 ).
Since K < 0, we can conclude that for b large enough,

K π 2 0 f (θ) (g(θ) -g(π -θ)) dθ > 0.
So for an appropriate choice of a, we see that χ(β) > 0 inducing that the process explodes almost surely for this β and a with b large enough.

A quick look at the associated deterministic process

The deterministic switched system can be introduced as follows:

x 0 ∈ R 2 \(0, 0) ẋt = A vt x t (15)
where (v t ) ∈ F(R + , {0, 1}) is the control function. The behaviour of these systems is well known, see [START_REF] Balde | A note on stability conditions for planar switched systems[END_REF] for more details. Consequently, for our choice of matrices, we know that there exist a worst trajectory, that is to say, a choice of the control function v that makes the system explode, and this control is periodic.

Let us define (v t ) a

1 u(1-u)β -periodical process satisfying: ∀t ∈ [0, 1 uβ ), v t = 0 ∀t ∈ [ 1 uβ , 1 u(1 -u)β ), v t = 1.
We denote χ d (β) = lim t-→∞ 1 t log xt x 0 . By simply solving the equation for the process (x t ) and calculating the matrices exponential we obtain:

x t = e -at B ξ N (t)+1 (a t )B ξ N (t) (τ N (t) ) • • • B ξ 1 (τ 1 )
where

B 0 (t) = e tA 0 = cos(t) b sin(t) -1 b sin(t) cos(t)
and B 1 (t) = e tA 1 = cos(t) Proof. Although this result is true for any u ∈ (0, 1), for sake of simplicity, we will prove it only for the case u = 1 2 where the computation is not heavy. In order to prove this result, we first notice that:

1 2nt log [B 0 (τ )B 1 (τ )] n X 0 X 0 -→ n→+∞ χ d (β)
where τ = 2 β . Let us calculate the matrix:

B 0 (τ )B 1 (τ ) = cos 2 (τ ) -b 2 sin 2 (τ ) C sin(2τ ) -C sin(2τ ) cos 2 (τ ) -1 b 2 sin 2 (τ )
where

C = 1 2 (b + 1 b ) > 1.
We calculate the characteristic polynomial of this matrix and we obtain:

X 2 + X(4C 2 sin 2 (τ ) -2) + 1 = 0.
By a simple analysis of this polynomial, we can show that there exists two real eigenvalues if 2C 2 sin 2 (τ ) -1 > 0 and two joint complex eigenvalues if 2C 2 sin 2 (τ ) -1 < 0, concluding that our matrix is diagonalisable in C. So there exists P ∈ GL 2 (C) such that:

(B 1 (τ )B 0 (τ )) n = P -1 λ n 1 0 0 λ n 2 P.
Let us now write:

1 2nτ log [B 0 (τ )B 1 (τ )] n X 0 X 0 = 1 2nt log λ n 1 0 0 λ n 2 P X 0 + o(1) = 1 2nτ log λ n 1 0 0 λ n 2 x y + o(1).
If the factor 4C 2 sin 2 (τ ) -2 = 0 then one of the eigenvalues is superior to the other in absolute value, for example

|λ 1 | > |λ 2 |. In this case, if x = 0 ie X 0 / ∈ E λ 2 the characteristic space of the eigenvalue λ 2 : 1 2nτ log [B 0 (τ )B 1 (τ )] n X 0 X 0 = 1 2τ log |λ 1 | + o(1). If x = 0 i.e. X 0 ∈ E λ 2 : 1 2nτ log [B 0 (τ )B 1 (τ )] n X 0 X 0 = 1 2τ log |λ 2 | + o(1).
If the factor 4C 2 sin 2 (τ ) -2 = 0 then for every X 0 :

1 2nτ log [B 0 (τ )B 1 (τ )] n X 0 X 0 = 1 2τ log |λ 1 | + o(1).

A way to obtain several blow up areas

One of the expectations of this article was to get a planar linear switched system whose χ function admits several blow up areas. A quick look at the deterministic case (4) allowed us to believe that it could be possible with our system. Unfortunately the simulation we can see in Figure 2 seems to show that we did not manage to succeed. We slightly modify the jump times in order to mimic the deterministic evolution (15). Let u = 1 2 (for sake of simplicity) and let n be a stricly positive integer. We now define the piecewise deterministic Markov process ((X n t , I n t )) t≥0 as follows: (I t ) t 0 is a continuous time Markov process defined on the state space {0, 1, 2, ..., 2n -1} with a constant jump rate nβ 2 and a jump mesure:

Q(i, .) = δ i+1 if i < 2n -1, Q(2n -1, .) = δ 0 .
And we define (X n t ) t≥0 as the solution of:

X t = X n 0 + t 0 A 1 I t <n X n s ds.
This process looks like (1). The difference lies in the fact that, instead of waiting for an exponential time of parameter β 2 before jumping from a matrix to the other, the process waits for a time following a Gamma law of mean 2 β and variance 4 nβ 2 . The next result compares the stochastic process to the periodic one.

Lemma 5.1. For every T > 0, the process (X n t ) 0 t T uniformly converges to the deterministic process (X t ) 0 t T solution of (15). Proof. Let us call N n (t) the number of jumps before t of the process (

I n t ), τ n 1 , τ n 2 , ...,τ n N (t)
are the interjump times, they follow a gamma law of mean 1 β and variance 1 nβ 2 . We denote

a n t = t - N (t)
k=1 τ n k . We finally call ξ n 1 , ξ n 2 , ...,ξ n N (t)+1 ∈ {0, 1} the states. By simply solving the equation for the process (X n t ) and calculating the matrices exponential we obtain:

X n t = e -at B ξ n N n (t)+1 (a n t )B ξ n N n (t) (τ n N n (t) ) • • • B ξ n 1 (τ n 1 ) where B 0 (t) = cos(t) b sin(t) -1 b sin(t) cos(t)
and B 1 (t) = cos(t)

1 b sin(t) -b sin(t) cos(t)
.

