Aurelia Vasile

The Classical Antiquity in the Romanian cinema. Ancient Rome: a precious enemy

Keywords: Romanian cinema, peplum, communist rule, nationalism

L'Antiquité, mais particulièrement la Rome Antique fut un sujet très important dans le cinéma roumain pendant le régime de Nicolae Ceauşescu. Certaines périodes de l'histoire romaine sont connectées à la civilisation dace et par conséquent, beaucoup de produits culturels roumains sont focalisés sur la relation entre les Romains et les Daces. Le régime de Ceausescu confère une importance particulière à cette période qui devient le berceau de la nation roumaine. La signification de l'histoire antique évolue d'une acception modérée dans les années 60, caractérisée par la synthèse de deux composants ethniques (le Daces et les Romains) à l'une radicale, pendant les années 80, lorsque l'élément romain est amoindri, tandis que l'autochtone est muni de tous les symboles fondateurs. Le but de ce travail est de révéler le rôle de Romains dans cette logique nationaliste et les moyens cinématographiques mis en oeuvre pour leur donner vie.

The Classical Era and especially the Ancient Rome was a very important subject in the Romanian cinema during Nicolae Ceauşescu's rule. Certain periods of Rome's history are connected to the Dacian civilization and consequently, many Romanian cultural products are focused on the relationship between the Romans and the autochthons (the Dacians).

Ceausescu's regime attaches an ideological importance to this period which becomes the cradle of Romanian nation. The significance of the antique history evolves from a moderate interpretation in the 60s, characterized by a melting of two ethnic components (the Dacians and the Romans) to a radical one, during the 80s when the Roman element is minimized while the autochthon is endowed with all the foundation symbols. The purpose of this work is to reveal the role of Romans in this nationalist logic and the filmic means to give them life.

After a transition period following the WWII, the Romanian cinematography was nationalized in 1948 and therefore, the production, the import, the distribution of films as well as the organization of cinemas' network are managed by the State institutions (The Committee of Cinematography, The National Center of Cinematography, Centrala "Româniafilm", or the ministerial institutions like the Committee State for Culture and Art (in the '60s) and the Council on Socialist Culture and Education (from the '70s), etc) in an extreme centralized way, following the political and ideological directives of the Communist Party. During the first decade, the organization of cinema and conception of films are dependent on the Soviet system and its vision of art. The topics, as well as the film aesthetic approaches come within the theory of "socialist realism", conceded to the rank of artistic criterion by Andrei Jdanov, the leader of the agitprop in USSR. Consequently, films made during the first years of the communist regime in Romania are characterized by a dogmatic rigidity and a simplistic and Manichean vision of the society. The privileged genres are especially social contemporary dramas, as well as war films. For this reason, the first socialist productions are constituted by austere films depicting and glorifying the proletariat's struggle toward socialist progress and communists glorious exploits during the war.

It is only from 1962/1965, in a climate of international thaw, that the party releases the pressure on filmmakers, but also on the conception of film and allows for the first forms of entertainment. In this context, the entertainment is at the same time a way to calm the protests and to gain support from the population, but also a financial source. At the beginning, the most exploited popular genres are the comedy and the musical, but thanks to the technical and economic means and especially to the explosion of the coproduction system during the '60s, the new blockbusters are "cape and sword" films and historical reconstitutions. After he became the leader of the party in 1965, Ceausescu pursues the mixed orientation "ideology and entertainment" and pleads for cultural diversity.

The interest of the Romanian cinema in classic period has a double meaning. On the one hand, the historic film is a form of integration in the world cinema in which a real leaning for peplum is to be noticed since the '50s. On the other hand, the reconstruction of the Antiquity is the result of the ideological thaw characterizing the Romanian politics and culture in the 1960s. For the first time, directors are authorized and even encouraged to abandon the eulogy of Soviet Union and to approach the Romanian past.

Classical Era in Romanian cinema and the ideological stakes

Three feature films of major importance for the Romanian cultural policy mainly constitute the filmic representation of the Classical Era: The Dacians (Dacii, Sergiu Nicolaescu, 1967), designating Dacia's inhabitants, the antique territory of a part of Romania,

The Column (Columna, Mircea Drăgan, 1968), name indicating the column built by the emperor Trajan (98-117) to celebrate his victory against the Dacians and Burebista (Gheorghe Vitanidis, 1980), the name of a Dacian king between 82 and 44 BC. These films are a part of a new genre invented in the 1960s: "the national film epic". Made during the golden period of coproduction in Rumania, the first two films were the result of international collaborations with French and West-German producers. The casting is the visible proof of this cooperation.

In The Dacians (Sergiu Nicolaescu, 1967) the audience can identify the French actors Pierre Brice, Marie-José Nat and Georges Marchal. In The Column (Mircea Drăgan, 1968), the foreign performers are also very important: Richard Johnson in the role of a Roman general, Antonella Lualdi, known for her roles in historical productions, acts here as a Dacian woman and Amedeo Nazzari, the romantic male lead of the Italian cinema during the fascist period, specialized in the international spectaculars in the '60s, interprets the role of the emperor Trajan.

The Dacians (Sergiu Nicolaescu, 1967) and The Column (Mircea Drăgan, 1968) tell the story of the Dacian people and their conflicts with the Roman Empire. Both films were conceived in a narrative continuity. The first one focuses on the Dacians' resistance and depicts Domitianus' campaign in 87 A.D. in Dacia, against the army of local king Decebalus.

The Romans lose the battle, but not the war. The following history was told by the second film. The Dacians were defeated and the story insists afterward on the relations between Romans and Dacians after the conquest of Dacia by the emperor Trajan in 106 AD. Dacia is transformed into a Roman province, while the autochthonous people hesitate between cooperation and resistance. The resistance lasts until the arrival of the barbarians who are even more threatening than Romans, and therefore, the Dacians becomes allied with their first enemy.

As for the film Burebista (Gheorghe Vitanidis, on 1980), the events are more ancient and present the story of Burebista, the Dacian king, who unified for the first time the Dacians tribes. He represents the same symbol as Vercingétorix for the French history, Boudicca for the English history, Ambiorix for the Belgium history or Arminius for the German history.

Although they had a audience (5 million spectators for The Column, 7 million spectators for The Dacians, 2.5 millions for Burebista), their importance is more ideological than artistic and commercial. These films are produced and distributed in different ideological contexts. The first two correspond to a period of recovery of the national values, while the third, Burebista, made in 1980, is a purely nationalist project answering the leader (Nicolae Ceausescu) legitimization need. The film was commanded to join the series of festivities dedicated to the celebration of 2050 years since the creation of "the unified and centralized State" of Burebista1 .

The essential purpose was to represent the origins of the Romanian people anchored, according to the Romanian collective imaginary, in Antiquity time of Dacians and Romans.

The origins' problem enhanced a lively interest especially in the political and cultural debates from the end of the 18th century2 . Both Dacians and Romans were constantly, sometimes alternatively, sometimes concomitantly, endowed with the founders' role they had to assume for the last two centuries3 . The nationalism characterizing the second half of the communist period fully exploits this theme. During 25 years of the Ceausescu's reign, the history view changed twice in a significant way. Starting from 1965, when he became the party leader, and until 1971, year of publication of the "July Theses"4 , the ideological orientation of the party was favorable to a recovery of the national values occulted during the previous period.

