Serge Klarsfeld

What We Talk About When We (Do not) Talk about the Holocaust in Bulgaria

This paper explores the history of the politics of Holocaust memory in Bulgaria. During World War II, Bulgaria, although an ally of the Reich, refused to deport its Jewish community. Until recently, the image of Bulgaria as a European exception has prevailedthough at a cost: this narrative omitted the almost total roundup of Jews in the Yugoslav and Greek territories under Bulgarian occupation between 1941 and 1944 and their deportation to Nazi-occupied Poland where they were exterminated. How to explain that, out of a complex past, a single facet -the non-deportation of the Bulgarian Jews, understood as a "rescue"has become the primary way of narrating the history of World War II in Bulgaria and beyond? Drawing on a rich body of archives, this article reconstructs the emergence and reconduction until today of this specific narrative. It traces the legal, political and cultural arenas, as well as the multiple local, regional and international spaces where these understandings of the past were formed, circulated and appropriated since 1944. Therby, it shows how Jewish wartime destinies became satellized around a broad range of (only faintly related) topics and cleavages. Ultimately, this paper brings Jews back into the experiencing, writing, and transmission of these historical events.

A dominant view of Jews destinies in wartime Bulgaria has long prevailed. This historical narrative centered on the "rescue of the Bulgarian Jews," id est the survival of the near total Bulgarian Jewish community of about 48,000 individuals. 1 Not that observers and public historians were not cognizant of the fact that in the territories of Yugoslavia (most of Vardar Macedonia and the Serb region of Pirot) and Greece (Western Thrace and Eastern Macedonia) occupied by Bulgaria, then an ally of Nazi Germany, between April 1941 and October 1944, an estimated 11,343 Jews had been rounded up, interned in transit camps, and deported to Nazi-occupied Poland where they were extermined. Even during the socialist era, the extermination of Jews from occupied Yugoslavia and Greece was never entirely muted (the extermination) in Bulgaria. However, it seems as if these facts thined away under the light that the non-deportation of the Bulgarian Jews threw upon them.

Certainly, several illustrious philosophers gave credence to such a historical interpretation of the events. In her classic piece on Eichmann in Jerusalem (1963), Hannah Arendt for instance wrote: "…not a single Bulgarian Jew had been deported or had died an unnatural death when, in August 1944, with the approach of the Red Army, the anti-Jewish laws were revoked. I know of no attempt to explain the conduct of the Bulgarian people, which is unique in the belt of mixed populations." 2 This specific sentence "not a single Bulgarian Jew had been deported or had died an unnatural death" -first penned by Bulgarian communist Jews in 1945 and circulated among Allied diplomatic milieus to plea for a lenient treatment of Bulgaria in the postwar era -would later feature in a flurry of scholarly pieces, essays and museum catalogs. 3 In 1999, Bulgarian-born French philosopher Tzvetan Todorov further elaborated on this point: "Two countries can remember their history with pride," he argued, "since the collective protection of Jews was assured even though they were under German control: Denmark and Bulgaria. " 4 (It may be worth recalling here that Bulgaria, unlike Denmark, joined the Tripartite Pact and was an ally to Nazi Germany. However, the Bulgarian authorities refused to declare war on the USSR and to send troops fight on the eastern front). Yet, as early as 1961, acclaimed Holocaust specialist Raul Hilberg offered a more cautious analysis: the Jews, he stated, had been a "pawn in the hands of an opportunistic power." 5 A combination between territorial ambitions, strategic prudence and delaying tactics explained the authorities' ultimate choices : "They [the Jews] were like a surplus commodity, to be traded for political advantage. The Reich could not completely destroy the Bulgarian Jews because it could not offer sufficient gain to the cautious Bulgarian rulers." Nevertheless, the American political scientist added: "It was as though the degree of involvement had already been predetermined. The operation was brought to a halt as if stopped by an invisible sign which said, 'So far and no farther.'" The phrase suggested something beyond understanding, a kind of mystery, perhaps even a miracle.

The present article offers a glimpse into a book I recently authored with a view to explaining the emergence and reconduction until today of this specific understanding of wartime events in the "old" (Bulgaria's pre-1941 boundaries) and "new" (post-April 1941 boundaries) kingdoms of Bulgaria. 6 The first section provides the factual background to public discussions about the fate of Jews in Bulgarian-held areas. Section two presents the investigation conducted across time (75 years) and space (six countries). Finally, in section three I expose the main research findings.

Jewish Wartime Predicament in Territories under Bulgarian Control: Facts and Public Disputes

2 Hannah Arendt, Eichmann in Jerusalem: A Report on the Banality of Evil, New York 2006 [1 st ed. in English, 1963], 188. 3 Bulgarian dictator Todor Zhivkov reiterated this phrase in 1977 in the catalog of the Sofia permanent exhibition dedicated to the "rescue of the Bulgarian Jews": Albert Cohen and Anri Assa (eds), Saving of the Jews in Bulgaria, 1941-1944, Sofia 1977 (page number missing in the catalog). 4 Tzvetan Todorov (ed.), La Fragilité du bien. Le sauvetage des juifs bulgares, Paris 1999, 9-10. In the English translation of the volume, the word "rescue" has been cautiously replaced with "survived": Tzvetan Todorov (ed.), The Fragility of Goodness: Why Bulgaria's Jews Survived the Holocaust, Princeton 2004. 5 See also Nissan Oren, "The Bulgarian Exception: A Reassessment of the Salvation of the Jewish Community," Yad Vashem Studies, 8, 1968, 3-106. 6 Nadège Ragaru, "Et les Juifs bulgares furent sauvés…" Une histoire des savoirs sur la Shoah en Bulgarie, Paris 2020.

What do we know about the historical facts? [START_REF]For an historical account of the events, see Frederick Chary, The Bulgarian Jews and the Final Solution[END_REF] In October 1940, the Bulgarian government, then headed by Germanophile prime minister Bogdan Filov, announced its intent to adopt anti-Jewish measures -a public statement, which provoked lively discussions among the political and intellectual elites, as well as the common citizenry. Two months later, the Bulgarian National Assembly voted anti-Jewish legislation providing for the identification, as well as the social and economic marginalization of Jewish citizens. [START_REF][END_REF] By the end of April 1941, Bulgaria, a member of the Tripartite Pact since March 1, was occupying most of Vardar Macedonia and the region of Pirot in Serbia (Yugoslavia), as well as Western Thrace and Eastern Macedonia (Greece) -a reward for letting the Wehrmacht use its territory to launch an assault on these two countries in the spring of 1941. The occupied lands were not formerly annexed; however, military, civilian and Church personnels were dispatched from the "old" kingdom to rule over them; Bulgarian legislation, currency and symbols were used in what was presented in official documents as "newly liberated territories," whereas king Boris was heralded as the unifier king (Tsar obedinitel). Bulgaria further accepted to take over Yugoslav debts (the Claudius-Popov agreement of April 24, 1941). Anti-Jewish measures were applied in the "new" as well as the "old" territories, leaving the Jewish municipalities (evreyski obshtini, those communal institutions subjected to ever greater state control) no alternative but to serve as transmission belts of orders given from Sofia.

Anti-Jewish policies intensified in the summer of 1942: in June, the Bulgarian executive was granted full powers on the "Jewish question" by Parliament; from then on, there would be little if any legislative control over state policies, all the more so as a number of decree-laws were not made public, nor were they timely subjected to parliamentary approval despite legal rule to the contrary. As a result of the June 10, 1942 decree-law on citizenship in the "new territories," Jews in Bulgarian-occupied territories were denied Bulgarian citizenship (unlike residents with a different ethnocultural background), and thus deprived of state protection. 9 The institutional framework of anti-Jewish policies quickly built up: pursuant the August 29, 1942 decree-law, a Commissariat for Jewish Affairs (KEV) was assigned the design and supervision of the anti-Jewish program. Reporting to the Minister of the Interior, the KEV enjoyed broad prerogatives. That Aleksandăr Belev, an antisemitic jurist who had taken an active part in the crafting of the 1940 anti-Jewish law, was appointed Commissioner for Jewish Affairs left little doubt about the future of Jews living in Bulgarian-controlled lands. Article 7 of the decree-law explicitly tasked the Jewish municipalities (evreyskite obshtini, pivotal communal institutions) with "preparing the deportation (izselvaneto) of the Jewish population." Article 29 envisaged the expulsion of Jews from Sofia "to the provinces or outside the Kingdom." 10 After some internal infighting between the Reich's Foreign ministry and the head of the Reich Main Security Office (Reichssicherheitshauptamt, RSHA), German request for the preparation of deportations was made explicit in mid-October 1942 through the German Legation in Sofia. 11 The Bulgarian authorities granted their approval by mid-November, provided that a section of the Jews would remain in Bulgaria for forced labor, and that a German advisor would assist in the process. 12 SS Hauptsturmführer Theodor Dannecker, infamously known for his role in the deportation of Jews from France, took up the part. One month after his dispatch to Bulgaria (January 21, 1943), he concluded with Commissioner Belev an agreement pertaining to the deportation of 20,000 Jews from the "new" territories: since the number of Jews in occupied Yugoslavia and Greece did not exceed 12,000, this accord opened the way for the deportation of about 8,000 "undesirable" Bulgarian Jews. The arrests, deportations, and appropriation of Jewish properties were authorized by a series of decrees adopted by the council of ministers at the beginning of March 1943.