Let us call N (t) the number of jumps of the deterministic process. We have:

X t = e -at B ξ N (t)+1 (a t )B ξ N (t) ( 1 β ) • • • B ξ 1 ( 1 β ).
We form the difference:

X n t -X t 1 N n (t)=N (t) B ξ n N (t)+1 (a n t ) • • • B ξ n 1 (τ n 1 ) -B ξ N (t)+1 (a t ) • • • B ξ 1 ( 1 β ) X 0 +1 N n (t) =N (t) B ξ n N n (t)+1 (a n t ) • • • B ξ n 1 (τ n 1 ) -B ξ N (t)+1 (a t ) • • • B ξ 1 ( 1 β ) X 0 .
We can prove by induction on N (t) that:

B ξ n N (t)+1 (a n t ) • • • B ξ n 1 (τ n 1 ) -B ξ N (t)+1 (a t ) • • • B ξ 1 ( 1 β ) -→ 0 in probability.
The case N (t) = 1 shows us what happens:

B ξ 2 (t -τ 1 )B ξ 1 (τ 1 ) -B ξ 2 (t - 1 β )B ξ 1 ( 1 β ) B ξ 2 (t -τ 1 ) B ξ 1 (τ 1 ) -B ξ 1 ( 1 β ) + B ξ 2 ( 1 β ) -B ξ 2 (τ 2 ) B ξ 1 (τ 1 ) C 1 |τ 1 - 1 β |.
By using the inequality of Bienaymé-Tchebychev, we gain the claimed convergence. Moreover we can say that this convergence also holds in L 1 as the random variable is bounded because our process does not explode in a finite time.

We prove now that P[N n (t) = N (t)] -→ 1 when n goes to +∞. First, let us write that N (t) = p and t = p β + η where 0 η < 1 β . We have:

P[N n (t) = p] = P [τ 1 + τ 2 + ... + τ p t < τ 1 + ... + τ p + τ p+1 ] = P   0 t - p i=1 τ i and t < p+1 i=1 τ i   = P p i=1 τ i - p β < η P   t < p+1 i=1 τ i | 0 t - p i=1 τ i   .
The first probability goes to 1 when n goes to +∞ thanks to the Bienaymé-Tchebychev inequality and the other probability is nonzero. This means that we have the claimed convergence: P[N n (t) = N (t)] -→ 1 when n goes to +∞. Back in the inequality, we take the expectation and using the previous results we prove that E[ X n t -X t ] goes to 0 uniformly on [0, T ].

Behaviour of the switched system with two centers of attraction

We slightly modify our initial system by shifting the centers of attraction of each flow. Assume that A 0 and A 1 are Hurwitz matrices and b 0 and b 1 in R d (with b 0 = b 1 ). For i = 0, 1 and

x ∈ R d , let t > 0 → ϕ i t (x) the solution of ẋt = A i (x t -b i ) with x 0 = x.
Remark 6.1. There exists C 1 and η > 0 such that, for all t 0,

ϕ i t (x) -b i Ce -ηt x -b i .
The goal of this section is to discuss the long time behavior of (X t ) if (I t ) t 0 jumps from 1 to 0 (resp. from 0 to 1) with rate λ 1 (resp. λ 0 ) i.e. (X t ) solution of (2). Let us couple two paths (X t , I t ) and ( Xt , Ĩt ) starting from (x, i) and (x, i) choosing the same jump times i.e. the same process (I t ) t 0 . Then, the difference

D t = X t -Xt is solution of Ḋt = A It D t .
Remark 6.2. If the initial discrete components are different, one has to wait for a random time T following an exponential law of parameter λ 0 + λ 1 and then, they can be chosen equal for all t T .

One can write D

t = R t U t with R t > 0 and U t ∈ S 1 . Recall that R t = R 0 exp t 0 A(U s , I s )ds . Let us define Ψ p (t) = sup u∈S 1 ,i=0,1 E (u,i) exp p t 0 A(U s , I s )ds 1/p .
Thanks to Markov property, the function Ψ p satisfies Ψ p (t + s) Ψ p (t)Ψ p (s). As a consequence, there exists χ p such that

χ p = lim t→∞ 1 t log Ψ p (t).
Moreover, Jensen's inequality ensures that χ p χ. As a consequence, it is possible that X t → 0 a.s. with χ p > 0 for some p > 0.

Remark 6.3. It is not easy to determine the sign of the Lyapunov exponent χ p . Lemma 6.4. When λ 0 , λ 1 and p are sufficiently small, there is convergence in Wasserstein distance for (D t ) after the coupling of the discrete components.

Proof. Once again, we assume that the initial discrete components are equal. Denote by N t the number of jumps for (I t ) t 0 before time t. One has I t = (-1) Nt I 0 .

D t = exp (t -T Nt )A T N t exp (T Nt -T Nt-1 )A I N t -1 • • • exp (T 2 -T 1 )A I T 1 exp (T 1 A I 0 )D 0 .
Using Remark 6.1, one has

D t C Nt+1 e -ηt D 0 .
If the jump rates of I are equal, (λ 0 = λ 1 = λ), then (N t ) t 0 is a simple Poisson process and

[E( D t p )] 1/p C exp -η - λ(C p -1) p t [E( D 0 p )] 1/p .
This ensures that when λ and p are sufficiently small one can establish convergence in Wasserstein distance, after the coupling of the discrete components (see [START_REF] Bardet | Long time behavior of diffusions with markov switching[END_REF]).

If the jump rates are not equal, (N t ) t 0 is not a Poisson process. For a fixed c ∈ R, let us define G i (t) = E i (c Nt ). Then, if T is the first jump time,

G 1 (t) = E 1 c Nt 1 {T t} + E 1 c Nt 1 {T >t} = t 0 cG 0 (t -s)λ 1 e -λ 1 s ds + e -λ 1 t .
As a consequence, thanks to an integration by parts,

G 1 (t) = cG 0 (0)λ 1 e -λ 1 t + c t 0 G 0 (t -s)λ 1 e -λ 1 s ds -λ 1 e -λ 1 t = cλ 1 e -λ 1 t + c -G 0 (t -s)λ 1 e -λ 1 t t 0 -c t 0 G 0 (t -s)λ 2 1 e -λ 1 s ds -λ 1 e -λ 1 t = -λ 1 G 1 (t) + λ 1 cG 0 (t).
Notice that, if λ 1 = λ 0 , one recovers that G 0 (t) = G 1 (t) = exp(λt(c -1)). In the general case, G 0 and G 1 are solutions of

y + (λ 0 + λ 1 )y + λ 0 λ 1 (1 -c 2 )y = 0 with respective initial conditions G i (0) = 1, G i (0) = λ i (c -1).
After solving this simple equation, it also comes that for λ and p small enough, we can establish convergence in Wasserstein distance.