Dacians and Romans, forgotten for a long time, make their reappearance on the historic scene as founding fathers of the Romanian nation. This is the context of release of The Dacians and The Column. The Dacians, as inhabitants of the territory of Romania, represent the ethnic basis and benefited from a more favorable glance, compared to the Romans who represent the conqueror. The autochthonous is an element more important for this period than the nobility and the glory of the winners, or the migratory population 5 . The national ideology of the last two centuries facilitated this type of foundation, anchored in the oldest roots, to the detriment of the traditional one, embodied by a character that came from elsewhere 6 . Furthermore, the expansionist attitude was condemned by Ceausescu during the invasion of Czechoslovakia in 1968 and his success leads to the proliferation of this type of speech even toward the past. In this way, the role of Romans loses of its greatness, and this is replaced by the autochthons' simplicity. Nevertheless, they are conferred certain missions and functions with an oscillated evolution during the communist period. Thus, the situation is different in the '80s, the period of Thracomania current, because the official ideology becomes more and more nationalist and the historical theories stress the agedness of Dacians, their advanced civilization. In that case, the Roman is strictly an enemy, without any concession for his contribution on the foundation of Romanians'.

The Roman: the conqueror

The filmic image of the Roman is mainly associated with that of the "conqueror". The Romanian cinema invents nothing with respect to the universal image of the "conqueror", which is diffused and reproduced in a multitude of speeches. Some major figures of generals, politicians or emperors (Julius Caesar, Scipio Africanus, Trajan) are good enough to illustrate this idea. But the point of view can change. Thus, the term "conquering" has a double meaning, because it indicates at the same time a virtue and a vice. The Roman of many Italian or American peplums is a remarkable conqueror, a model of bravery. In the Romanian film, this value has negative connotations and from this perspective he joins the type of representation conveyed in the French film about the same subject, such as Vercingétorix (Jacques Dorfmann, on 2001) or the popular series Astérix. 5 For the relation between "Autochthonus" and the "Migratory Populations" see Gheorghe-Alexandru Niculescu, "Nationalism and the Representation of Society in Romanian Archaeology" in Nation and National Ideology. Past, Present and Prospects. Bucharest, New Europe College 2002, pp.209-234 6 The problem of the Origins in the imaginary of the modern and traditional societies was treated by Mircea Eliade, Le mythe de l'éternel retour, Paris, Gallimard, 1949; idem. Nostalgie des origines : méthodologie et histoire des religions, Paris, Gallimard, 1971 The film The Dacians focuses from the beginning on the Roman power. Produced during a very favorable period to the pageant, economically, artistically and politically, the film benefits from important financial advantages. The film direction is given to Sergiu Nicolaescu who is going to become, afterward, the specialist of spectaculars in Romania. He wants to impress from the beginning of the film. Before even the credit titles, the Roman army submerges the screen. A succession of static shots, presented gradually, beginning by a long shot of the army and until a close-up on a detail of the Roman uniform creates the image of crushing, invincible and especially uncountable strength. This sensation is highlighted by the orderly alignment of the soldiers in the field, by the low-angle shot of their commander and by the jerky music of brass band. This is taken over, at the linguistic level, by a verbal exchange between the two opposite camps, the Romans and the Dacians, showing the arrogance of the first ones and the determination of the second ones: "Open the door. We give you the life and the freedom / Who are you? / The masters of the world. / You will succeed, only if we die". (The Dacians, 1967).

Furthermore, this vision is evoked by Decebalus, the first character-king of the Romanian peplum. He dreads the strength of Romans: "they lost battles, but they were never defeated. They are stubborn, powerful, numerous, rapacious" (The Dacians, 1967). This approach exceeds the frame of the simple conflicting relationship between two opposite strengths and contains a mythical dimension. The Roman is not any more a common rival, he is the absolute enemy.

The same image is transmitted in Burebista. The description of Caesar as ambitious and invincible consolidates the portrait of the Roman invader. He is represented in front of his solders after the victory against the Gauls. Lacking financial means, the film only briefly evoke the strength of the Roman legions and their conquest ambition in a sequence where Caesar speaks to the army. The scriptwriter Mihnea Gheorghiu attributes to Caesar a sentence that summarizes in few words his international strategy: "My future book will be called De bello Dacico" (Burebista, 1980). From his part, the director uses his expression means, to bring his tribute to the conquerors. A low-angle shot of Caesar pedestrian puts him on a pedestal of invincibility. His parade in front of the army is supposed to show the greatness of the winner, but the director chooses to suggest the power and the monumentality of the Roman army by the cheering of the soldiers. The audience is forced to content itself with this substitute and to imagine their number by the frequency of the sound. The final result was a cheep ridiculous scene. We can assume that the cause of this disappointing result was the very poor film budget, the effects of which are obvious throughout the film. Nevertheless, the intention to present the Roman Empire as a powerful, invincible force counts more for this study than the technical success. This picture is completed at the narrative level by an atmosphere of Dacia as a besieged estate, a permanent target of the Roman attacks. So, the council of the State is on alert. Some characters, gladiators and soldiers in the Spartacus' army found refuge near Burebista with the aim to avenge the death of their leader. Burebista organizes war preparations in training fields in case of a conflict with Romans. The war seams threaten constantly the Dacian civilization.

The main problem of the directors was the absence in the history of a big battle between Dacians and Romans, and therefore the impossibility to build a dramatic framework around the fight and the impossibility to show Dacians' skills. In absence of an important military conflict, the directors invent a permanent state of alert within the Dacian population, ready at any time to resist the enemy attacks: the farmers train for battle; the manufacturers make weapons; the Council of State stays vigilant. By all this details, the film reaches two objectives: first, it shows the qualities of Dacians, with whom the public was supposed to identify itself and second, indirectly, it hints at the dimension of the Roman threat.

The image of the conqueror is often accompanied with the cruelty, the frivolity, the cowardice and the arrogance. Several sequences insist on these aspects, and present them as inevitable. For instance, the report of the military and political situation of Romans underlines the barbaric nature of the offensive: "Romans rise towards us with envy for slavery and for plunder" (Burebista, 1980). Their presence in the Greek towns of the Black Sea is shown as a permanent aggression against the inhabitants. This collective image of the Roman is doubled by the individual characters. Clearly, the emperor Domitian and his general, Cornelius Fuscus in The Dacians or Gaius Antonius Hybrida proconsul of Macedonia and Julius Caesar in Burebista, are the main negative characters of the stories. In The Dacians, the emperor Domitian and his Praetorian Guard Cornelius Fuscus hold the role of frivolous and cowardly Roman leaders. This typology results from the information selected by the scriptwriter, Titus Popovici, from the biography of these two personalities. The reign of Domitian was one of the most controversial subjects, but the choice of the authors went to the dark aspects of his career. Thus, the emperor in The Dacians is an ambitious, vain, immoral, treacherous and cruel person. The origin of this representation is without no doubt The Lives of Twelve Caesars by Suetonius who depicts Domitian in very negative colors, which was the perfect occasion for the scriptwriter to stress the antithesis between the wrong reasons of Romans enterprise and the defensive approach of Dacians.