At this point, however, the destinies of Bulgarian and non-Bulgarian Jews bifurcated. In early March 1943, the Bulgarian authorities organized round-ups and the temporary internment of Jews from northern Greece (March 4), Vardar Macedonia (March 11) and Pirot (March 12), with the support of the Bulgarian police, army and administration. Deported by train (from Skopje) and by boat (from Lom, along the Danube) -together with the German police -, an estimated 11,343 Jews were later exterminated in Treblinka. Their properties were subsequently confiscated, and liquidated. In the "old" kingdom, by contrast, the public protests of elected officials, orthodox dignitaries and notables, coupled with German military defeats in the East, convinced the authorities to suspend, postpone and finally renounce the deportations of Bulgarian Jews. In May 1943, Commissioner Belev submitted a new deportation plan. When the government and King Boris III refused to apply the plan and "contented themselves" with authorizing the expulsion of Jews living in Sofia and other Bulgarian cities into the provinces, German demands were not considered sufficiently pressing by the Bulgarian executive to move ahead with the deportation of Jews of Bulgarian nationality.

Thus, in the last resort about 48,000 Bulgarian Jews were not deported; 11,343 Jews from the occupied territories were exterminated. Such was the toll in the fall of 1944 when the Red Army invaded/liberated (the wording depends on the political views of the observer) Bulgaria and helped a communist led coalition, the Fatherland Front, to overthrow the wartime regime. Trying to make sense of Jewish experiences during the world conflict represents a daunting task -all the more as, in these multicultural regions long part of the Ottoman empire, many a Jewish lineage extended beyond recently created nation-state borders. The descendants of those who survived the war and those who did not grew up with painfully divided memories and knowledge of World War Two.

Public disputes over the 1943 events started prior to the end of the war. They have continued for nearly eight decades. Until recently, however, these discussions focused on those individuals or groups who deserved to be credited for the "rescue of the Bulgarian Jews." 13 11 A copy of the German diplomatic note annotated by prime minister Filov is preserved in the Bulgarian archives: Central State Archives (Centralen Dărzhaven Arhiv, CDA), Fond (Fund, F) 176k, Inventory (opis, o) 1, Archival Unit (arhivna edinica, ae) 256, Page (list, l.) 43. Filov mentions the pending arrival of a German advisor. 12 See the Bulgarian verbal diplomatic note to the German Legation in Sofia dated november 12, 1942: CDA, F 176k, o 1, ae 256, l. 31, 32. 13 For a synthetic view, see Stefan Troebst, "Salvation, Deportation or Holocaust? The Controversy over the Fate of Bulgaria's Jews in World War II -before and after 1989," in Muriel Blaive, Christian Gerbel, Thomas Lindenberger (eds), Clashes in European Memory. The Case of Communist Repression and the Holocaust, Innsbruck 2011, 37-52; Nadège Ragaru, "Nationalization through Internationalization. Writing, Remembering, Gradually, literary and visual patterns were woven onto a loom that has remained rather indifferent to political, social and cultural transformations, as well as to changes in the memorial regimes in Bulgaria, on the European scale, and worldwide. No matter the protagonists and episodes chosen, a single story has continued to be told, one that outsources crimes to individuals while nationalizing virtues as collective. Depending on the political allegiance of the speaker, his/her citizenship and the time of the assessment, the stress falls on the Bulgarian Communist Party and antifascist resistance, dictator Todor Zhivkov, the Orthodox Church, King Boris III, several conservative political figures (chief among them Dimităr Peshev, once vice-president of the National Assembly), some intellectuals, common citizens, the Bulgarian people, et cetera.

In the 2010s, a consensus slowly crystallized in Bulgaria, which bridged the (otherwise deep) divide between the former and the anti-communists: the list of the deserving was progressively expanded so as to accomodate the preferences of the two political contenders. Ultimately, the virtues of a few individuals were transferred onto the entire Bulgarian society and indexed upon a supposed unique national tradition of tolerance towards minorities. On the occasion of the 75th anniversary of the March 1943 events, the Bulgarian National Assembly unanimously adopted a resolution, which stated that "northern Greece and the kingdom of Yugoslavia were then under German jurisdiction" and regretted that "the local Bulgarian administration was not able to stop [...] the criminal act undertaken by the Hitlerite command." 14 Most strikingly, all accounts of the Jewish destinies in wartime Bulgaria -both before and since the end of socialism in 1989 -share two common tropes. First, the Jews are "spoken," in the passive voice: "they were saved", "we were saved". Such are the formulations encountered in nearly all retellings of the past, including those authored by Jews. 15 It is a well known fact that, in the wake of World War Two, internationally the Jews were often portrayed as passive victims of the Reich and its allies. From the 1970s onwards, a new generation of scholars has unearthed and demonstrated the active responses of Jews in the face of persecution, as well as their role in the writing of their destruction. Surprisingly, this trope has persisted in Bulgaria to this day.

The second feature common to most narratives of the Holocaust in Bulgarian-held territories resides in the euphemization of Jewish suffering. The transport of Jews from the occupied territories is readily presented as the sine qua non condition of the "rescue of the Bulgarian Jews," and as a concession intended to preserve as many lives as the unequal power relations between Bulgaria and the Reich would allow. Not only does this representation of the events fail to properly account for the destruction of the Jews living in areas occupied by Bulgaria. It also casts a shadow upon anti-Jewish policies in the "old" kingdom: aside from the Jewish expulsions from Sofia and other cities in May 1943, the enforcement of anti-Jewish policiesidentification of the Jews; professional exclusions and Aryanization of property; political, economic, and social marginalization; detention and internal exile; forced labor, et ceteraremains a territory almost uncharted by public historians and state-sponsored commemorative initiatives. 16 and Commemorating the Holocaust in Macedonia and Bulgaria after 1989," Südosteuropa. Journal of Politics and Society, 65 (2), 2017, 284-315. 14 Available at: https://www.parliament.bg/bg/news/ID/2582 (accessed on March 9, 2021). 15 One of the most notable qualities of the testimony-based book recently published by Jacky with Martha Aladjem Bloomfield resides in their endeavor to give Jews a voice in the reconstitution of their wartime experiences: Jacky Comforty with Martha Aladjem Bloomfield, The Stolen Narrative of the Bulgarian Jews and the Holocaust, Lanham 2021. 16 For an exception, see the major piece by Avramov, "Spasenie" i padenie.

An Investigation: The Social Lives of the "Rescue"

The enduring and alluring power of these tropes led me to embark on a ten year long investigation across Bulgaria, North Macedonia, Serbia, Germany, Israel and the United States. With several axial questions in mind: how to explain that, out of a complex and contradictory past, a single facet -the non-deportation of Jews from the "old" kingdom, understood as a "rescue" -has become the primary way of narrating and transmitting the history of World War Two in Bulgaria and beyond? How, more precisely, to fathom that information on these deportations has seemed barely relevant to reconstructing the history of World War Two ? These interrogations, in turn, are grounded in a preoccupation that may actually constitue the central node in a twenty-five year research career dedicated to Southeast Europe: seeing and failing to see. 17 Seeing is recognizing, French anthropologist Gérard Lenclud once convincingly argued; 18 otherwise put, no cognition is devoid of pre-cognition. Adressing these questions led me to reconstruct the social production and circulation of what has long been considered as true facts -"true" because they were widely believed.

In so doing, my purpose was not to substitute one new narrative to the previous one, even less to tell the "truth" of the past. The point was to identify the social actors and arenas where understandings of World War Two had been formed, circulated and appropriated. Most importantly, this research aimed to bring Jews back into the story of their "rescue"/extermination, and in the writing, transmission and commemoration of the war period. For that purpose, to the notion of memory -which has, in the past decades, been used to the point of exhaustion -, I preferred that of traces of an incorporated past, and recused a study confined to the realms of academic or textbook history. For one basic reason: writing the Holocaust has never been the sole province of professional historians. During and following World War Two, awareness of the extermination of the Jews progressed at a crossroads between academics, experts and activists, and among historical, legal, and political fields -including as these clashed with each other. Therefore, I believe that the scope of any investigation into the sites, modes, and agents that produced knowledge on the fate of the Jews in areas governed by Bulgaria, needs to embrace a broader range of social spaces and professional worlds, chiefly judicial courthouses, documentary and feature films, museums, as well as political scenes. Otherwise, many of the ways in which ideas about the past come to innerve senses and sensibilities are obfuscated.