We are now interested in the invariant measure, if it exists, of our process (X t ) t 0 solution of (2). Let us call T 0 = 0, T 1 , ..., T n the jumping times of the process (X t ) and τ 0 , τ 1 , ..., τ n , the interjump times following an exponential law of parameters λ 0 and λ 1 alternatively. Let us define Y n = X T 2n . By simply solving the equation 2 between each jumping times we obtain:

Y n = e τ n+1 A 1 e τnA 0 Y n-1 -e τ n+1 b 0 + b 1 .
Applying Kesten's renewal theorem ([11]) to the discrete process (Y n ) will give some informations about the tail of the invariant measure of (X t ). Theorem 6.5. If there is no deterministic control that makes the deterministic process associated to (Y t ) explodes, it means that there exists an invariant measure µ for our process and the support of µ is necessarily bounded.

In the other hand, we assume that χ < 0 and that there exists a deterministic control such as:

∃k ∈ 2N + 1 such as ∀x ∈ S d-1 , ∃t 0 , t 1 , ..., t k > 0 such as e t k A I k ...e t 1 A I 1 e t 0 A I 0 x > x . ( 16
)
We call B = e τ k A I k ...e τ 1 A I 1 e τ 0 A I 0 and B 0 , B 1 , ..., B n n random variables i.i.d following the law of B. We also assume that:

) 18
where λ is the Lebesgue measure on R n 2 . Under these assumptions, there exists a unique invariant measure µ for the process (Y t ) and it has an heavy tail:

∃p 1 > 0 such as ∀0 p < p 1 E µ [ Y p ] < +∞ and ∀p > p 1 E µ [ Y p ] = +∞.
Proof. Let x ∈ S d-1 . There exist t 0 , t 1 ,... ,t k > 0 such as: e t k A I k ...e t 1 A I 1 e t 0 A I 0 x > x .

Consequently, by continuity, there exist > 0 and δ > 0 such as: ∀s i ∈ J i = (t i -δ, t i + δ), e s k A I k ...e s 1 A 1 e s 0 A 0 1 + .

Let us write B = e τ k A I k ...e τ 1 A 1 e τ 0 A 0 , where τ 0 , τ 1 ,... ,τ k are k + 1 random exponential variables.

We have:

E [ B p ] E [1 τ k ∈J k ...1 τ 1 ∈J 1 1 τ 0 ∈J 0 B p ]
P(τ k ∈ J k )...P(τ 1 ∈ J 1 )P(τ 0 ∈ J 0 )(1 + ) p

As τ 0 , τ 1 ,... ,τ k are k + 1 exponential random variables, P(τ k ∈ J k )...P(τ 1 ∈ J 1 )P(τ 0 ∈ J 0 ) > 0.

Consequently:

E [ B p ] -→ p→+∞ +∞,
meaning that there exists x 1 > 0 such as E [ B x 1 ] = 1. Consequently, if the process (Y t ) does not explode, [1] ensures that under the invariant measure µ:

P( Y > r) ∼ c r x 1 .
It implies that µ has finite moments of order p < x 1 . Remark 6.6. In dimension 2, hypothesis ( 16) can be verified using the criteria in [START_REF] Balde | A note on stability conditions for planar switched systems[END_REF] which gives the existence of an explosive control for the switched system. In higher dimension, no general result gives this information. Remark 6.7. Hypothesis (17) and (18), directly extracted from a version of Kesten's renewal theorem in [1], tells us, basically, that the process (Y t ) creates density according to the Lebesgue measure and its projection on the sphere S d-1 is a recurrent irreducible process. These hypothesis are not easy to check in the general case. Corollary 6.8. If there exists a deterministic control that makes the system explode from one of the stable point b 0 or b 1 , then, since A 0 and A 1 are Hurwitz, the invariant measure has an heavy tail. Example 6.9. In dimension 3, it is possible to create an example satisfying Theorem 6.5 by picking the following matrices:

A 0 =    -a b 0 -1 b -a 0 0 0 -1    and A 1 =    -1 0 0 0 -a b 0 -1 b -a    .
When we consider each system Ẋt = A i X t separately, one can check that, when a < 1, the projection of (X t ) on the sphere tends to a different ecuador. Consequently, using results in [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF], it is possible to show that the process (X t ) solution of (2) satisfies hypothesis (17) and (18).

Figure 1 :

 1 Figure 1: Example of a trajectory of the process (1).

Figure 2 :Theorem 1 . 1 .

 211 Figure 2: The function χ for a = 0.15, b = 3 and u = 0.5.

Figure 3 :

 3 Figure 3: Sign of the function χ for a = 0.10, b = 2.5. Positive inside and negative outside.

Figure 4 illustrates

 4 Figure 4 illustrates Proposition 4.1.

Figure 4 :

 4 Figure 4: In blue the general shape of χ d for a = 0.1, b = 2 and u = 0.5. In red, x 0 is in E λ 2 for β = 1.

Figure 5 :

 5 Figure 5: For a = 0.15, b = 3 and n = 50, a second blow up area appears in χ.

max n 0 P

 0 B n ...B 0 x B n ...B 0 x ∈ U > 0 for all x ∈ S d-1 and any open ∅ = U ⊂ S d-1 . (17) P(B n 0 ...B 0 ∈ .) γ 0 1 Bc(Γ 0 ) λ for some Γ 0 ∈ GL d (R), n 0 ∈ N and c, γ 0 > 0. (

Acknowledgments. I acknowledge financial support from the French ANR project ANR-12-JS01-0006-PIECE.