Less than his master, Cornelius Fuscus reinforce the dark iconography of the Roman leader. It is likely is that Titus Popovici used the few notice about Fuscus in the work of Tacitus, Histories ("Finding his pleasure not so much in the rewards of peril as in peril itself, to assured and long acquired possession he preferred novelty, uncertainty, and risk" Book II, 86) and in a Juvenal Satire ("Fuscus who planned battles in his marble halls" Satire 4, 112), he adds some dramatic situations illustrating and exaggerating the comment of classic authors, creating a despicable character. In the film, he is presented like an incompetent coward and ambitious commander. This description of the Roman leader in The Dacians equally applies to the characters presented in Burebista. Hybrida, mentioned by the historical sources as a cruel military and politician (Quintus Tullius Cicero, De petitionis consularibus, 2; Dion Cassius, Roman History, XXXVIII, 10) is the key character of the film. He was the perfect alibi to accuse the Romans and in the same time to build a credible and truthful intrigue, based on the antic sources. The scriptwriter, Mihnea Gheorghiu catches this opportunity and creates around him one of the essential intrigues of the film. He supplies the perfect model to embody the evil.

One of the characters qualifies him as "looter and killer of the Gets of Scythia Minor" (Burebista, 1980). Moreover, he proves it himself afterward. His frivolity is revealed during the orgies organized in his residence and the indecent comments to the leader of the city: "The games for Dionysus could convince your women to be more kind with my soldiers after so many days of fasting" (Burebista, 1980).

But after all, the image of Romans is only a way to highlight the qualities of Dacians.

Unlike the other two films, the authors of Burebista reduce the Romans to the role of a reverted mirror, to bring to light the place and the importance of Dacians and their king, which reveals the personality cult, encouraged in Romania and arrived at its highest limit in 1980.

The Roman: the civilizing

If the first cinematic attribute of the Roman is linked to his victories and to his political and territorial domination, to which the filmmakers added a filter of condemnation, the second one, as paradoxical it may seem, concerns the same characteristics, except they are glorified.

Representing the Roman in a dual way proves his importance in the historiographical view of the formation process of the Romanian people. From this point of view, the war between Dacians and Romans does not constitute any more a completely tragic event. Nevertheless, the balance between the two parts was never even. During the regime of Ceausescu, considering the official historiographical speech which privileged the autochthonous roots, the weight of Dacians was more important than that of Romans.

Moreover the filmic representation of the Romanian antiquity helps illustrating the official ideology. The Dacians are presented like a people endowed with unlimited virtues, which demonstrates the sympathy of the filmmakers towards them, in disfavor of the Romans.

Nevertheless, the later ones are not excessively demonized because they represent the other half foundation of the Romanian people. As ancestors of the Romanians, they do not accomplish the function of "the bad guy" up to the end. This one is well-balanced, toneddown, improved.

The characters of Severus in The Dacians and Tiberius in The Column, are the perfect models. Result of a dramatically-heroic combination made by the scriptwriter Titus Popovici, both Romans weigh heavily in the balance between "good and bad". Severus and Tiberius are presented being very cultivated and educated, with a high sense of morality, which determines sometimes the audience to feel inclined towards "the enemy".

The perfect model to illustrate this idea is The Column, almost completely dedicated to Romans. The accent is put on the possibilities of cohabitation between Dacians and Romans.

While Dacians accepts with difficulty the defeat and the death of their king, their rivals want peace. The emperor Trajan declares it from the beginning of the film, as answer to a suggestion of his generals to exterminate the population: "I conquered Dacia to have peace in this place of the world, and it will be peace. A Roman peace. With them. Together." (The Column, 1968) For the first time, Romans hold the role of peacemakers, whereas Dacians leads a campaign of resistance against the occupants. Moreover, a part of the Dacians community work with their conquerors for the construction of the roman fort, which is a mark of their friendship and the pacifist cohabitation after the war. Completed by the love story between Andrada, a Dacian woman and Tiberius, the Roman general and followed by the birth of their child, the idea of cohabitation symbolize the emergence of a new people, the Romanians.

Besides, in The Column, the Romans illustrate for the first time the idea of civilization. Tiberius defines himself not only as a soldier, but also as a man with a noble education. The Romans embody the civilizing mission and bring, once Dacia conquered, new technologies: aqueducts, trenches of defense around the roman fort. They suggest typically Roman cultural events. In the end, even vanquished Dacians are obliged to recognize their defeat and the importance of the military and cultural experience of their opponents.

An analysis of this phenomenon to Florentin Dragoş NECULA, « Comunism in Dacia : Burebista contemporanul nostru » in Analele Universităţii Bucureşti, istorie, XLII-XLIII, Bucuresti, Editura UniversitatăţiiBucureşti, 1993Bucureşti, -1994, pp. 37-51 , pp. 37-51

Antoine Roger, Les fondements du nationalisme roumain(1791-1921), Genève, Droz, 2003

The role of the two founders people in the Romanian imaginary, during the last two centuries was analyzed by Lucian Boia, Istorie şi mit în conştiinţa românească,Bucuresti, Humanitas, 2002, (1997 for the first edition), pp. 135-179 (English version History and myth in Romania consciousness, Budapest, New York, Central European University Press, 2001)

Scînteia 7 July 1971, "Propunerile de măsuri prezentate de tovarăşul Nicolae Ceauşescu Comitetului Executiv al CC al PCR, pentru îmbunătăţirea activităţii politico-ideologice, de educare marxist leninistă a membrilor de partid a tuturor oamenilor muncii". [Exposition made by comrade Nicolae Ceauşescu on the Program of the RCP for the improvement of ideological activity, for raising the general level of knowledge and the socialist education of the masses, for grounding the relation in our society on the principles of socialist and communist equity.]

Approximately ten years later, in Burebista, the perspective of Romans changes fundamentally. They are almost eliminated from the diegesic space, whereas narrative, they get the status of the absolute enemy without any concessions. They become "nations' looters and the conqueror of the universe" (Burebista, 1980). The time of the syntheses between Romans and Dacians is over and the new political and cinematic view insists on the ancient local roots, on Dacians' superiority on Romans, enforced despite the professional historians' warnings 7 . This phenomenon is widespread starting from the end of the 70s, when is launched the concept of protochronisme 8 .

In conclusion, the embellishment of the conquerors' image has two main objectives. On the one hand, the few virtues of Romans are natural, taking into account their representation as an important ethnic component of the Romanian people. On the other hand, this technique is meant to bring to light, once again, the qualities of Dacians. As far as the official historiography identifies the Romanians mainly to Dacians, this procedure allows, while praising or stigmatizing the "other", to describe itself 9 .