The architecture of the book reflects this analytical perspective: throughout the chapters, I linger over the production of legal knowledge about Jewish wartime life and death during the Bulgarian trials for anti-Jewish crimes of 1945; the negotiation, in the late 1950s, of an "Eastern European" way of rendering the historical facts during the making of a Bulgarian-East German film coproduction, Sterne/Zvezdi (dir. Konrad Wolf); the travels -across the East-West divide and between 1943 and the late 1980s -of a unique visual archive, namely film footage of the March 1943 Jewish deportations from northern Greece, and its changing status (documentary source, courtroom evidence, testimony with memorial resonance); and, 17 For two recent illustrations, see Nadège Ragaru (ed.), "Écritures visuelles, sonores et textuelles de la justice: une autre histoire des procès à l'Est," Cahiers du monde russe, 61/3-4, 2020, 275-498; Nadège Ragaru, "Dialogues avec le visible. L'historien et le regard," Critique internationale, 2015/3, 68, 9-20. 18 Gérard Lenclud, "Quand voir, c'est reconnaître," Enquête, 1, 1995, 113-129. finally, the knowledge produced after 1989 within the framework of public controversies. Such a broad focus does not entail eliding differences in the nature, span and use of these multiple modes of grasping the past. Nor do I want to suggest that such composite materials ever aggregated into a compact whole. Nevertheless, among these heterogeneous productions, situated differently in time and space, we can find references and correspondences which, without producing a single unified vision, did outline a realm of the conceivable. It is within this space that differences in understanding the events have been and continue to be formulated.

The Holocaust in Bulgaria: Debating Other Issues by Proxy

The first finding, perhaps, resides in the temporal origins of the "rescue" narrative. I initially associated the crafting and circulation of this specific representation of the past with the 1960s, the moment when dictator Todor Zhivkov worried about the rehabilitation of King Boris III among Bulgarians in exile, on the one hand, and used patriotism and nationalism as a substitute for the declining appeal of the socialist project, on the other. Contrary to this initial supposition, the notion that the Bulgarian Jews were "rescued" emerged in the very last months of World War Two. One needs to picture this complex historical moment, when the overthrow of the "bourgeois regime," purges and acts of personal vengeance, and uncertainty about the future intersect with the discovery of the breadth of Jewish destruction across Europe. Week after week, the Bulgarian Jews who were expelled from Sofia and other Bulgarian cities in May 1943 and who now are painstakingly regaining their homes, read in the Jewish press about the suffering of their coreligionaries. In Hungary, they learn, an estimated 440,000 Jews were deported in the spring of 1944; out of a number of 80,000 prewar Jews in Yugoslavia, about 20,000 might still be alive. Through the Soviet and international media, they get a glimpse of the unfathomable and remain dumbfounded: how to explain that they should still be alive? Among Jews of communist persuasion, there emerges the belief that, had it not been for the military losses of the Reich, the red Army and the regime change, their fate would have been similar to that of other European Jews. This conviction acquires particular salience at the time of the trial of several dozen former bureaucrats of the Commissariat for Jewish Affairs before chamber seven of the (highly politicized) People's Courts. In his final speech delivered on March 29, 1945, prosecutor Mancho Rahamimov, a communist defense lawyer by profession, offers an eloquent presentation of this view: "(...) thanks to the energetic and stubborn intervention of the Bulgarian society and the Fatherland Front, which -then in illegality -prepared Bulgarian peasants and workers to defend the Jewry, thanks to the democratic Bulgarian people who harbored warm feelings towards them, we were saved. And I, here, from this place, as prosecutor, as a delegate of the Central Consistory of the Jews of Bulgaria, warmly thank the government of the Fatherland Front and the entire valiant Bulgarian people for our rescue. " 19 A few months later, on August 4, 1945, Bulgarian Jewish artist, Marko Behar, publishes in the leftwing Jewish News (Evreyski vesti) a caricature featuring a man wearing a yellow star, and about to be hanged, with the following caption: "The Fatherland front saved us from certain death. " 20 By that time, the "rescue" has acquired its narrative structure: the Bulgarian Jews were saved (passive voice); they need to express their gratefulness for their savior; anti-Jewish measures are blamed on a narrow group of individuals (the king, his entourage and a "clique of fascists"), whereas the vast majority of the Bulgarian people, sharing in a tradition 19 CDA, F 1449, o 1, ae 185, l. 45. 20 Evreyski vesti, 40, August 4, 1945, 1. of national tolerance, dissociated themselves from the "fascist" rulers and expressed feelings of solidarity towards the local Jews. As for the deportations from Yugoslavia and Greece, they bear a German stamp and were prepared with such secret rapidity that the Bulgarians could not halt them.

The second finding concerns the satellization of a broad range of topics, interests and cleavages around debates over Jewish wartime destinies. As noted earlier, in Bulgaria the destruction of Jews from the occupied territories was never taboo per se. The reasons behind this configuration are simple: first, the discussion of the Holocaust in Bulgaria was systematically translated into a retelling of the "rescue of the Bulgarian Jews." Second, since the end of the war the Holocaust has been constantly associated with debates over other (more or less loosely) related questions. Otherwise put, talking about the persecution of Jews allowed to discuss other issues by proxy. These parallel questions included the building of socialism; the role of the monarchy; the glory of the Communist Party, as well as the relationships beteween Jews in the diaspora and in Israel. Let us briefly consider how the intertwining between the Jewish predicament and this flurry of topics developed and evolved over the course of decades.

In 1945, denouncing in public speech the acts committed against the Jews served to demonstrate the scale of "fascist crimes" in the country, to rally a politically divided Jewish community to the project of the Fatherland Front, and to propel revolutionary momentum. In the diplomatic realm, the communist-led coalition in power was desperately trying to lend credibility to the notion of Bulgarian opposition to the defunct pro-Nazi regime, and thus solicit leniency from the victorious powers ahead of the negotiation of peace treaties. These are among the reasons why some alleged authors of anti-Jewish crimes faced legal proceedings in the spring of 1945. At the end of the 1950s, invoking the Holocaust within the context of Bulgarian-East German discussions over the making of a feature film dedicated to the deportation of Jews from Bulgarian-occupied northern Greece became a way for elites from the two countries to draw on distinct symbolic reservoirs for legitimizing their (ambivalent) pasts, and arm themselves for contemporary struggles (mostly against the Federal Republic of Germany, FRG). In representing the fates of Greek Jews imprisoned in a Bulgarian detention camp, they painstakingly set the terms for simultaneously establishing a national identity and a common belonging to the Eastern bloc. Meanwhile, their representational choices betrayed their position within a global moment in which certain modes of symbolizing the Holocaust (women as allegories of Jewish suffering, the use of Christian symbols to depict the Jewish tragedy, etc.) crossed the divide between East and West.

By the middle of the 1960s, the "rescue of the Bulgarian Jews" acquired renewed visibility. The 15 th anniversary of the March 1943 events (1958) had coincided with initial steps towards the installation in the Jewish Cultural House of a permanent exhibition dedicated to the "rescue," which was finally inaugurated in 1964. Internationally, the question of the judicial treatment of war crimes against Jews had been put on the agenda with the Eichmann trial in Jerusalem (1961) and a new wave of NS crimes trials in West Germany (as well as in the USSR, Poland and East Germany). Meanwhile, the supposed warmongering attitude of the FRG was endlessly stigmatized in Eastern European media. As early as 1960, the leaders of the Bulgarian Jewish community were informed of the start of an investigation against wartime minister plenipotentiary in Sofia, Adolf-Heinz Beckerle, in the Land of Hesse (a Land in Germany, what should I do? Land:Länder is the name they use). The former SA officer and diplomat was indicted for complicity in the murder of 11,343 Jews from the Bulgarian-occupied territories in Yugoslavia and Greece. While legal professionals, Jewish organizations, and Holocaust survivors from West Germany, Israel, the United States and Yugoslavia engaged in complex collaborative efforts, the Bulgarian authorities waited until 1966 (and Soviet encouragement to do so) before they accepted to help the west German investigators locate incriminating evidence against Beckerle. The sending of Bulgarians to testify before the district court in Frankfurt was not devoid of advantage. It offered an arena where the Bulgarian communists could publicize their vision of World War Two and oppose the reading of the war promoted by anti-communist Bulgarians in exile and Jews settled in Israel. Once more, the treatment of anti-Jewish persecutions mediated other contentious issues, namely interpretative battles over the role of the king during wartime.

At the turn of the 1980s, references to the "rescue," notably through museum exhibitions, scholarly publications, documentary and feature films, became increasingly interwoven with praise for the Communist Party, for its leader Todor Zhivkov, and for a state whose external image suffered from suspicions of its involvement in the May 1981 assassination attempt on Pope John Paul II, followed by the dull reception of Gorbachev's perestroika. The promotion of a heroic narrative of World War Two constituted one aspect of the patrimonialization of history, a practice shared by several Eastern European states during late socialism. This endeavor was accompanied with a heralding of national unity, which would play out against other Bulgarian minorities, id est the Turks and the Bulgarian-speaking Muslims who were subjected to forced assimilation campaigns in the early 1970s and even more brutally the second half of the 1980s. 21 Interestingly, there is no evidence that this policy was fostered, or even considered with a positive eye, in the USSR.

From the 1990s to the early 2000s, finally, Jewish destinies became one arena on which partisan identities and national roots were both fostered and contested. In the Republic of Macedonia born out of the Yugoslav wars (today North Macedonia), Jewish suffering became a metaphor for a fate of national non-recognition; rediscovered in Bulgaria, the Jewish predicament was marshaled to denounce the pretense of the anti-communists that the pre-Communist era had been faultless, allegedly embodying at once civilization and modernity. In dialogue with an expanded range of actors who felt empowered to speak their truth of the past (politicians, memory entrepreneurs, non-governmental activists, Jewish survivors and their descendants, etc.), the events of World War Two turned into battlegrounds.