Philippe Vergne

Super Low Traction under EHD and Mixed Lubrication Regimes

Introduction

Superlubricity, near frictionless sliding and super low traction

After the pioneered experimental works on superlubricity by Martin et al. on MoS 2 [1], Hirano et al. on tungsten and silicon [2] and the further confirmation by Dienwiebel et al. on graphite [3], many groups around the word investigated the occurrence of near frictionless sliding contacts. This large mobilization of tribologists, material sciences specialists and physicists has lead to emerging solutions involving new materials and coatings, the most promising being carbon based like graphite, diamond, carbon composites or diamond-likecarbons. Some of them are currently used in practical applications.

However the introduction of a fluid between two contacting surfaces remains the traditional and still the most efficient way to prevent contact failures when the operating conditions generate high contact pressures, large thermal dissipation or when the presence of worn films or particles is prohibited. In the field of lubrication, super low traction doesnot probably have the same significance compared to superlubricity of carbon based materials which gives friction coefficients lying within 5% (near atmospheric conditions) to almost 0.1% (under vacuum) whereas values encountered under classical "dry" conditions are almost always greater than 10%-20%. The situation is different especially in EHL: the highest friction coefficients are close to 10% when traction fluids are involved, i.e. fluids that have especially designed to transmit the highest friction, and they vary within 3-6% for the rest of lubricants. The range of variation is consequently very narrow and these typical values are really low compared to those obtained in dry contacts: as a consequence the gain expected from a super low traction regime (defined in section 2.2) in lubrication will be probably more limited, especially in the case of experiments conducted at the meso or macro scales. This weak perspective could be one explanation on the relatively low number of articles in recent literature dealing with lubricated superlubricity in the above conditions.

Nevertheless there is still strong interest in this topic and more generally in the fundamental understanding of friction between lubricated surfaces. Dowson and Ehret [START_REF] Dowson | Proc. Instn. Mech. Engrs[END_REF] have recalled that typical EHD films were about one micron thick when the first solutions to the elastohydrodynamic problem were proposed. But this situation has changed with time and nowadays EHD films are of nanometer rather than micrometer proportions. This has been possible thanks to numerous contributions -both experimental and numerical -on film thickness build-up mechanisms published during the last 20 years that improved our knowledge on very thin EHL films, the influence of surface features etc. Nevertheless, very few of these publications also deal with friction. A second interest concerns industrial applications that are developed with increasing demands for higher energy efficiency, durability, and environmental compatibility. Since friction is one of the main sources of lost energy in mechanical elements, it becomes a matter of urgency to propose innovative solutions to control and to optimize this parameter. An intermediate step would be a better understanding of the friction mechanisms under lubricated conditions and a significant improvement of friction prediction.

Chapter objectives and summary

In this chapter we will report and discuss the experimental appearance of super low friction forces that occurred in EHL or in mixed lubricated applications, i.e. tribological situations far away from those prevailing during nano or micro tribotests or during lubricated wear experiments. It means that we simulated lubricated contacts as those existing in real life, involving engineering surfaces and materials, applying representative speeds and normal loads. This domain is also called conventional tribology. The title of this contribution mentions both elastohydrodynamic and mixed lubrication regimes. Compared to dry conditions, we specifically focused on the lubricant response according to two directions: -From the rheological point of view to ensure that its behavior could favor super low traction under full EHD separation. -Based on the classical Striebeck diagrams that normally present a minimum friction in the EHL regime, to analyze the transition region between EHL and mixed lubrication where lubricated superlubricity could occur.

Traction versus super low traction

Generalities on EHD traction

Friction or traction in highly loaded lubricated contacts results from complex and coupled phenomena that are not yet totally understood. Formally one would have to account for two distinct contributions: rolling friction that comes from the inlet pressure rise and shearing friction that results from a velocity difference of the contacting surfaces. However both numerical and experimental previous works [START_REF] Gohar | [END_REF][6] showed that the contribution of the former term is generally negligible compared to the latter. Consequently we will consider in the following that traction only results from the lubricant strength to slip and/or spin motions occurring in the high pressure region of the conjunction. Slip, small film thickness and high contact pressure contribute to generate very important shear rates and very high shear stresses, as viscosity strongly increases with pressure this effect being one of those that allow the elastohydrodynamic lubrication (EHL) mechanisms to take place. Rheological and thermal effects can occur simultaneously. Moreover some unusual contact features reported in the literature suggest the occurrence of interfacial or boundary effects [START_REF] Dowson | Proc. Instn. Mech. Engrs[END_REF].

Compared to solid or dry lubrication, friction coefficient under EHL regime varies over a quite reduced range, from few % (see figure 1) to a maximum rising 10 to 12% in the typical case of traction fluids (figure 2). The traction coefficient (=friction coefficient) is usually evaluated and plotted as a function of the slide to roll ratio (SRR see equation 1) defined by the ratio of the sliding velocity that generates lubricant shearing (and hence friction) to the mean entrainment velocity that is an essential parameter in separation build-up.

e U U SRR / ∆ = (1)
where ∆U = U 1 -U 2 is the sliding velocity, and U e = (U 1 + U 2 )/2 is the mean entrainment velocity.

The question of rating "super low traction" compared to "common traction" is developed in the next section together with the main related experimental issues.

Super low traction and experimental issues

Compared to the two typical cases plotted in figures 1-2, super low traction characterizes not only situations where lower friction coefficients are encountered but also when different shapes of the traction curves are reported. A very steep friction increase from SRR = 0 is characteristic of lubricated contacts working under EHL conditions. Traction coefficients close to the maximum friction value are found for low slide to roll ratios, typically in the range 1 < SRR < 5%. However different behaviors can be found according to the operating conditions and the lubricant properties. Among them one can find the super low traction response that corresponds to lower friction values than those reported in figures 1-2. It can occur whatever the lubrication regime is, neither EHL nor mixed and gives a progressive friction increase with SRR. This peculiar behavior will be described and discussed in a next section.