It still remains that the representation of the Roman Empire is more than a reconstitution of battles and costumes or the mise en scène of some classical quotations. The Romanian peplum is a manifesto for identity in the communist period that evolved from a double root in the 60s to a single root in the 80s. This change reveals at the end the ideological conditions of film making, the rediscovery of antique origins and a moderate treatment of historical values and the sliding towards extreme nationalism of Ceausescu in the 80s. 7 National Archives of Film (ANF), File of production "Burebista"; Constantin PREDA, "Report in the scenario Burebista", 18 June 1980 : "We cannot assert that in Rome it is used wooden plows, whereas geto-dacian would have had iron superior plows" 8 For a detailed analysis of this phenomenon see Verdery (1994:) Compromis şi rezistenţă. Cultura română sub Ceauşescu. Transl. by Mona Antohi and Sorin Antohi. Bucharest: Humanitas 1994, pp. 152-204. (First published as National Ideology under Socialism. Identity and Cultural Politics in Ceausescu's Romania. Berkeley: University of California Press, 1991) 9 This technique is also available for the filmic representation of the next historical periods and its own enemies: the Ottomans and Western Europe. For a detailed analysis, see Aurelia Vasile, « L'étrangerun ennemi nécessaire. Représentation de l'Autre dans le cinéma roumain pendant le régime national-communiste de Nicolae Ceausescu » in Sciences Humaines Combinées, Revue électronique de l'ED LISIT, nr.2 Identités, images et représentations de l'autre, Dijon, 2008


Harry Dankowicz

Jenny Jerrelind

Control of near-grazing dynamics in impact oscillators

Keywords: piecewise smooth systems, grazing trajectories, grazing bifurcations, discontinuity mappings, control

A method is presented for controlling the persistence of a local attractor near a grazing periodic trajectory in a piecewise smooth dynamical system in the presence of discontinuous jumps in the state associated with intersections with system discontinuities. In particular, it is shown that a discrete, linear feedback strategy may be employed to retain the existence of an attractor near the grazing trajectory, such that the deviation of the attractor from the grazing trajectory goes to zero as the system parameters approach those corresponding to grazing contact. The implementation relies on a local analysis of the near-grazing dynamics using the concept of discontinuity mappings. Numerical results are presented for a linear and a nonlinear oscillator.

Introduction

The onset of impacting motions of mechanical systems under variations in a system parameter m is known to be associated with dramatic and abrupt changes in system response [START_REF] Nordmark | Non-periodic motion caused by grazing incidence in an impact oscillator[END_REF][START_REF] Nordmark | Universal limit mapping in grazing bifurcations[END_REF][START_REF] Fredriksson | Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators[END_REF][START_REF] Dankowicz | On the origin and bifurcations of stick-slip oscillations[END_REF][START_REF] Molenaar | Mappings of grazing impact oscillators[END_REF]Dankowicz et al. 2002;[START_REF] Dankowicz | Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators[END_REF]. Specifically, bifurcations in the long-term system response occur as m increases past some critical value m grazing , at which value there exists a periodic nonimpacting oscillation of the mechanical system that achieves zero-relative-velocity contact with an internal or external mechanical limit. In a state-space description of the dynamics of the system, such zero-relative-velocity contact corresponds to a grazing contact between a state-space trajectory and a discontinuity surface, for example, representing the sudden changes in the velocities of the system components that result from an impact. In contrast to periodic trajectories in smooth systems, the local description in the vicinity of a grazing trajectory is well known to be non-differentiable with dramatic implications to the stability of the grazing trajectory and to its persistence under further parameter variations. Indeed, as there is no advance warning of this instability, any local description must account for the non-smooth character of the flow near the grazing trajectory.

Analysis of the local dynamics in the vicinity of a grazing trajectory for mzm grazing is made possible through the introduction of a discontinuity mapping that (i) captures the local dynamics in the vicinity of the grazing contact including variations in time-of-flight to the discontinuity and the impact mapping; (ii) can be entirely characterized by conditions at the grazing contact; (iii) is non-smooth in the deviation from the point of grazing contact and (iv) can be studied to arbitrary order of accuracy [START_REF] Nordmark | Non-periodic motion caused by grazing incidence in an impact oscillator[END_REF][START_REF] Dankowicz | On the origin and bifurcations of stick-slip oscillations[END_REF][START_REF] Di Bernardo | Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems[END_REF][START_REF] Dankowicz | Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators[END_REF]. Properly formulated, the discontinuity mapping thus introduces the correction to the otherwise smooth dynamics that is due to the brief interaction with the discontinuity.

As shown by [START_REF] Fredriksson | Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators[END_REF] but see also [START_REF] Nordmark | Non-periodic motion caused by grazing incidence in an impact oscillator[END_REF][START_REF] Nordmark | Universal limit mapping in grazing bifurcations[END_REF], necessary and sufficient conditions for the persistence of a local attractor near the grazing periodic trajectory for mzm grazing may be formulated in terms of a condition on the linearization of the mechanical system about the grazing trajectory in the absence of impacts. In this paper, we rely on this insight for the formulation of a discrete, closed-loop control strategy that controls the persistence of a local attractor by appropriate real-time changes to the position of the mechanical limit. In particular, we illustrate the possibility of dramatically changing the bifurcation scenario associated with grazing contact with a minimum of control.

The paper is organized as follows. Section 2 presents an analysis of the neargrazing dynamics of a piecewise smooth formulation of a linear, forced impact oscillator and the formulation of a feedback strategy for controlling the persistence of a local attractor near the grazing periodic trajectory. Section 3 illustrates the generality of the analysis and control methodology by considering an application to a nonlinear oscillator. A concluding discussion is presented in §4.

A linear oscillator (a ) Mathematical model

Consider the response of a linear oscillator to a periodic excitation in the presence of perfectly elastic impacts of the oscillator with a mechanical limit at some critical displacement. In particular, suppose that the evolution of the oscillator is governed by the second-order differential equation

d 2 q dt 2 ðtÞ C 2g u dq dt ðtÞ C 1 u 2 qðtÞ Z A 1 u 2 K 1 cos t K 2g u sin t ; ð2: 1Þ
for qRq c , u, g, AO0, and such that if In terms of the state vector

q lim t/t
x Z ð x 1 x 2 x 3 x 4 x 5 Þ T Z ð q _ q t mod 2p A q c Þ T ; ð2:4Þ
the corresponding dynamical system is given by dx dt ðtÞ Z f ðxÞ