In the last resort, it is my contention that the centrality of anti-Jewish persecutions since the end of the Second World War has precisely resulted from their incessant reformulations. They, in turn, were made possible as a result of the intimate links that joined the national appropriations of history to a trans-nationalizing of Holocaust memory, one to which the Jewish communities originating in the Balkans, as well as the non-Jewish Balkan diasporas, have contributed in important ways. At the risk of overstatement, one may argue that this transnational process has constituted one mode of, if not condition for, the increasing nationalization of academic as well as non-professional historical writings. A couple of years ago, there was room for some hope that a change in the pattern depicted in this article was in the offing. In March 2018, prime minister Boyko Borisov (GERB) attended a commemorative ceremony in tribute to Macedonia's Jews on the occasion of the 75th anniversary of their deportation in March 1943. Although the head of government failed to make a statement regarding Bulgaria's co-responsibility in the round-ups and deportation, his sheer presence -the first of such standing since 1945 -could be viewed as a positive sign. In October 2018, Bulgaria was admitted as a full member in the International Holocaust Remembrance Alliance (IHRA), an initiative launched in 1998 to foster knowledge and remembering of the Holocaust worldwide. Memory controversies about the role of wartime Bulgaria in Europe, Israel and the United States had invited an ever greater number of voices to join in the discussion of the past, and encouraged better documentation of the historical facts. Meanwhile, Bulgarian scholars had unearthed and published important archival documents about the wartime events. Alongside members of civil society, human rights activists, journalists, intellectuals, and descendants of Jewish survivors, these scholars had fostered a reconsideration of the official discourse about the (non)-deportations.

By the fall of 2020, however, against the backdrop of a political and social crisis exacerbated by the Covid-19 pandemic, the Bulgarian executive chose to play the national card ahead of the spring 2021 legislative elections: in a surprising turnabout, in December of the same year the very government, which had supported North Macedonia's aspirations for membership in the European Union (EU) in 2018-2019, blocked the opening of the country's EU talks on account of the alleged failure of the Macedonian authorities to deliver on the promises made at the time of signing a bilateral friendship treaty (August 2017), especially in the field of history writing and commemoration. On December 11, 2020, the leadership of the Macedonian Jewish Community reacted sharply to the Bulgarian stance. In an open letter, they called for recognition by Bulgaria of its complicity in the Holocaust of Macedonia's Jews: "in view of the continuous attempts to impose only one 'historical truth' in relation to historical events that happened on the territory of today's Republic of North Macedonia during World War II, the Jewish Community in the Republic of Macedonia and the Holocaust Fund of the Jews from Macedonia remind everyone of the indisputable and accepted historical facts regarding the role of the Kingdom of Bulgaria in the Holocaust and the deportation of the Jews from today's North Macedonia." 22 Since then, other worrying developments have occurred. On February 16, 2021, three days after the Sofia municipality banned the so-called "Lukov march," 23 a yearly event in tribute to pro-Nazi general Lukov, the Sofia City Court refused to cancel the registration of the Bulgarian National Union "Edelweiss" (Bӑlgarski natsionalen sӑyuz "Edelvays", BNSE), the organizer of the March. [START_REF]Sӑdӑt othvӑrli iska na prokuraturata za zabrana na organizatorite na Lukov march[END_REF] First held in 2003, the gathering had been drawing an increasing number of radical nationalist participants from all over Europe in the second half of the 2010s. In the fall of 2017, the World Jewish Congress launched a petition asking for the annulment of the 2018 edition of the march. The petition collected 175,000 signatures and was submitted to prime minister Borisov by the vice-president of the WJC, Robert Singer, on February 2, 2018. [START_REF]WJC Petition Signed by 175,000 and Calling for Ban on Neo-Nazi March Delivered to Bulgarian PM[END_REF] In February 2020, and again in 2021, Sofia mayor Yordanka Fӑndakova forbid the manifestation, with a significant number of policemen being deployed along the participants' route.

The decision of the Sofia City Court came as a result of a long process. In January 2020, the Bulgarian General prosecutor's office had instructed the State Agency "National Security" to investigate possible anti-state activities on the part of the Bulgarian National Union "Edelweiss". [START_REF]Geshev se zahvashta i s Lukov marsh (Geshev Takes up the Case of the Lukov March too)[END_REF] In August 2020, the prosecutor's office asked the Sofia City Court to cancel the registration of the Union on the grounds that the organization instilled "racial, ethnic and religious hatred," and had created a paramilitary structure. Shortly thereafter, a renowned historian, Mihail Gruev, acting director of the Bulgarian national archives, was asked by the court to submit a historical expertise on general Lukov, the Legionaries and several other figures celebrated by the Bulgarian National Union "Edelweiss". The expertise stated that general Lukov, in his capacity as minister of Defense (1935-1938), had played a significant role in the rearming of Bulgaria through a rapprochement with Nazi Germany. 27 From 1942 onwards, the Bulgarian officer also headed the National Union of Bulgarian Legions (Sӑyuz na bӑlgarskite natsionalni legioni, SBNL), an organization of about 15,000 members, which promoted nationalistic, pro-Nazi and antisemitic views and practices. [START_REF]Sotsialno nalyavo. Natsionalizmӑt -napred. Programi i organizatsionni dokumenti na bӑlgarski avtoritaristki nacionalisticheski formatsii (Socially to the Left. Nationalism -Forward[END_REF] Although general Lukov himself was not known to have made anti-semitic statements, historian Gruev asserted that the slogans and practices of the organization created an atmosphere conducive to the persecution of Jews. Furthermore, as head of the Legionaries, general Lukov had shown ever greater proximity to the Reich's Legation in Sofia until he was killed by members of the underground resistance movement on February 13, 1943. [START_REF]A similar interpretation may[END_REF] The court rejected the request. Most surprising were the explanations for the ruling. Arguing that such a decision would have created an unwelcome precedent (the first ban on a nongovernmental organization since the democratic changes on November 10, 1989), judge Veneta Tsvetkova offered her own view of historical "truth": "both Lukov's personality and the nature of SBNL", she claimed, "are not unambiguous and are multi-layered in historical aspect, evolving over the years, due to which the automatic association of the historical analysis of the SBNL to anti-Semitic activity and to an imitation of the Nazi model in terms of organization and uniforms is unfounded. On the one hand, given Lukov's merits as a military expert and his contribution to strengthening the Bulgarian army in the period between the two wars, as well as the lack of any evidence of anti-Semitic activity or Nazi propaganda by the same person, the general, as indicated by witnesses, was mythologized precisely because of his military merits in preserving the territorial integrity of the state and providing protection through the strengthening of the Bulgarian army. On the other hand, the SBNL are also associated with the bearing of patriotic ideas for the unity of the nation and the state territory. Therefore, one cannot automatically assumed that the organized processions and celebrations are a conduit of forbidden ideologies and incite ethnic, racial or religious hatred." 30 Judging by this new encounter between politics, justice and the writing of history, the palimpsest of retellings of the past in Bulgaria may have a future more protracted than expected. 30 See the decision at: https://legalacts.justice.bg/Search/GetActContentByActId?actId=lIaKcXoQS5Q%3D&fbclid=IwAR0go0HJwtj WWzPCC9lNNAbS41ChL4J5mOzSL9Ea3-DlLtnj_QyhwJkEEMI (accessed March 13, 2021)

Conclusion

  21 Dӑrzhavna Agentsiya Arhiv (DAA), "Vӑzroditelniyat protses." Bӑlgarsktata dӑrzhava i bӑlgarskite turtsi (The "Revival Process." The Bulgarian State and the Bulgarian Turks), Sofia 2009, 2 vol.; Mihail Gruev and Aleksey Kalyonski, Vӑzroditelniyat protses. Myusyulmanskite obshnosti i komunisticheskiyat rezhim ((The Revival Process. The Muslim Communities and the Communist Regime), Sofia 2008.

"North Macedonian Jews: Bulgaria has moral obligation to admit responsibility for WWII atrocities," December 11, 2020, at: https://eurojewcong.org/news/communities-news/north-macedonia/north-macedonian-

See also a report by Karl Klingenfuss, then employed with Referat D III of the Reich's Foreign Office: Scientific archive (Nauchen


Pascal Floquet

On the linearization of Antoine's law

A multivariate linearized Antoine's equation (MLAE) is presented in this short note. To our knowledge, it is the first time this multi-linear equation is proposed in the literature. Compared with initial null parameters or « Clapeyron » type initial parameters, that do not take into account the highly non-linear correlation between the parameters B and C, our procedure gives a very good initial guess for the nonlinear regression of vapor pressure versus temperature. Fortunately, the multi-linear expression of linearized Antoine's law is obtained in a simple, non iterative manner. Examples relevant from thermodynamical literature are used to show the quality of our MLAE parameters in order to converge to nonlinear regression solution of Antoine's model and to compare our approach with the classical ones.