Since under classical EHL conditions important friction values are obtained for even low SRR values, more attention has been paid in the past to control very accurately the velocities of the two specimens. This was also motivated by the existence of low SRR in ball bearings (typically few percents) that could generate much more traction variations than in the cases where SRR > 5%. However, in the super low traction regime the experimentalist has to face another practical problem: the challenge is now to measure very low friction forces with adequate accuracy. This difficulty can be illustrated by the following practical example. For a given normal load -let's say 25N -it was classically relevant to assume a friction force sensibility of at least ±0.2% of the normal load (in our example ±0.05N). For the purpose of super low traction study, the requirement becomes more demanding and an acceptable sensibility should be 5 or 10 times higher than the value presented in the above example. In the super-low traction regime, this leads to sensibility lying in the range ±0.01 to ±0.005N, these values being considered as minimum ones. Facing this requirement, it becomes evident that numerous devices will be no longer adapted to study super low traction, as for instance conventional large twin disk machines or basic ball-on-disk set ups. This new demand has been already claimed by several authors in order to achieve significant breakthroughs in the understanding of superlubricity. Realistic devices have to be both wellcontrolled and relevant to operating machinery [7]. In their recommendations for future endeavors, Perry and Tysoe [8] mentioned that both nanoscale devices (like AFMs) and macro-tribometers need to be improved. They underlined the importance of technical points that are rarely discussed in papers: reproducibility, reliability, calibration, uncertainty analysis, sample and surface preparation, environmental control etc… They also pleaded for the improvement of the capacities of tribometers by a more precise control of forces and speeds.

Experimental conditions

Compared to micro scale or nanoscale investigations, we applied operating conditions closer to those found in real lubricated mechanisms like in rolling bearing elements or in automotive components: concentrated circular contacts, medium to high contact pressures, continuous motion of both specimens, variable slide to roll ratios, smooth surfaces, controlled lubricant feeding flow… These operating conditions were fulfilled by using a ball-on-disk test rig, similar to those designed to measure film thickness in EHD contacts and already described elsewhere [START_REF] Molimard | Proc. of the 25 th Leeds-Lyon Symp. on Tribology[END_REF][START_REF] Hartl | [END_REF][11].

A polished one inch ball of AISI 52100 bearing steel is loaded against a flat disk and both specimen are driven independently to allow for any desired slide to roll ratio. The ball and disk velocities are controlled with high precision and the cumulated geometrical defects are adjusted to minimize any fluctuation within the contact. The bottom of the ball dips into the reservoir containing the lubricant, ensuring fully flooded conditions. The contact and the lubricant are thermally isolated from the outside and heated (or cooled) by an external thermal controlling system. A platinum resistance probe monitors the lubricant temperature in the test reservoir within ± 0.1°C. Parts in contact with the lubricant are made from chemically inert alloys and any type of material likely to react with the fluid (rubber, elastomer) has been inhibited. The balls and the disks were carefully polished and cleaned following a three-solvent procedure to ensure minimum surface contamination.

Traction forces and normal load were recorded via a multi-axis strain gauge sensor. It combines a broad range of measurable forces, appropriate sensibilities over the different directions and high stiffness. This facility is directly positioned between the main frame of the test ring and the vertical assembly that includes the brushless motor, couplings, the shaft and its bearings and finally the disk. This design provides several important advantages: -High linearity and sensibility due to a continuously applied prestressed state along the 3 directions. -Only static parts are involved, leading to high signal to noise ratio compared to measuring systems attached to moving elements.

Several materials have been used for the disks: BK7 glass, pure synthetic sapphire and AISI 52100 bearing steel leading to composite RMS roughnesses of the undeformed surfaces of respectively 4, 5 and 9 nm. High pressure rheology and/or film thickness measurements have been carried out on most fluids investigated in this chapter. These previous experiments provided appropriate data for the evaluation of the λ parameter as the lubrication regime will be an important factor in our analysis. λ is defined as follows: σ λ

/ min h = (2)
where h min is the minimum film thickness, σ the composite rms roughness of the contacting surfaces.

Furthermore:

λ > 3
gives full EHL separation,

3 > λ > 2
EHL regime but local contacts may occur,

2 > λ > 1 mixed lubrication regime, λ < 1
severe mixed lubrication regime.

Lubricated super low traction

This part reports and discusses experimental results where super low traction coefficients have been encountered. Several lubrication regimes (from EHL to mixed lubrication) and different types of lubricants will be considered. A table reported in annex summarizes the main rheological properties of these fluids together with their chemical composition and structure.

In the two first subsections, super low traction will be investigated under thick EHL conditions. Two really different lubricants will be studied and we will show that superlubricity can easily be explained thanks to the rheological behavior of each fluid.

The last subsection will concern mineral base oils, with and without additives. We will focus on the transition region between full EHL separation and mixed lubrication where film thickness becomes close to surface roughness. In this context special attention will be paid to the influence of the lubricants chemical structure.

Newtonian isothermal piezoviscous behavior

In this first section, the objective is to show how very low traction can be achieved using a simple viscous fluid that obeys in the simplest manner to EHD contact conditions. With this in mind we considered glycerol, a pure tri-alcohol characterized by its very compact molecular structure and low pressure viscosity coefficient. Results reported in figures 3-6 were obtained from various operating conditions varying the entrainment speed and the normal load and by changing the disk material from glass to sapphire and then to steel. Newtonian isothermal piezoviscous estimations of the friction coefficient based on the mean contact pressure P 0 (=P H .2/3, P H being the Hertzian pressure) and Barus law are also plotted in these figures with dashed or full lines. Experimental results are represented by symbols only.