Z x 2 K 1 u 2 x 1 K 2g u x 2 C x 4 1 u 2 K 1 cos x 3 K 2g u sin x 3 1 0 0 0 B B B B B B B B B @ 1 C C C C C C C C C A ; ð2:5Þ
as long as h D ðxÞ Z def x 1 K x 5 R 0 and the discontinuous change in velocity at impact is given by the jump map

g D ðxÞ Z x 1 Kx 2 x 3 x 4 x 5 0 B B B B B B B B @ 1 C C C C C C C C A : ð2:6Þ
As long as x 4 (0)%Kx 5 (0), there exists a family of asymptotically stable periodic solutions

xðtÞ Z x 4 ð0Þcos t Kx 4 ð0Þsin t t mod 2p x 4 ð0Þ
x 5 ð0Þ
0 B B B B B B B B @ 1 C C C C C C C C A : ð2: 7Þ
Indeed, when x 4 ð0ÞZKx 5 ð0ÞZKx 5 Ã O 0, a grazing periodic trajectory is obtained that achieves grazing contact with the discontinuity surface D given by the zero-level surface of the event function h D ðxÞ at the point

x à Z ð x 5 à 0 p Kx 5 à x 5 Ã Þ T , since h D;x ðx à Þ$f ðx Ã Þ Z 0: ð2:8Þ
In the following we are particularly concerned with bifurcations associated with changes in x 4 (0) and x 5 (0) away fromHx 5 Ã , respectively. To visualize the results of the numerical and theoretical study, introduce a Poincare ´section P corresponding to the zero-level surface of the event function h P ðxÞZ h D;x ðxÞ$f ðxÞZ x 2 , for x 2 increasing. For notational convenience, consider the following definitions (cf. figure 1 (Note that P h DZ D 0 .) In the absence of impacts, points on P correspond to local minima in the value of x 1 along system trajectories (since _ x 1 Z x 2 ). Since g D maps D K to D C , trajectories that reach D K experience an instantaneous jump across P to D C without intersecting P (as the incoming velocity x 2 !0 is changed to an outgoing velocity Kx 2 O0). We represent such crossings by the virtual point of intersection with P of the corresponding forward trajectory segment in the absence of the jump in velocity.

(b ) Near-grazing dynamics

We wish to associate a Poincare ´mapping P with the Poincare ´section P introduced above. Ignore, for a moment, the jump map associated with the discontinuity D and assume that the dynamics are governed entirely by the vector field f. Suppose that the forward trajectory based at a point where F is the smooth flow corresponding to the vector field f. Here, the latter condition corresponds to the requirement that x 2 be increasing, i.e. that the acceleration of the oscillating mass be positive. where the discontinuity mapping D maps x to some point on P in such a way that the subsequent dynamics respect those of the corresponding actual trajectory.

x
To arrive at an expression for D consider the trajectory segments shown in figure 1 (cf. [START_REF] Nordmark | Non-periodic motion caused by grazing incidence in an impact oscillator[END_REF][START_REF] Dankowicz | On the origin and bifurcations of stick-slip oscillations[END_REF][START_REF] Di Bernardo | Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems[END_REF][START_REF] Dankowicz | Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators[END_REF]. Here, an incoming trajectory governed by the vector field f reaches the discontinuity surface D at a point x in 2D K , experiences a jump to a point x out Z g D ðx in Þ 2D C , and then continues to flow under the vector field f. The trajectory segments on the far side of the discontinuity surface D correspond to a flow governed by the vector field f from x in forward in time until reaching P at a point x 0 , and from x out backward in time until reaching P at a point x 1 . The sought correction to the smooth flow given by F is then obtained by mapping x 0 to x 1 , as this correctly accounts for the effects of the jump map.

Thus, given an initial point xzx * on P, such that h D ðxÞ! 0, we define D as the composition of the following steps:

(i) flow for a time t 1 !0 with the vector field f until reaching D K ; (ii) apply the jump map g D ; (iii) flow for a time t 2 !0 with the vector field f until reaching P.

To arrive at a functional expression for D, we seek to express the flow times in terms of the corresponding initial conditions in state space and the corresponding parameter values. Suppose, in particular, that there exists a point x * , such that

h D ðx Ã Þ Z 0; ð2:21Þ h P ðx Ã Þ Z 0; ð2:22Þ and a à Z def h P;x ðx à Þ$f ðx Ã Þ Z f 2 ðx Ã Þ Z Kx 5 à O 0: ð2: 23Þ
Clearly, x * 2P is a point of simple grazing contact with D corresponding to a local minimum in h D along a trajectory segment of the vector field f based at x * .

Step 1. Suppose that h D ðxÞ! 0 and consider the function As the smooth function t (1) (x, y) is implicitly defined by equation (2.29), we can compute arbitrary partial derivatives of t (1) at (x * , 0) using implicit differentiation and demanding that all partial derivatives of the left-hand side of (2.29) must vanish at (x * , 0, 0). Now let D 1 ðx; yÞ Z Fðx; Kt ð1Þ ðx; yÞÞ: ð2:30Þ

E
Since F is smooth, we can compute arbitrary partial derivatives of D 1 at (x * , 0).

Step 1 is completed by expanding D 1 (x, y) to desired order in the deviation from (x * , 0) and substituting yZ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kh D ðxÞ p .

Step 2. Since g D is smooth, it can be expanded to desired order in the deviation from x * , after which the result of step 1 may be substituted for x thus completing step 2.

Step 3. Finally, consider the function

E
Kh D;x ðx à Þ$ðx K x Ã Þ q ; ð2:38Þ where b Z f ðx Ã Þ h P;x ðx à Þ$g D;x ðx à Þ$f ðx Ã Þ a à K g D;x ðx à Þ$f ðx Ã Þ ffiffiffiffiffi 2 a à r Z K2 ffiffiffiffiffi 2 a à r ð 0 0 1 0 0Þ T : ð2:39Þ
For small deviations from x * in the direction of negative values of h D , this term results in a large stretching in a direction relative to x * given by the image of the vector b under the Jacobian P smooth,x (x * ) of the smooth Poincare ´map. Consider the sequence xZ fx n g N nZ1 defined by

x n Z h D;x ðx à Þ$ P smooth;x ðx à Þ$ /$P smooth;x ðx Ã Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} n times $b: ð2:40Þ
Then, as argued by [START_REF] Fredriksson | Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators[END_REF], a necessary condition for the persistence of a local attractor near the grazing trajectory when x 4 (0)TKx 5 (0) is that x is a non-negative sequence (but see comments in the next section). Indeed, such a local attractor can be shown to lie within a region whose size grows as the square root of the absolute value of the deviation of x 4 from x 4 Ã . Here, we refer to such a scenario as a continuous grazing bifurcation. In contrast, if x n !0 for some integer n, no local attractor will persist. Instead, such a discontinuous grazing bifurcation is associated with a sudden jump to an impacting system attractor that persists also for x 4 (0)(Kx 5 (0).

For the present example, it is straightforward to show that

x 1 Z x 5 à uðl 2 1 K l 2 2 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi g 2 K 1 p ffiffiffiffiffi 2 a à r ;
ð2:41Þ

where

l 1 Z exp½Kpðg C ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi g 2 K 1 p Þ=u; l 2 Z exp½Kpðg K ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi g 2 K 1 p Þ=u: ð2:42Þ
It follows that a necessary condition for the persistence of a local near-grazing attractor is that gO1.