Introduction

Historically, many different formulas and graphical representations have been suggested for the vapor pressure-temperature relationship for a pure component [START_REF] Wisniak | Historical Development of the Vapor Pressure Equation from Dalton to Antoine[END_REF]. Clapeyron's formula proposed in 1834 and thermodynamically stated by Clausius in 1864 gave birth to the well-known Clausius-Clapeyron differential relationship This assumption, or a stronger one of constant heat of vaporization, was soon found to be one of the main reasons why the Clausius-Clapeyron equation would only fit the experimental data in a limited temperature range. A common way to improve the fitting ability is to use the Antoine's equation for the vapor pressure of a saturated liquid which introduces a third parameter C in the approximated Clausius-Clapeyron equation

) 3 ( log C T B A P    
Classically, the three coefficients A, B and C have to be determined thanks to experimental data. From a numerical point of view, the equation ( 2) is clearly linear with the coefficients A and B, in terms of log P° and 1/T variables; thus, if a substance behaves according to the Clausius-Clapeyron model, a plot of log P° against 1/T should yield a straight line with slope -B. On the contrary, Antoine's equation ( 3) is a non-linear equation with respect to parameters A, B and C and for the same variables log P° and 1/T. This equation is a well-known non-linear model in the field of Chemical Engineering education and often used as an example for identification purpose of non-linear model [START_REF] Corriou | Process Control. Theory and Application[END_REF]. We propose, in this work, to promote a change of variables leading to a linearized Antoine's equation. It is easy, with this model, to compute the above coefficients A, B and C via a classical linear regression method. Thus, these values of coefficients are good candidates for initial points of a non-linear regression of complete non-linear Antoine's law. To prove the adequacy of a such initial guess, three examples have been chosen with experimental value (T exp , P exp ) of temperature and vapor pressure: carbon tetrachloride CCl 4 [Hildebrand and Mc Donald in Boublik et al., 1973], di-isobutyl hexahydrophtalate (DIBE) [START_REF] Liu | Isobaric Vapor-Liquid Equilibria of the Binary System Maleic Anhydride and Di-isobutyl Hexahydrophtalate at 2.67, 5.33 and 8.00 kPa[END_REF] and 2,6,10,14-tetramethylpentadecane (Pristane) [START_REF] Bourasseau | Measurement and Prediction of Vapour Pressures of 2,6,10,14-tetramethylpentadecane (pristane)[END_REF].

Several Non-linear Regressions on Antoine's Law

The non-linear Antoine's model (3) can be fitted with experimental data via a lot of distinct criteria to be minimized. Classical ones are the following:

2 exp 1 exp 2 1 exp exp 2 1 exp exp 2 1 exp 2 1 exp 2 1 exp exp exp 2 1 exp exp 2 exp 1 exp 2 1 exp ) ) )( ((log (BC) criterion . Boublik (RC) criterion Relative ) ( ) ( (
n i n i C T B A n i calc n i C T B A calc n i n i n i calc n i calc n i                                                                                                                    
All these criteria deal with least-square minimization problems that are not exactly equivalent in terms of parameter solutions or final criterion value. In this work, all the least-square fitting are performed with lsqnonlin program proceeding from Matlab ® Optimization Toolbox, based on the Levenberg-Marquardt method with line search for medium-scale problem [START_REF] Levenberg | A Method for the Solution of Certain Problems in Least-Squares[END_REF][START_REF] Marquardt | An Algorithm for Least-Squares Estimation of Nonlinear Parameters[END_REF]. For comparison purpose, the tolerances associated with these problems (termination tolerance on the function value  F and on variable change  X1 ) are set to 10 -7 .

Interaction between C and D coefficients

The difficulties of non-linear regression procedures (LAC, LRC, AC, RC, BC or other criteria) arise from the correlation between B and C coefficients and from the number of local minima of the optimization problem. As an example, it can be seen, for the carbon tetrachloride CCl 4 [Boublik et al., 1973] example and for a fixed value of coefficient A (A = 15), the response surface (criterion AC) versus B and C values. It can be observed than the "minimum valley" is quite narrow and may explain the numerical difficulties encountered.

Several Initial Points for the Non-linear Regression

All the non-linear regressions are performed with the following initial points:

 "Null" initial point: all the initial coefficient values A.

                  
This equation ( 4) is linear with the coefficients A/C, -1/C and (A -B/C), in terms of transformed variables logP°, T and TlogP°. It is then possible to linearly identify these coefficients and, thus, the original ones A, B and C, by fitting logP°e xp with T exp and T exp logP°e xp .

If we note X the following (n exp x 3) matrix

                  exp exp exp
                   C C A C B A 1  by  = (X T X) -1 X T Y Then                  3 3 1 2 3 2 3 2 1        C B A
The adequation of the linearized model can be seen in the following figure that gives the response logP° with T and TlogP° for the Pristane example. Scatter points are experimental points and surface corresponds to the multi-linear model. The table 1 shows the complete non-linear regression results for all the criteria and examples, in terms of coefficient initialization. It is easy to see that our MLAE initialization procedure gives in all cases better results compared with other initialization methods. The convergence of the non-linear regression to optimal solution with the different initializations is shown, for the DIBE example and the LAC criterion, in the following figure. As foreseen, the proposed MLAE procedure is quite good for quickly converge to optimal solution. The table 2 shows the literature results and the criterion chosen. In all the case, the initialization proposed and the non-linear regression end to the same or better solution.

Conclusion and Significance

A new linearized Antoine equation is proposed to initialize the well-known Antoine's regression problem. This equation is easy to compute and allows to obtain quickly good results to the non-linear regression of vapor pressure versus temperature of pure component.

Notations

  , P°, v and T respectively the vapor pressure, the corresponding volume and the temperature.

Figure 1 :

 1 Figure 1: Values of AC criterion versus coefficients B and C (A = 15) for the CCl 4 example A good initial point of the regression procedure provides us a less difficult path tracing to the solution and a clear advantage in terms of global minimum localization.

  B and C are set to zero;  "Clapeyron" initial point: initial coefficient values A and B are set to the solution of the linear Clausius-Clapeyron equation (2); the initial value of C is equal to zero;  "Linearized Antoine" initial point: all the initial coefficient values A, B and C are set to the solution of the linearized Antoine equation below. This new linear initial point we proposed is obtained by the following transformation of Antoine equation (

  multi-linear least-square regression gives the coefficient array

Figure 2 :

 2 Figure 2: Values of response logP° with T and TlogP° for the Pristane example

Figure 3 :

 3 Figure 3: Convergence versus Initialization for the DIBE example and the RC criterion (Null Initialization: left Y axis, Clapeyron and MLAE Initializations: right Y axis)

Table 1 :

 1 Comparison of initialization techniques on Antoine regression problem with several criteria (* maximal number of function calls)

	P°

Table2: Literature results (norm of the criterion) on Antoine regression problem with several criteria (in grey, the criterion chosen by the authors, if known)

Three termination criteria have been applied: non evolution on variables (evolution less than  X ), non evolution of leastsquare function (evolution less than  F ) and first-order optimality conditions less than  F .


Aneel Tanwani
email: aneel.tanwani@laas.fr.

Hyungbo Shim
email: hshim@snu.ac.kr

Lyapunov Functions for Singularly Perturbed Hybrid Systems with Frequent Jump Dynamics

Keywords: Hybrid system, Singular perturbation, Practical stability, Heterogeneous agents, Coupled nonlinear oscillators, Sampled-data system

This article considers the stability analysis for a class of hybrid systems with the focus being on the frequently occurring jump dynamics. The system class is modelled as a singularly perturbed hybrid system where the singular perturbation parameter governs the frequency of jumps. Consequently, this results in a quasi steady-state system modeled by a differential equation without any jumps, and the boundary-layer system described by purely discrete dynamics. By imposing appropriate assumptions on the quasi steady-state system and the boundary-layer system, we derive results showing practical convergence to a compact attractor when the jumps occur frequently often. Our system class is motivated by the control design problem in a network of second-order continuous-time coupled oscillators, where each agent communicates the information about its position to the neighbors at discrete times. As a corollary to our main result, we show that if the information exchange between the agents and their neighbors is frequent enough, then the oscillators achieve practical consensus.

I. INTRODUCTION

Hybrid systems, which comprise continuous and discrete dynamics, have found utility in modeling of a broad class of physical systems and algorithms. Conventional results on stability of hybrid systems, like [2, Theorem 3.18], require the continuous and discrete dynamics to be asymptotically stable. However, it is natural to develop stability conditions which provide an interplay between stability of continuous and discrete dynamics. Taking switched systems as an example of hybrid systems, it is seen that if the continuous dynamics are stable but discrete dynamics are not, then we can obtain stability conditions by constraining the frequency of switching times in the form of upper bound on average dwell-time (ADT) [START_REF] Liberzon | Switching in Systems and Control[END_REF]Chapter 3]. On the other hand, if the continuous dynamics are not stable but the discrete dynamics compensate for the growth of the state during continuoustime evolution, then the overall system is asymptotically stable if the number of discrete jumps over each bounded interval is large enough [START_REF] Hespanha | Lyapunov conditions for input-to-state stability of impulsive systems[END_REF]. This article considers an interconnected hybrid system where one subsystem has stable continuous dynamics, and the other subsystem has stable discrete dynamics only. The particular structure that we consider in our work is analyzed using singular perturbation methods [START_REF] Khalil | Nonlinear Systems[END_REF]Chapter 11].