The traction response of this fluid is really different from that of the two boundary cases showed in section 2.1, especially for the results plotted in figures 3-5. Firstly friction coefficient never exceeds 1% over the classical EHL range (0 < SRR < 50%) whatever the experimental conditions were. Secondly a linearly increasing friction coefficient is measured when the slide to roll ratio is increased. Finally a fair agreement is found between experimental results and those obtained from our basic numerical model. However increasingly noticeable deviations appear when contact pressure and/or entrainment speed are increased, as it can be seen in figure 4 (P H = 0.87GPa, 0.08m/s), in figure 5 (P H = 0.95GPa) and in most cases plotted in figure 6. Film thickness measurements -not reported here -showed that under pure rolling conditions (from 5mm/s to 5m/s, steel-glass contact, 40°C) glycerol behaved like a Newtonian piezoviscous fluid, as expected considering its simple and compact molecular structure. Shear thinning as may occur with polymers cannot occur with this fluid, only inlet shear heating has induced a drop on film thickness 0.8m/s. experiments of had weak influence on film thickness: thickness reduction of -7% at most for 0.38m/s, SRR=180% and 0.52GPa. This further study confirmed the idea that the applied power input and more specifically the pressure are likely the main causes of the observed deviations. A Hertzian pressure of 0.95GPa corresponds to a normal load that is 3 and 6 times the ones required to generate respectively 0.64GPa and 0.52GPa for a steel-steel contact. The term α.P 0 is sometimes used in the EHL community to describe the lubricant response: with glycerol, the deviation from the Newtonian isothermal piezoviscous behavior occurs when α.P 0 becomes greater than 3.1. Increasing the entrainment velocity also favors the deviation from Newtonian piezoviscous isothermal behavior. The transition occurs when SRR exceeds 60% at low speed (figure 5) and above 15% at 0.38m/s (figure 6). It is thus possible to estimate the in-contact mean shear rate by sliding to central film thickness. leads values of 4.10 +5 and 2.10 +5 s -1 at respectively 0.075 and 0.38m/s; in rheological terms it means that the deviation from Newtonian isothermal behavior appears more rapidly when the entrainment velocity increases. However one has to keep in mind that under the operating conditions of figures 5 and 6, glycerol also shows super low traction behavior. Measured friction coefficients remain lower than 1% when SRR varies from 0 to 50%.

Shear thinning

Shear thinning is probably the simplest and most frequent rheological effect that can reduce both film thickness and friction in an EHD conjunction lubricated by a conventional fluid. Shear thinning here concerns a steady shear rate dependence of viscosity. However it has been frequently combined with other effects like thermal heating and/or non linear high shear stress behavior, leading to certain confusion between the actual mechanisms that might influence friction. Compared to the glycerol results reported in this section were obtained with a very different fluid, in terms of molecular structure, rheological properties and tribological response. The main objective in studying this lubricant model was to demonstrate that shear thinning that affects both film thickness and traction can be described by a unique ordinary non Newtonian relationship of the power law type [12]. This was accomplished by accurate measurements in viscometers (carried out by S. Bair at Georgia Tech) of the shear response of a high molecular weight polyalphaolefin (HMW PAO) and in-contact accurate measurements (performed at INSA de Lyon) of film thickness and traction under conditions which accentuate the shear thinning effect. This fluid possesses a very low critical stress for shear thinning, high viscosity at ambient temperature and pressure viscosity coefficient close to those of commonly used formulated lubricants. Moreover it also showed super low traction behavior. In spite of these properties, experimental values of traction coefficient (black symbols in figure 7) are unusually low and sometime (at low speed and low slide-to-roll ratio) near the threshold of resolution of the sensor. In figure 7, the full line represents the Newtonian isothermal piezoviscous estimation of friction for the entrainment speed of 0.13m/s. Friction coefficients predicted by a numerical model assuming that shear thinning follows Carreau equation are plotted with dashed lines. Carreau equation is written: The Newtonian isothermal piezoviscous calculation greatly overestimates the actual traction behavior of this lubricant while the Carreau shear thinning model gives a good agreement with the measured data. We also showed [12] that central and minimum film thicknesses were insensitive to sliding and well described using the power-law relationship previously mentioned as equation (3). The in-contact central film thickness and traction were thus entirely predictable from the rheological properties obtained from viscometers using simple calculations. This proved that shear thinning -occurring mainly in the contact inlet -was the dominant effect that affected the shearing response of this fluid, in the absence of measurable thermal heating. As a consequence, the very low traction coefficients reported in figure 7 were attributed to this rheological behavior enhanced by the lubricant molecular nature.

2 / ) 1 ( 2 ) . ( 1
- •       + = n γ λ µ η (3)
Results discussed so far were obtained under operating conditions that generated thick EHL films: in figures 3-7, the values of the λ parameter defined in equation ( 2) varied in the range 6-150.