(c ) Control strategy

We now seek to introduce a feedback control to render x n O0 for all n by making discrete changes to the value of x 5 , when the trajectory intersects the zero-level surface C of the event function h C ðxÞZ x 2 for x 2 decreasing. Denote by x ÃÃ Z ðKx 5 Ã 0 0 Kx 5 Ã x 5 Ã Þ T the intersection of the grazing periodic trajectory with C. Now, let the proposed changes be governed by the jump map

g C ðxÞ Z x 1 x 2 x 3 x 4 x 5 ÃÃ C c 1 ðx 1 K x 1 ÃÃ Þ C c 3 ðx 3 K x 3 ÃÃ Þ C c 5 ðx 5 K x 5 ÃÃ Þ 0 B B B B B B B B @ 1 C C C C C C C C A : ð2: 43Þ
It follows that

P smooth Z P C/P smooth B g control B P P/C smooth ; ð2: 44Þ
where P P/C smooth and P C/P smooth correspond to the Poincare ´maps from P to C and from C to P, respectively. The Jacobian of P smooth can then be written as a product of the Jacobians of P P/C smooth , g control and P C/P smooth evaluated at the appropriate points along the grazing trajectory. Since

P C/P smooth;x ; P P/C smooth;x Z Ã Ã Ã Ã 0 0 0 0 0 0 Ã Ã Ã Ã 0 0 0 0 1 0 0 0 0 0 1 0 B B B B B B B @ 1 C C C C C C C A ; ð2:45Þ it follows that P C/P smooth;x $g C;x $P P/C smooth;x Z Ã Ã Ã Ã 0 0 0 0 0 0 Ã Ã Ã Ã 0 0 0 0 1 0 Ãc 1 C Ãc 3 Ãc 1 C Ãc 3 Ãc 1 C Ãc 3 Ãc 1 C Ãc 3 c 5 0 B B B B B B B @ 1 C C C C C C C A ; ð2:46Þ
where * refers to a non-trivial coefficient. Appropriate choices of values for the control parameters c 1 , c 3 and c 5 may thus be sought that will render x n O0 for all n. Since, for fixed values of c 5 each x n is linear in the remaining control parameters, allowable values for c 1 and c 3 lie in a region bounded by straight lines in the c 1 , c 3 two-dimensional parameter space. Similarly, values outside such a region may be employed to render the grazing bifurcation discontinuous.

(d ) Comments on the continuity criterion

As argued in [START_REF] Fredriksson | Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators[END_REF], a sufficient condition for the persistence of a local attractor is that the sequence x L Z fx L n g N nZ1 , where where u n is a unit vector in the direction of

x L n Z h D;x ðx à Þ$ L$ /$L
P smooth;x ðx à Þ$ /$P smooth;x ðx Ã Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} n times $b; ð2:49Þ
is bounded from below by a positive number.

As formulated above, the quantities x n /0 as n/N for fixed values of the control parameters, since the eigenvalues of P smooth,x lie within the unit circle. As a consequence, it may become numerically difficult to resolve whether x n O0 for large enough values of n. In contrast, since xn must be bounded from below by a positive number for all n, there is no numerical difficulty in ascertaining whether the sufficient condition for the persistence of a local attractor is satisfied for a given set of control parameters. On the other hand, while the x n are linear in c 1 and c 3 (for a given values of c 5 ), the xn are nonlinear in the control parameters. In the numerical discussion below, we therefore seek values of the control parameters for which x is a non-negative sequence, and only after the fact do we confirm that x is bounded from below by a positive number.

The continuity criterion that x is bounded from below by a positive number is directly related to the local stability of the grazing periodic trajectory. Indeed, as long as the continuity criterion is satisfied, a small initial perturbation from the grazing trajectory such that x 4 ð0ÞZ x 4 Ã ZKx 5 Ã will result in at most one impact with the mechanical limit, and thus the stability of the grazing trajectory for

x 4 Z x 4
à is entirely determined by the non-impacting dynamics. As shown in [START_REF] Fredriksson | Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators[END_REF], for values of x 4 away from x 4

à the continuity criterion implies that the number of iterations of the composite Poincare ´map between successive impacts goes to infinity as x 4 / x 4 à , in essence that impacts are rare and far apart for small deviations from the grazing trajectory.

The implications of this rarity of impacts is that a control strategy that realizes the continuity criterion is particularly well suited for controlling transient dynamics that involve occasional impacts. Indeed, if it were only required that a local attractor persist as x 4 would increase beyond x 4 Ã , one might naively assume that it would suffice to choose values for c 1 , c 3 and c 5 such that the mechanical limit would remain away from the non-impacting periodic trajectory. In the case of the linear oscillator, this would imply that

x 5 K x 5 Ã !Kðx 4 K x 4 Ã Þ; ð2:50Þ where x 5 K x 5 Ã Z c 1 ðx 4 K x 4 Ã Þ C c 5 ðx 5 K x 5 Ã Þ; ð2:51Þ i.e. that c 1 ! c 5 K 1; ð2:52Þ
independently of the value of c 3 . To keep the mechanical limit away from the non-impacting periodic trajectory for x 4 ! x 4 Ã , however, would result in the opposite inequality. It follows that to retain a local attractor near the grazing periodic trajectory on an open neighbourhood of x 4 Ã , values for c 1 , c 3 and c 5 must be chosen to satisfy the continuity criterion. For example, the continuity criterion implies that the largest-in-modulus eigenvalue of P smooth,x (x * ) must be real and positive. But c 5 is one of the eigenvalues of P smooth,x (x * ). Thus, in situations where the largest-in-modulus eigenvalues of P smooth,x (x * ) in the absence of control are complex conjugates, one must choose a value for c 5 that is greater than the modulus of these eigenvalues.

(e ) Numerical results

Consider the linear oscillator, such that u Z 10; g Z 0:2; x 5 Ã Z K1: ð2:53Þ

Then, in the absence of control, i.e. when c 1 Zc 3 Zc 5 Z0, we find x 1,2 O0 but x 3 !0, i.e. the grazing bifurcation is discontinuous. Figure 2 shows the impacting attractor to which an initial condition in the vicinity of the grazing trajectory converges for x 4 (0)R1 and which also persists for x 4 (0)(1.

In order for x n O0 for all n, the largest-in-modulus eigenvalue of P smooth,x must be real and positive. Since, here The shaded region in figure 3 shows a subset of the region of values for c 1 and c 3 , such that xn O 0 for n%90 when c 5 Z0.8819 and c 5 Z0.89. A bifurcation diagram showing the persistence of a local attractor in the near-grazing dynamics under variations in x 4 is shown in the upper panel of figure 4, where c 1 Z0, c 3 Z5.8 and c 5 Z0.89. Furthermore, the lower panels of figure 4 show the local attractor with control for x 4 K x 4 Ã Z 10 K5 as obtained from direct numerical simulations and as predicted by the composite Poincare ´mapping (for the corresponding discontinuity mapping, see appendix A).