Singular perturbation techniques have been conventionally studied for analysis of ordinary differential equations with slow and fast dynamics [START_REF] Khalil | Nonlinear Systems[END_REF]. The basic idea is to see the system as a perturbation of a nominal system that comprises the boundary-layer system (describing the limit of fast dynamics) and the quasi steady-state system (described by slow system with limiting value of fast dynamics). More recently, we have seen the use of singular perturbation methods for a class of hybrid systems where fast evolution appears in continuoustime dynamics only [START_REF] Sanfelice | On singular perturbations due to fast actuators in hybrid control systems[END_REF], [START_REF] Wang | Analysis for a class of singularly perturbed hybrid systems via averaging[END_REF]. The solutions resulting from a continuous-time boundary-layer system generate an average vector field which approximates the slow subsystem. Using the stability of the average dynamics, with appropriate hypotheses on system data, the authors show semiglobal practical stability for original system.

Moreover, we see the use of singular perturbation methods for analysis in different applications. One such application of these methods, which is also studied in this article, is the synchronization problem in networked systems. For multiagent systems with diffusive coupling and continuous-time heterogeneous agents [START_REF] Kim | Robustness of synchronization of heterogeneous agents by strong coupling and a large number of agents[END_REF], [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF], it is observed that the agents achieve practical convergence if the coupling strength is large enough. One can view the coupling parameter as a singular perturbation parameter [START_REF] Lee | A tool for analysis and synthesis of heterogeneous multi-agent systems under rank-deficient coupling[END_REF], [START_REF] Maghenem | Singular-perturbations-based analysis of synchronization in heterogeneous networks: A case-study[END_REF]. More recently, we have also seen the use of discontinuous coupling for synchronization of networked system in [START_REF] Coraggio | Distributed discontinuous coupling for convergence in heterogeneous networks[END_REF]. Our work actually proposes singular perturbation methods for analysis of networked systems where the discontinuity in coupling arises due to time-sampled information exchange between neighbors.

Based on this last remark, another research direction which is particularly relevant for our paper is the stability of dynamical systems under fast switching or frequent impulses. A related application in this direction is studied in [START_REF] Maheshwari | Stabilization under round robin scheduling of control inputs in nonlinear systems[END_REF], where the authors propose the use of fast switching roundrobin protocol for stability of a nonlinear controlled system. However, to the best of authors' knowledge, the use of singular perturbation methods for analysis of fast switching/impulses has not appeared in the literature, and this paper makes a contribution in this direction. An exception in this direction is the paper [START_REF] Yun | Singular perturbation for sampleddata systems with fast subsystems[END_REF], where the authors take sampling period as the singular perturbation parameter but the resulting analysis is carried out entirely in discrete domain, which is completely different from the approach adopted in this paper.

This article proposes singular perturbation techniques for a class of hybrid systems, where frequency of jump dynamics is associated with the singular perturbation parameter. In particular, as the parameter gets smaller, the frequency of jumps increases. We construct an appropriate Lyapunov function for the interconnection between slow continuous dynamics and fast discrete dynamics, and show practical convergence under mild assumptions on system data. Such constructions are inspired from the analysis of sampleddata systems [START_REF] Tanwani | Stabilization with event-driven controllers over a digital communication channel with random transmissions[END_REF]. As an application of our main result, we study a heterogeneous multi-agent system where each node is described by a second order nonlinear oscillator. Each agent communicates one of its states (the output) to its neighbors at discrete times. We show that if the frequency of time instants, at which agents exchange information, is large enough then the oscillators achieve practical consensus.

II. SINGULAR PERTURBATION THEOREMS FOR

A CLASS OF HYBRID SYSTEMS In this article, we are interested in studying a class of hybrid systems, parameterized by a scalar ∈ (0, 1],

     ẋ = f x (x, y, ), ẏ = f y (x, y, ), τ = σ, for (x, y, τ ) ∈ C x × C y × [0, 1],      x + = x, y + = g y (x, y), τ + ∈ [0, γ], for (x, y, τ ) ∈ C x × C y × {1}, (1)
where x ∈ R n , y ∈ R m , and τ ∈ R. The sets C x ⊂ R n and C y ⊂ R m are assumed to be closed, and contained in open and connected domains D x ⊂ R n and D y ⊂ R m , respectively. The vector fields f x : D x ×D y ×R ≥0 → R n and f y : D x ×D y ×R ≥0 → R m , and the function g y : D x ×D y → R m are assumed to be continuously differentiable in their arguments. The constant γ is assumed to belong to [0, 1), and σ is a positive constant. In the terminology of hybrid systems used in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF], it is seen that the flow set is

C x × C y × [0, 1]
and the jump set is C x × C y × {1}.

For the problems studied in this paper, the parameter appearing in (1) is taken to be small, which implies that the jumps occur frequently, and thus, the evolution of y relies highly on the jump dynamics. Since the jumps correspond to rapid evolution, we call x the slow variable and y the fast variable. The evolution of the timer τ , determined by the parameters , γ, and σ, describes the interplay between the flow and the jump. The evolution of system (1) guarantees that, over an interval [0, T + τ (0)], there are at least T σ/ jumps, and after every jump, the system flows for at least (1-γ) /σ units of time. Therefore, there is no accumulation time of jumps (i.e., there are finitely many jumps in any finite continuous-time interval) and the time domain of every complete solution has unbounded continuous-time element.

If is sufficiently small, the behavior of the solution 1

1 According to the framework of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF], one writes the solution at a given time instant as (x(t, j), y(t, j)) using the hybrid time index (t, j). In our case, we find it convenient to write (x(t), y(t)), which implicitly refers to (x(t, j), y(t, j)) for some j ∈ N ≥0 . The choice is motivated by the fact that, in this paper, the complete solutions have unbounded t. Whenever t is a jump instant, we indicate the value of y after the jump with y(t + ).

(x(t), y(t)) can be approximated by two different dynamic systems. The first system is called the boundary-layer system given by, for y ∈ C y ,

y + = g y (x, y) for a fixed value of x ∈ C x . (2)
When is sufficiently small, there are many jumps within a small continuous-time interval. This means that, from an initial condition (x(0), y(0)), while the solution x(t) remains close to x(0), the solution y(t) obeys y + = g y (x, y) which is approximated by [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]. If (2) has a unique equilibrium for each fixed x ∈ C x , then we may consider a continuously differentiable function h : R n → R m such that

g y (x, h(x)) = h(x), ∀ x ∈ C x . (3)
If, in addition, the equilibrium h(x) is asymptotically stable, then we can imagine that the solution y(t) rapidly approaches h(x(0)) after some jumps, and remains near h(x(t)) as time goes on. In order to approximate the behavior of x(t), we define the quasi steady-state system given by

ẋ = f x (x, h(x), 0), x ∈ C x (4)
which basically approximates the evolution of x(t) on the slow manifold {(x, y) ∈ R m × R n : y = h(x)}. In the following, let us clarify the intuition discussed so far.

A. Main Results

The basic problem studied in this paper relates to the stability of system [START_REF] Coraggio | Distributed discontinuous coupling for convergence in heterogeneous networks[END_REF]. Toward this end, we work under the assumption that the quasi steady-state system (4) and the boundary-layer system (2) are both stable. As a principal result, we show that by choosing small enough, system (1) achieves stability in an appropriate sense under aforementioned assumption.

Assumption 1: The boundary-layer system (2) is uniformly (in the fixed variable x) exponentially stable at the equilibrium h(x) ∈ C y ; that is, there exists a continuously differentiable function V y :

D x × D y → R such that b|y -h(x)| 2 ≤ V y (x, y) ≤ b|y -h(x)| 2 , V y (x, g y (x, y)) -V y (x, y) ≤ -b|y -h(x)| 2 , |∇ y V y (x, y)| ≤ l y |y -h(x)|, |∇ x V y (x, y)| ≤ l x |y -h(x)| 2
for all (x, y) ∈ C x × C y where b, b, b, l y , and l x are positive constants.

Theorem 1: Suppose that Assumption 1 holds and the quasi steady-state system (4) admits an asymptotically stable compact attractor A ⊂ C x , in the sense that there exists a continuously differentiable function

V x : D x → R such that, for all x ∈ C x , α(|x| A ) ≤ V x (x) ≤ α(|x| A ), (5)
∇ x V x (x)f x (x, h(x), 0) ≤ -α(|x| A ), (6)
where α and α are class K ∞ functions, and α is a class K function. Then, for any compact sets K x ⊂ C x and K y ⊂ C y and for every ρ > 0, there exists * > 0 such that, for any 0 < ≤ * and for any x(0) ∈ K x , y(0) ∈ K y , and

τ (0) ∈ [0, γ],

B. Comparisons with Existing Literature

Let us draw some comparisons between the system class (1) and some related frameworks from the literature.