In this section, we focus now on mineral base oils submitted to thin film tribological experiments performed under operating conditions where λ was lower than 7, and in most cases even lower than 3. All the tests were conducted on steel-steel contacts. Several objectives were followed: to check if these fluids studied under these specific conditions could give super low traction and to pursue investigations on the role of the lubricants molecular structure on their frictional behavior. Mineral base oils of different structures were studied under 3 operating conditions: high contact pressure and high entrainment speed (0.95GPa, 3m/s, 40°C, figure 8), medium contact pressure but associated with lower entrainment speed at the same temperature (0.64GPa, 1.8m/s, 40°C, figure 9) and finally same pressure and speed conditions as in the second case but at higher temperature (0.64GPa, 1.8m/s, 70°C, see results in figure 10). These conditions lead to thinner and thinner film thicknesses and consequently they permit to increase the severity of the contact conditions Except MIN and ISO, the lubricants exhibit similar rheological properties (see table in annex) leading to almost constant λ values for the results reported in figure 8 correspond to λ equals to 7 (full separation) for MIN, 2.1 for ISO and 2.9 for the others fluids. Apart from MIN that gives friction coefficients of 3% for SRR absolute values of 50%, ISO, PN and ARO fluids show moderate traction values and a progressive friction increase when SRR is increased. Base oil containing aromatic fractions (ARO) gave the lowest traction, then the mixture of paraffinic and naphtenic fractions (PN), the isoparaffinic base oil (ISO) and finally the more viscous fluid considered here (MIN). Even if data reported in figure 8 show that both relatively low friction coefficients and smooth variations with SRR could be encountered, these preliminary results seem still far away from super low traction. The cases plotted in figure 9 represent λ values of 5.4 for MIN and respectively 2.2 for PN and ARO: ISO traction results are not reported in this figure because this fluid (100% isoparaffinic) produced much higher traction coefficients (6 to 8%) than the other lubricants. This deviation is likely due to the inability of this base oil to maintain a low friction when operating conditions become more severe ( λ=1.7 → mixed regime). This has been confirmed by further experiments where friction coefficients higher than 10% were found applying the same speed and normal load but increasing the temperature and consequently decreasing λ. It should be noticed that under the operating conditions of figure 9, the low traction behavior of PN and ARO follows an almost linear increase when increasing the absolute value of the slide to roll ratio. A similar response has been found for the MIN base oil at 70°C where a maximum friction lower than 1% has been measured at SRR= 50%, 0.64GPa and 1.8m/s. LP traction results (see figure 1) have been obtained under the same operating conditions than those mentioned in figures 8-9. By comparison with base oils studied before and in spite of its low molecular weight, viscosity and pressure viscosity coefficient, the traction response of this 100% linear paraffinic fluid follows the classical EHL shape. Contact pressure has a dominant influence: rheological investigation carried out on this linear hydrocarbon showed the appearance of phase changes at very low hydrostatic pressures. Under the contact dynamic conditions all occurs as if the fluid was frozen within the conjunction and gave an almost constant friction coefficient only dependent on the slide to roll ratio Moreover this behavior is consistent with solidification theories and visco-plastic models described in EHL literature. It also should be noticed that friction slightly increases when the pressure and the entrainment speed are decreased. This variation denotes a change from full EHL separation (λ≈3) to mixed lubrication regime (λ≈2).

These results showed that under mild contact pressures but under very thin film conditions (2<λ<3) super low traction can occur with simple mixtures of mineral fractions. This superlubricity regime should be considered as an optimum compromise that occurs over a quite narrow range of operating conditions. Further experiments run at higher temperature have confirmed that friction may significantly increase when λ is approaching 1, according the lubricants chemical structure. It is not possible to advance a physical explanation on the near frictionless behavior of the mixtures, probably a favorable compromise between the paraffinic parts chain length and the lubricity contribution of naphtenic and aromatic fractions. However it is easier to understand why pure linear paraffinic (pressure induced rheological effect) and isoparaffinic (low viscosity and piezoviscosity coefficient, lower coverage of rubbing surfaces) base oils are unable to produce such interesting friction properties.

Additive influence

Results presented in the previous section were mainly obtained when 2<λ<3, i.e. when local contacts between the specimen surfaces may occur that corresponds to the transition between full EHL separation and mixed lubrication. In this situation additives are usually introduced to extend the acceptable working range towards real mixed regime. Here we chose to add 8% w/w of lauric alcohol (noted LA in figure 10), actually a mixture composed of almost 70% of dodecanol and 30% of tetradecanol. This additive is considered as a lubricity improver (= friction reducer) and is used especially in the field of metal forming. Results will be discussed according the chemical structure of the base oils. In presence of lauric alcohol, friction given by MIN is slightly lower than the one measured on the neat base oil whatever the operating conditions (see figure 10,70°C). The additive has in this peculiar case (full separation even at 70°C) a rheological effect in reducing a little the film thickness. At 70°C a super low traction regime is achieved in both cases, with and without lauric alcohol. Friction generated by ISO is considerably reduced when LA is added, however the traction coefficients remain higher than those found with the other lubricants and above the typical yield value that defines super low traction. Even if the additive seemed to have a negligible influence on ARO friction at 40°C, the results obtained at 70°C show a significant but however limited increase (from 1.7 to 2.7%). The presence of aromatic fractions in the base oil suggests that an induced antagonistic effect between lauric alcohol molecules and the aromatic chains is the cause of the friction increase. The tendency found with PN is totally opposed to the last one (ARO). Without additive we observe a friction increase when the temperature varies from 40 (figure 9) to 70°C (Figure 10), due to transition from soft to more severe mixed regime (λ varies from 2.2 to 1.4). Traction drops when lauric alcohol is added. As for MIN+AL, one can consider that a super low traction regime is reached when PN+AL is used at this temperature. Lauric alcohol addition permits to reduce NP friction for the three imposed operating conditions, but in any case values representative of super low traction regime were measured. has been shown that a super low traction regime may occur. The apparition of such lubrication regime depends on the lubricant structure and rheological properties. For different reasons pure isoparaffinic and pure linear paraffinic fractions gave relatively high traction coefficients whereas mixtures of paraffinic and naphtenic chains showed superlubricity, enhanced when fatty alcohols were added. Furthermore it should be underlined that operating conditions that permit to observe super low traction coefficients correspond to λ values lying in the range 1-3. This means that in some cases the classical Striebeck curve that usually presents a minimum friction value that coincide with EHL regime could be modified to take into account an almost frictionless behavior occurring at the transition between full film (EHL) and mixed lubrication regimes.

Discussion and conclusion

When trying to understand in detail the tribological mechanisms and interactions occurring in real contacts it would be useful to conduct separate on three different scales, the macro, micro and nano scale and to study separately the mechanical and the tribochemical changes taking place in the contact. However much of the current information is fragmented, with linkages between individual results need have yet to be established.