l 1 Z 0; l 2 Z 2; l 3;4 Z e Kpð1Gi2

A nonlinear oscillator

Now consider the response of a nonlinear oscillator to a periodic excitation in the presence of elastic impacts of the oscillator with a mechanical limit at some critical displacement. In particular, suppose that the evolution of the oscillator is governed by the second-order differential equation d 2 q dt 2 ðtÞ C g dq dt ðtÞ C qðtÞ Z A cos 2 ut ð1 K qðtÞÞ 2 ; ð3:1Þ for q!q c !1, u, g, AO0, and such that if where e represents a coefficient of restitution. In terms of the state vector the corresponding dynamical system is given by dx dt ðtÞ Z f ðxÞ Z

q lim t/t K Ã t Z q c and
x Z ð x 1 x 2 x 3 x 4 x 5 Þ T Z ð q _ q ut mod p A q c Þ T ; ð3:4Þ 4 6 8 1 0 2 0 0 1 2 3 4 x 1 -x 5 (¥10 -2 ) 0 1 2 -6 -4 -2 0 2 4 6 x 1 -x 5 (¥10 -2 ) 0 1 2 x 1 -x 5 (¥10 -2 ) x 4 -x 4 (¥ 10 -5 ) * x 3 -x 3 (¥ 10 -3 ) x 3 -x 3 (¥ 10 -3 ) * * -6 -4 -2 0 2 4 6 (a) (b) (c)
x 2 Kx 1 K gx 2 C x 4 cos 2 x 3 ð1 K x 1 Þ 2 u 0 0 0 B B B B B B B B @ 1 C C C C C C C C A ;
ð3:5Þ as long as h D ðxÞ Z def x 5 K x 1 R 0 and the discontinuous change in velocity at impact is given by the jump map

g D ðxÞ Z x 1 Kex 2 x 3 x 4 x 5 0 B B B B B B B B @ 1 C C C C C C C C A : ð3:6Þ
Following the discussion in the case of the linear oscillator, introduce a Poincare śection P given by the zero-level surface of the event function h P ðxÞZ h D;x ðxÞ$f ðxÞZKx 2 for x 2 decreasing. Again, we may analyse the dynamics in the vicinity of a grazing trajectory by composing the smooth Poincare ´map P smooth associated with P in the absence of impacts with a discontinuity mapping D that (i) captures the local dynamics in the vicinity of the grazing contact including variations in time-of-flight to the discontinuity and the impact mapping; (ii) can be entirely characterized by conditions at the grazing contact; (iii) is non-smooth in the deviation from the point of grazing contact; and (iv) can be studied to arbitrary order of accuracy. Properly formulated, the discontinuity mapping thus introduces the correction to the otherwise smooth dynamics that is due to the brief interaction with the discontinuity.

Suppose that gZ0.04, uZ0.56, eZ0.8. Then, a grazing periodic trajectory is obtained for

x à Z ð 0:5 0 1:541 053. 0:283 159. 0:5 Þ T : ð3:7Þ In this case, x 1 !0 in the absence of control and the corresponding grazing bifurcation is thus discontinuous. Indeed, the left panel of figure 5 shows the impacting periodic attractor to which an initial condition in the vicinity of the grazing trajectory converges for x 4 ð0ÞR x 4 à and which also persists for x 4 ð0Þ( x 4 à . Here, the largest-in-modulus eigenvalues of P smooth,x (x * ) for c 5 Z0 are complex conjugate with modulusz0.8939. Let c 5 Z0.95. Then, the right panel of figure 5 shows a subset of the region of values for c 1 and c 3 , such that xn O 0 for n%90. As an example, figure 6 shows the grazing bifurcations scenario that is obtained when c 1 Z0 and c 3 ZK1.4. Specifically, while the upper panel shows the predictions of an analytical expansion of P to linear order in the deviation xKx * , the lower panel shows the results of applying the control to the original dynamical system over a larger range in the deviation x 4 K x 4 à .

Conclusions

This paper has presented a linear, discrete, closed-loop control strategy for ensuring the persistence of a local attractor in the near-grazing dynamics of an impact oscillator. Although the methodology has been presented specifically for forced, single-degree-of-freedom oscillators, there is nothing in the approach that limits it to this low dimensionality or to systems that originate in second-order differential equations. Indeed, the discontinuity-mapping approach as well as the continuity criterion formulated here are general to near-grazing dynamics in arbitrary-dimensional piecewise-smooth dynamical systems. Of course, the higher the dimension of the underlying dynamical system, the more control parameters would enter into the expression for the feedback strategy. Again, however, for fixed values of the control parameter coupling the change in the position of the mechanical limit to the current position of the mechanical limit (c 5 in the above analysis), the resultant expressions for the elements of the sequence x would be linear in the remaining control parameters. One would thus, again, expect that allowable values for the remaining control parameters would lie in a region in parameter space bounded by hyperplanes. As noted above, in a situation where the largest-in-modulus eigenvalues of the Jacobian of the smooth Poincare ´map in the absence of control were complex conjugate, it would be necessary to select the control parameter coupling the change in the position of the mechanical limit to the current position of the mechanical limit to be larger than the modulus of these eigenvalues (but always less than one).

The control of system dynamics in the presence of impacts using the position of the system discontinuity has previously been employed by the authors (Dankowicz & Piiroinen 2002;Jerrelind & Dankowicz in press). There, however, the analysis focused on the stabilization of impacting dynamics in the presence of large-impactvelocity motions, far from the near-grazing region. In that case, the local dynamics were described by a linear Poincare ´map and the analysis focused on the control of the linearized system about an impacting reference trajectory. In contrast, in the situation considered here, the stabilization of a particular trajectory is practically impossible. Instead, the aim of the present work has been to change the bifurcation behaviour in the near-grazing region ensuring, not the stability of a single trajectory, but the weaker notion of stability pertaining to the persistence of a local attractor. The proposed control strategy affords a means of suppressing the catastrophic loss of stability of low-impact-velocity motion to high-impact-velocity motion that is a distinct possibility in impact oscillators. Instead, at the expense of small, real-time and closed-loop adjustments to the position of the system discontinuity, low-impact-velocity motions may be sustained over a large interval about the point of grazing contact. The implications of this to practical mechanical and/or electrical systems will be explored in future work.

This material is based upon work supported by the National Science Foundation under grant no. 0237370 and by the Swedish Science Council, Division of Engineering Mechanics, grant numbers 2003-3699 and 2004-6342.

Appendix A. Explicit expansions for the discontinuity mappings

As suggested in the main text, implicit differentiation may be employed to obtain expressions for the coefficients in the various Taylor expansions in the construction of the discontinuity mapping D. As an example of the result of such computations, the formulae below correspond to the discontinuity mappings for the linear and nonlinear oscillator, respectively, for xzx * . Since D maps P to P and since x 4 and x 5 are constant under the smooth flow, we only report the non-trivial 1 and 3 components of D.

Specifically, in the case of the linear oscillator with u Z 10; g Z 0:2; x à Z ð K1 0 p 1 K1 Þ T ; ðA 1Þ we find

D 1 ðxÞ zx 1 Ã C Dx 1 ; ðA 2Þ
D 3 ðxÞ zx 3 Ã K 2 ffiffi ffi 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Dx 5 K Dx 1 p K 4 25 ðDx 5 K Dx 1 Þ C Dx 3 ; ðA 3Þ
where DxZxKx * and we have included terms up to linear order in the deviation Dx.

Similarly, in the case of the nonlinear oscillator with g Z 0:04; u Z 0:56; e Z 0:8;

x à Z ð 0:5 0 1:541 053. 0:283 159. 0:5 Þ T ; ðA 4Þ

we find D 1 ðxÞ zx 1 Ã C 0:6400Dx 1 C 0:3600Dx 3 ; ðA 5Þ D 3 ðxÞ zx 3 Ã K 2:0180 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Dx 1 K Dx 5 p K 0:2107ðDx 1 K Dx 5 Þ C Dx 3 : ðA 6Þ Note that to obtain an expansion for the composite Poincare ´mapping to first-order in the deviation from the point of grazing contact, it is necessary to compute expansions of the corresponding smooth Poincare ´mappings to second-order. This is a straightforward application of the definition of the smooth Poincare ´mapping and the solution to the first and second variational equations for the smooth flow.