1) Singularly perturbed hybrid systems: We observe that, by letting τ = τ , τ -dynamics can be similarly written as

˙ τ = σ, τ ∈ [0, ] τ + ∈ [0, γ ], τ ∈ { }.
Because we work with very small value of the parameter , it is natural to think of (1) as a singularly perturbed hybrid system, as studied in [START_REF] Wang | Analysis for a class of singularly perturbed hybrid systems via averaging[END_REF]. The quasi steady-state system is a hybrid system, described as,

ẋ = f x (x, h(x), 0), x ∈ C x x + = x, x ∈ C x .
The boundary-layer system is also a hybrid system, with x ∈ C x fixed, and described as

ẏ = f y (x, y, 0), y ∈ C y ˙ τ = σ, τ ∈ {0} y + = g y (x, y), y ∈ C y τ + = 0, τ ∈ {0}.
In other words, this reformulation is consistent with our earlier observation that the quasi steady-state system is purely continuous-time, and the boundary-layer system is purely discrete-time. This particular observation differentiates our system class from the one studied in [START_REF] Wang | Analysis for a class of singularly perturbed hybrid systems via averaging[END_REF].

2) Impulsive systems with reverse average dwell-time: One can also think of (1) as an impulsive system as studied in [START_REF] Hespanha | Lyapunov conditions for input-to-state stability of impulsive systems[END_REF]. We find stability conditions for impulsive systems in terms of average dwell-time and reverse average dwell-time associated with the sequence of impulse times. For every solution of system (1), if N (t, s) denotes the number of jumps on the interval (s, t], then

N (t, s) ≥ σ(t -s)
which essentially associates a reverse average dwell-time property to our system. However, rapid switching does not stabilize x-dynamics, and the slow switching may lead to instability of y-dynamics. Due to this reason, the tools from [START_REF] Hespanha | Lyapunov conditions for input-to-state stability of impulsive systems[END_REF] cannot be directly applied here.

In other words, our system (1) is an interconnection of a hybrid system with state x ∈ R n having asymptotically stable continuous-time dynamics but not necessarily stable discrete dynamics, and another hybrid system with state y ∈ R m having stable jump dynamics but possibly unstable continuous dynamics. From this viewpoint, small gain theorems for hybrid systems from [START_REF] Liberzon | Lyapunov-based smallgain theorems for hybrid systems[END_REF] are more suited for stability analysis of (1). However, for the applications studied in this article, imposing input-to-state stability assumptions on x and y subsystems is restrictive, and for this reason we propose analysis based on singular perturbation methods which results in practical stability rather than asymptotic stability.

III. NETWORK OF HETEROGENEOUS OSCILLATORS

As a by-product of our theoretical result in Theorem 1, we study the problem of synchronization of coupled nonlinear oscillators. We consider N continuous-time controlled oscillators, which are connected to each other by an undirected and connected graph. The control action is chosen as a function of the states of their neighbors at discrete times. If this control is updated sufficiently often, then we show that all the oscillators converge to each other in practical sense.

Each of the N oscillators (the agent) has a two dimensional state (ω i , ξ i ) ∈ R2 , for i = 1, . . . , N , whose dynamic equation is written as

ωi = -ω i + ξ i ξi = φ i (ω i , ξ i ) := (1 -f i (ω i ))(-ω i + ξ i ) -g i (ω i ) (9a)
where the functions f i , g i are assumed to be smooth. For example, if the agent i has f i (ω i ) = µ i (ω 2 i -1) and g i (ω i ) = ω i with µ i > 0, then the agent i is a well-known Van der Pol oscillator [START_REF] Khalil | Nonlinear Systems[END_REF]. Now we suppose that the agents are connected through an undirected and connected graph, and the state ξ i is updated from time to time as a functions of its neighbors' state ξ j , j ∈ N i , where N i is the index set of neighboring agents of the agent i. In particular, we describe the update model as

ω + i = ω i ξ + i = j∈Ni w ij ξ j (9b)
where w ij is the (i, j)-th entry of a doubly stochastic matrix W ∈ R N ×N . If the network is modeled by a symmetric adjacency matrix A, and D is the diagonal matrix that describes the degree of each node, then W is given by I -ϕD + ϕA, for which every row and column sums to 1 by construction and the weight ϕ > 0 is chosen small enough such that the matrix W becomes non-negative. The time when the update (9b) occurs is governed by a timer modeled as

τ = σ/ , τ ∈ [0, 1], (9c) τ + ∈ [0, γ], τ ∈ {1}, (9d)
where ∈ (0, 1], γ ∈ [0, 1), and σ > 0. This model implies that all the states ω i , ξ i , and τ are updated by (9b) and (9d) whenever τ (t) becomes 1, and after the update, the timer state τ (t + ) is chosen arbitrarily in [0, γ]. Such a framework for updating the state of agents at discrete times has been proposed in [START_REF] Tanwani | Suboptimal filtering over sensor networks with random communication[END_REF]. Now, let R ∈ R N ×(N -1) be a matrix such that R R = I and 1 N R = 0. Then, it is seen that Λ := R W R ∈ R (N -1)×(N -1) is a positive definite Schur stable matrix. With the matrix R, we can write [START_REF] Liberzon | Lyapunov-based smallgain theorems for hybrid systems[END_REF] in the form of [START_REF] Coraggio | Distributed discontinuous coupling for convergence in heterogeneous networks[END_REF]. Let us introduce the new coordinates (s, ζ) for ξ := col(ξ 1 , . . . , ξ N ) ∈ R N as

s = 1 N N i=1 ξ i ∈ R, ζ = R ξ ∈ R N -1 .
Then, the system (9) can be written as

ωi = -ω i + s + R i ζ, i = 1, . . . , N, ṡ = 1 N N i=1 (1 -f i (ω i ))(-ω i + s + R i ζ) -g i (ω i ) ζ = R φ(ω, 1 N s + Rζ) τ = σ (10a)
with the flow set R 2N × [0, 1], and

ω + i = ω i , for i = 1, . . . , N, (10b) s + = s, ζ + = Λζ, τ + ∈ [0, γ] (10c)
with the jump set R 2N × {1}. In ( 10), R i denotes the i-th row of R, and with ω := col(ω 1 , . . . , ω N ),

φ(ω, 1 N s + Rζ) =    φ 1 (ω 1 , s + R 1 ζ) . . . φ N (ω N , s + R N ζ)    .
Here, if we treat (ω 1 , . . . , ω N , s) = (ω, s) as the slow variable and ζ as the fast variable with both the flow set and the jump set for (ω, s, ζ) are the same as R 2N , then the system is now in the form of (1). On the other hand, the quasi steady-state (QSS) system is

ωi = -ω i + s, i = 1, . . . , N, ṡ = 1 N N i=1 (1 -f i (ω i ))(-ω i + s) -g i (ω i ), (11)
which corresponds to (4). The following assumption and lemma assures that this QSS system3 satisfies the assumption of Theorem 1.

Assumption 2: The second order dynamical system

χ = -χ + s ṡ = (1 -f (χ))(-χ + s) -g(χ) (12)
where

f (χ) := 1 N N i=1 f i (χ), g(χ) := 1 N N i=1 g i (χ) (13)
has a locally exponentially stable attractive compact set A 0 ⊂ R 2 (i.e., a stable limit cycle).

Lemma 1: Under Assumption 2, the QSS system (11) has a locally asymptotically attractive compact set A ⊂ R N +1 , which is a stable limit cycle for [START_REF] Maheshwari | Stabilization under round robin scheduling of control inputs in nonlinear systems[END_REF], given by

A = {(χ, . . . , χ, s) ∈ R N +1 : (χ, s) ∈ A 0 }. (14)
Moreover, there exists V that satisfies ( 5) and ( 6) on a neighborhood Ω of A.

Proof: The basic idea for the proof comes from [6, Theorem 1]. Let

χ := 1 N 1 N ω, χ := R ω so that ω i = χ + R i χ. Then, we obtain χ = -χ + s, χ = -χ, ṡ = 1 N N i=1 (1 -f i (χ + R i χ))(s -χ -R i χ) -g i (χ + R i χ) = (1 -f (χ))(-χ + s) -g(χ) + 1 N N i=1 ψ i (s, χ, χ)
where ψ i vanishes when χ = 0 because

ψ i = f i (χ + R i χ)R i χ + (f i (χ + R i χ) -f i (χ))(χ -s) + (g i (χ) -g i (χ + R i χ)).
By continuous differentiability of φ i , there is a continuous function

Ψ(s, χ, χ) ∈ R 1×(N -1) such that (1/N ) N i=1 ψ i (s, χ, χ) = Ψ(s, χ, χ) χ. By Assumption 2, there is a Lyapunov function W (χ, s) and a constant c > 0 such that c 1 |(χ, s)| A0 ≤ W (χ, s) ≤ c 2 |(χ, s)| A0 , Ẇ | along (12) ≤ -c 3 |(χ,
V (s, χ, χ) = W (χ, s) + k| χ| 2 (15)
where

k := c 2 4 M 2 /(2c 3 ). Then, V ≤ -c 3 |(χ, s)| 2 A0 + c 4 |(χ, s)| A0 M | χ| -2k| χ| 2 ≤ - c 3 2 |(χ, s)| 2 A0 -k| χ| 2
on the set

Ω := (s, χ, χ) : V (s, χ, χ) < min{c, kC 2 } . (16)
This completes the proof.