In our case, results and subsequent analyses were based on macro scale tribological simulations. As an illustration, we showed how the rheological behavior can influence friction generated by the different fluids. Moreover, we may consider that the micro scale is of minor interest mainly because very smooth surfaces have been used. The nano scale analysis appears much more promising. The results from Krim [13] for instance, showed that liquid layers being more flexible and therefore slightly more commensurate with the surface, exhibited higher friction than their solid counterparts. However this finding has to be considered as a boundary case of our simulations for two reasons: it entailed liquid monolayers confined between extremely smooth and clean substrates made of high purity materials whereas our experiments involved at least several tens molecular layers sliding between bearing steel surfaces. With the objective to merge information gained on the nano scale with that observed at the macroscopic scale, nonequilibrium molecular dynamic (NEMD) simulations may appear relevant. Jabbarzadeh et al. [14] showed the correlation between the degree of branching of C 30 alkane isomers and many important flow properties. They simulated Couette shear flow of thin (≈7nm) films submitted to very high shear rates. However, in spite of a great interest in the knowledge of lubricants high shear rate behavior, they assumed constant thickness and ambient pressure conditions. Bair et al. [15] proposed an approach that combined high shear rate NEMD simulations and high pressure rheological experiments to calculate EHD traction forces based on a Carreau simulated and experimental traction was successful but validated for only squalane. More recently Jabbarzadeh et al. [16] found that dodecane could exhibit a very low friction state when confined between perfect surfaces. However it appears that a confusion between shearing and friction exists and somewhat limits the contribution of this approach and the nanorheology one to a better knowledge of mechanisms that occur in lubricated macro contacts. Another direction to make substantial progress in the understanding of friction and super low friction concerns the influence of the fluid structure. In [14], it is reported that linear alkanes should give lower friction (apparent viscosity), higher layering (and shear thinning) and lower slip than branched fluids of the same carbon chain length. Based on molecular interaction considerations and with the objective of molecular design of efficient traction fluids, some authors [17] showed that the lowest traction coefficients were obtained with fluids that could not interlock each other when they passed through the contact, like aromatic compounds. These tendencies qualitatively confirm some of our experimental results obtained on mineral base oils. However some weakness in the arguments (effect of contact pressure, real lubricant composition, engineering surfaces influence…) do not allow a formal link between nano and macro approaches.

To conclude on super low traction, it is expected that this concept will generate a similar scientific passion in the tribology community than thin film lubrication did in the early 90's. This desire is at first justified as friction reduction remains a major challenge to reduce energy losses, to improve durability of manufacturing goods and to meet constantly renewed environmental requirements. Another motivation for this lies in the fact that much more numerous applications will operate under conditions where lubricated superlubricity could occur. The use of low viscosity lubricants is one but an example: low viscosity could be induced by increasingly working temperatures (like in automotive engines) or an intrinsic property of the fluids like in applications where fuels have to lubricate mechanisms. Concerning bridging the gap between nano and macro scale analyses and the improvement of friction understanding, it is expected that these issues can also contribute to a more quantitative prediction of traction coefficients. Obviously super low traction is included in these perspectives which could lead to extremely efficient solutions like for instance the one recently described by Kano et Note that the pressure viscosity coefficient α * is actually the reciprocal asymptotic isoviscous pressure defined by Blok [19]. This parameter permits to account the pressure viscosity coefficient variation with pressure. When introduced in the classical EHL relationships, it gives the best agreement with measured film thicknesses.

Additional references

Figure 1 :

 1 Figure 1: Low traction values given by linear paraffinic mineral base oil (LP) at 40°C.

Figure 2 :

 2 Figure 2: Typical high friction results obtained with a traction fluid (Santotrac 50, 25°C)

Figure 3 :Figure 4 :

 34 Figure 3: Traction curves measured with glycerol at 40°C, steel -glass contact.

Figure 5 :Figure 6 :

 56 Figure 5: Traction curves measured with glycerol at 40°C and 0.075m/s, steel -steel contact.

Figure 7 :

 7 Figure 7: Super low traction obtained with a HMW PAO (75°C, 0.53GPa, steel-steel contact).

4. 3 . 1 .

 31 Mineral base oilsThe nomenclature chosen for these fluids is detailed in annex: ARO, PN, ISO and LP present typical carbon chain length in the range C 11 -C 14 as MIN contains longer chains (C 17 -C 21 ).

Figure 8 :

 8 Figure 8: Traction curves obtained with different mineral base oils at 0.95GPa and 3m/s.

Figure 9 :

 9 Figure 9: Low traction behavior obtained with 3 mineral base oils under more severe conditions than in figure 8.

Figure 10 :

 10 Figure 10: Comparison of traction coefficients obtained on base oils with and without lauric alcohol at 70°C.

  19. H. Blok, Proc. Int. Symp. on Lubrication and Wear, Houston 1963, McCutchan Pub. Corp., Berkeley, 1. 20. S. Bair, personnal communication.

  al.[18].

				the
	Name	Type	Content	Structure	Viscosity (Pa.s)	α* α (GPa -1 )
	Santotrac	Traction	Dicyclohexyl
	50	fluid	alkane	C 18 H 34	0.056 at 25°C	36 at 25°C
			+ additives		[20]	[20]
	Glycerol	Trialcohol		C 3 H 8 O 3	0.293	5.4 at 40°C
					at 40°C	[20]
	HMW PAO	Synthetic	Polyalpha	M≈30000	1.42	14.8
		hydrocarbon	olefine	kg/kmole	at 75°C	at 75°C
	MIN	Mineral	paraffinic +	C 17 -C 21	0.0047	13.5
		base oil	naphtenic		at 40°C	at 40°C
	ARO	Mineral	paraffinic
		base oil	+ naphtenic	C 12 -C 14	0.0015	10.2
			+ aromatic		at 40°C	at 40°C
	PN	Mineral	paraffinic +	C 12 -C 13	0.0015	10.5
		base oil	naphtenic		at 40°C	at 40°C
	ISO	Mineral	100% iso-	C 11 -C 14	0.0011	8.7
		base oil	paraffinic		at 40°C	at 40°C
	LP	Mineral	100% linear	C 13 -C 14	0.0015	8.4
		base oil	paraffinic		at 40°C	at 40°C

thank several colleagues who accepted to experimental data (S. Bair, Georgia Tech, Atlanta, USA) or who contributed to measurements on mineral base oils (F. Wiltord, former PhD student at LaMCoS) and on glycerol (K. Yagi, post-doctoral fellow at LaMCoS). Special acknowledgments to J.-M. Martin (LTDS, Ecole Centrale de Lyon, France) who inspired the study on fatty alcohols and to Guillermo Morales (SKF-ERC and visiting professor at LaMCoS) for his valuable comments on the manuscript.