Figure 2 .Figure 3 .

 23 Figure 2. Chaotic impacting attractor in the near-grazing dynamics. (a) Intersection of attractor with P ; (b) intersection of attractor with D.

Figure 4 .

 4 Figure 4. Near-grazing dynamics in the presence of control. (a) Bifurcation diagram under variations in x 4 . Lower panels show local chaotic attractor for x 4 K x 4 Ã Z 10 K5 (b) as obtained from direct numerical simulations and (c) as predicted by the composite Poincare ´mapping.

Figure 5 .

 5 Figure 5. (a) Near-grazing dynamics in the absence of control and (b) subset of values of c 1 and c 3 , such that xn O 0 for n%90 when c 5 Z0.95.

Figure 6 .

 6 Figure 6. Near-grazing dynamics in the presence of control. Bifurcation diagrams under variations in x 4 (a) as predicted by the composite Poincare ´mapping and (b) as obtained from direct numerical simulations.

  If we reintroduce the non-trivial jump map g D associated with D, the above expression is still valid as long as h D ðxÞR 0. If, instead, h D ðxÞ! 0, we recognize that the point x corresponds to a virtual point of intersection that cannot actually be reached by the piecewise smooth dynamical system. It would, nevertheless, be convenient if we could again define the Poincare ´mapping P by the above formula, but possibly including an initial correction to account for the

	It follows that		x 1 Fðx ref ; t ref Þ Z 0	ð2:15Þ
	and	h	D F t ðx ref ; t ref ÞO 0: x 0	ð2:16Þ
	x out The implicit function theorem implies that there exists a unique smooth function x in t(x) defined on a neighbourhood of x ref , such that
			g tðx ref Þ Z t ref	ð2:17Þ
	and
			Fðx; tðxÞÞ h0;	ð2:18Þ
	h i.e. t(x) is the time of flight from x back to P. A smooth Poincare ´mapping
	P smooth can now be defined on a neighbourhood of x ref by the expression Figure 1. Schematic illustrating the correction to the smooth flow in the presence of impacts introduced through the concept of the discontinuity mapping D. P smooth ðxÞ Z Fðx; tðxÞÞ: ð2:19Þ
	Now introduce the function
		Fðx; tÞ Z h P ðFðx; tÞÞ:	ð2:14Þ

virtual nature of the initial point x. To this end, consider the Poincare ´mapping P defined by PðxÞ Z P smooth ðDðxÞÞ; ð2:20Þ

  ð2Þ ðx; tÞ Z h P ðFðx; KtÞÞ:It follows from the definition of E (2) that t 3 ZKt (2) (x).As the smooth function t (2) (x) is implicitly defined by equation (2.35), we can compute arbitrary partial derivatives of t (2) at x * using implicit differentiation and demanding that all partial derivatives of the left-hand side of (2.35) must vanish at x

						ð2:31Þ
	Then,
			E ð2Þ ðx à ; 0Þ Z h P ðx Ã Þ Z 0	ð2:32Þ
	and
			E t ðx à ; 0Þ Z Ka à ! 0: ð2Þ	ð2:33Þ
	The implicit function theorem again implies the existence of a unique smooth function t (2) (x) on an open neighbourhood of x * , such that
			t ð2Þ ðx Ã Þ Z 0		ð2:34Þ
	and
			E ð2Þ ðx; t ð2Þ ðxÞÞ h0:	ð2:35Þ
	It follows that DðxÞ Z	(	x D 2 ðg D ðD 1 ðx;	ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Kh D ðxÞ p	when h D ðxÞR 0; ÞÞÞ when h D ðxÞ! 0:	ð2:37Þ

* . Now let D 2 ðxÞ Z Fðx; Kt ð2Þ ðxÞÞ: ð2:36Þ

Since F is smooth, we can compute arbitrary partial derivatives of D 2 at x * .

Step 3 is completed by expanding D 2 (x) to desired order in the deviation from x * and substituting the result of step 2 for x.

Clearly, the presence of the square-root term renders the resulting discontinuity map non-differentiable at a point x, where h D ðxÞZ 0, i.e. a point of grazing contact.

The lowest-order-in-x Kx * , non-trivial term in the expansion of D for xzx * is a term of the form b ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi


Marie Capron

Sandrine Bardet

K C Sujan

M Matsuo

H Yamamoto

One-dimension visco

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

One-dimension visco-elastic modelling of wood in the process of formation to clarify the Hygrothermal Recovery behavior of tension wood M. Capron † ‡* , S. Bardet 2 , K.C. Sujan †ˠ , M. Matsuo † and H. Yamamoto † † Graduate School of Bio-agricultural Sciences, Nagoya University, marie.capron@esrf.fr 2 Laboratoire de Mécanique et de Génie Civil (LMGC), Université Montpellier, sandrine.bardet@umontpellier.fr ‡ ESRF -The European Synchrotron ˠ Institute of Wood Technology, Akita Prefectural University, Graduate School of Bioresource Sciences Wood production on stem by deposit of concentric layers on its periphery are going along with the setting up of growth stress. Growth stress has two origins: (1) loading due to weight of the structure is applied progressively when the tree is growing;

(2) cell maturation, which happened at the end of the deposit of a new layer, causes an expansion, called maturation deformation, which can't happen freely due to the previous layer and lead to the creation of initial growth stress [START_REF] Archer | Growth Stresses and Strains in Trees[END_REF]. The growth stress can be released during cutting and also during hygrothermal treatment (HT), it can be called Hygrothermal Recovery (HTR) [START_REF] Kubler | Growth stresses in trees and related wood properties[END_REF]. In this work, an one-dimension rheological modelling of wood in the process of formation is proposed to clarify the HTR behavior of Tension Wood (TW). This study focuses on the longitudinal dimensional changes. The rheological analogy is made of a series of four elements (Fig. 1): an elastic element represented by a spring of compliance S0 which is equal to that of mature wood; a deformation mechanism, α, representing the expansion tendency during maturation; and two visco-elastic elements represented by a spring of compliance S1 and S2 in parallel with a dashpot also called Kelvin-Voigt model. In order to find a coherent modelling parameters, a fitting with experimental data has been done using different samples from three species konara oak (Quercus serrata Murray), urihada maple trees (Acer rufinerve Siebold et Zucc.) [START_REF] Sujan | Continuum contraction of tension wood fiber induced by repetitive hygrothermal treatment[END_REF] and keyaki (Zelkova serrata Makino). The modelling is able to fit the different parts of HTR for TW (Fig. 2).

Figure 1 :

 1 Figure 1: Rheological analogy representing the modelling using of 2 Kelvin-Voigt with α the maturation strain, S0 the elastic or mature compliance and S1 and S2 the delayed compliance.

Figure 2 :

 2 Figure 2: Longitudinal strain change as a function of cumulative treatment duration experimental data for Urihada maple (o), Konara oak (x) and Keyaki (▽) (TW) and their modelling using two visco-elastic elements (-).