We can now establish the following result about the networked system (9). Theorem 2: Under Assumption 2, suppose that the initial condition of the multi-agent system (9) is such that [START_REF] Wang | Analysis for a class of singularly perturbed hybrid systems via averaging[END_REF]. Then, for each η > 0, there is * > 0 such that, for each 0 < ≤ * ,

(ω 1 , ω 2 , . . . , ω N , (1/N ) N i=1 ξ i ) ∈ Ω given in
lim sup t→∞ |(ω i (t), ξ i (t)) -(ω j (t), ξ j (t))| ≤ η, lim sup t→∞ |(ω i (t), ξ i (t))| A0 ≤ η, ∀i, j = 1, . . . , N.
The proof follows from Theorem 1 and the fact that, if ζ ≡ 0 then ξ i = s and lim t→∞ |ω i (t) -ω j (t)| = 0 by (10a) for all i, j = 1, . . . , N . Indeed, the boundary-layer system is ζ + = Λζ, which corresponds to (2) and satisfies Assumption 1 (with the function h being the zero function). The stability of QSS system (11) follows from Lemma 1.

Example 1: We have simulated four Van der Pol oscillators (9a) with f i (ω i ) = µ i (ω 2 i -1) where µ 1 = 1, µ 2 = 1.5, µ 3 = 2, and µ 4 = 2.5, and g i (ω i ) = ω i for i = 1, . . . , 4. The oscillators are connected through an undirected and connected graph, for which the adjacency matrix A is 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 and the matrix W in (9b) is W = 0.6I + 0.2A.

For the timer, we work with, σ = 1, γ = 0.9, and study different cases for ∈ {0.05, 0.2, 1}. At every jump time, τ + is randomly chosen according to a uniform distribution on [0, γ]. The state trajectories of all the agents are plotted in Figure 1 for two different values of .

IV. ANALYSIS AND PROOFS

For convenience of the proof, let us define z := y -h(x) and consider the closed set

C xz := {(x, y -h(x)) : (x, y) ∈ C x × C y }.
With the state (x, z, τ ) ∈ C xz × [0, 1], we can rewrite the flow dynamics as

ẋ = f x (x, h(x), 0) + [f x (x, z + h(x), ) -f x (x, h(x), 0)] =: f x (x, h(x), 0) + F x (x, z, ), ż = f y (x, z + h(x), ) - ∂h ∂x (x)f x (x, z + h(x), ) =: F z (x, z, ) τ = σ,
and the jump dynamics, for (x, z, τ ) ∈ C xz × {1}, as

x + = x, z + = g y (x, z + h(x)) -h(x) =: g z (x, z), τ + ∈ [0, γ].

A. Proof of Theorem 1

For the stability analysis of the hybrid system in the translated coordinates, let us consider the Lyapunov function candidate

V , in an open neighborhood of [0, 1] × C xz , V (τ, x, z) := e cx τ V x (x) + e -czτ V z (x, z) (17)
where V z (x, z) := V y (x, z + h(x)), and the positive scalars c x and c z are chosen to be sufficiently small (less than 1) and will be specified later. The maximum value * of the parameter is also to be chosen less than or equal to 1. Since τ ∈ [0, 1], the function V satisfies the following inequalities:

α(|x| A ) + e -1 b|z| 2 ≤ V (τ, x, z) ≤ e 1 α(|x| A ) + b|z| 2
and hence the lower and upper bounds on V are class K ∞ functions with respect to distance from the set [0, 1] × A × {0}. In particular, for any η > 0, define the following compact sets:

Ω η := (x, z) ∈ C xz : |x| A ≤ α -1 η 2e , |z| ≤ η 2b Ω η (τ ) := {(x, z) ∈ C xz : V (τ, x, z) ≤ η} Ω η := (x, z) ∈ C xz : |x| A ≤ α -1 (η), |z| ≤ eη b .
Then, an immediate consequence is that

Ω η ⊂ Ω η (τ ) ⊂ Ω η ⊂ C xz , ∀τ ∈ [0, 1].
Now, pick κ large enough so that K xz ⊂ Ω κ where K xz := {(x, y -h(x)) : (x, y) ∈ K x × K y }. Then, we restrict our Lyapunov analysis to the compact set Ω κ .

With the given ρ, take η = min{κ, α(ρ), ρ 2 b/e}. From now on, we show that V < 0 and ∆V := V (τ + , x + , z + ) -V (τ, x, z) < 0, for every (x, z) ∈ Ω κ \ Ω η , and every τ ∈ [0, 1]. This allows us to conclude that the state variables (x, z) converge to Ω η as long as the maximal solution exists (see [2, Theorem 3.18]), and thus, proves Theorem 1. Now, by the compactness of Ω κ , there exist constants M x and M z such that

|f x (x, z + h(x), )| ≤ M x , |F z (x, z, )| ≤ M z , |∇ x V x (x)| ≤ M V for all (x, z, ) ∈ Ω κ × [0, 1]
. Also, by invoking Lemma 2 in Appendix, there exist N x and N such that

|F x (x, z, )| ≤ N x |z| + N for all (x, z, ) ∈ Ω κ × [0, 1].
To analyze ∆V along the jump dynamics, we observe that

∆V = V (τ + , x + , z + ) -V (τ, x, z) ≤ e cx γ V x (x) -e cx V x (x) + V z (x, g z (x, z)) -e -cz V z (x, z) ≤ -(e cx -e cx γ )V x (x) + V z (x, g z (x, z)) -V z (x, z) + (1 -e -cz )V z (x, z) ≤ -(e cx -e cx γ )α(|x| A ) -b|z| 2 + (1 -e -cz )b|z| 2 .
Let the positive scalar c z be sufficiently small such that

(1 -e -cz )b < b.
Then, since γ < 1, for the chosen c z and for any c x > 0 and > 0,

∆V < 0, ∀(τ, x, z) ∈ [0, 1] × Ω κ .
Next we compute the bound on the derivative of V along the flow dynamics as follows: Proof: For a given δ > 0, pick a positive s * ≤ s such that α 1 (s * ) ≤ δ, and take

V = c x σe cx τ V x (x) + e cx τ ∇ x V x • f x (x, h(x), 0) + e cx τ ∇ x V x • F x (x, z, ) - c z σ e -czτ V z (x, z) + e -czτ ∇ z V z • F z (x, z, ) + ∇ x V z • f x (x, z + h(x)
v 12 = max s∈[s * ,s] α 1 (s) α 2 (s)
which is well-defined. We immediately get α 1 (s) ≤ δ, 0 ≤ s < s * , v 12 α 2 (s), s * ≤ s ≤ s which is the desired inequality.

  s)| A0 , and |∇W (χ, s)| ≤ c 4 |(χ, s)| A0 on the set {(χ, s) : W (χ, s) ≤ c}. With any C > 0, let M := max W (χ,s)≤c,| χ|≤C |Ψ(s, χ, χ)|, and let

Fig. 1 .

 1 Fig. 1. Plot of (ω i (t), ξ i (t)) for four agents. Numbers are time stamps, and all trajectories circulate clock-wise. (Top left): Case of no coupling; Started from four different initial conditions (time stamps '0'), their motions are not synchronized, and the black agent does not have a stable limit cycle so that it converges to the origin. (Top right): Coupled with = 1; Jumps are observed and they approximately synchronize. (Bottom left): Coupled with = 0.2. (Bottom left): Coupled with = 0.05. The animated version can be seen at https://youtu.be/f_TYWumAD-w.

2 + c z σ 2 eM x l x |z| 2 + 1 -κ 0 |z| 2 +Lemma 2 :

 222122 , ) ≤ c x σe cx α(|x| A ) -α(|x| A ) + e cx M V (N x |z| + N ) -c z σ e -cz b|z| 2 + M z l y |z| + M x l x |z| 2 .Using Lemma 3 in Appendix, with s = max (x,z)∈Ωκ |x| A , it follows that for every o x > 0, there existsv x > 0 such that α(|x| A ) ≤ v x α(|x| A ) + o x .Using this bound and Young's inequality, it follows thatV ≤ (c x σe cx v x -1)α(|x| A ) + c x σe cx o x -c z σ e -cz b|z| -cz b|z| 2 + (e cx M V N x + M z l y ) 2 2c z σe -cz b + e cx M V N ≤ -(1 -c x σe cx v x )α(|x| A ) -κ κ c + c x σeo xwhere κ 0 := M x l x , κ 1 := c z σe -cz b z /2, andκ c := (eM V N x + M z l y ) 2 /(4κ 1 ) + eM V N . With m := max 1 2 α α -1 η 2e , η 4b , the choice of * and c x such that * ≤ min 1, κ1 κ0+1/2 , m 2κc , c x ≤ min 1, 1 2σevx , m 2σeox makes V ≤ -(1/2)α(|x| A ) -(1/2)|z| 2 + m < 0 if |x| A > α -1 (η/(2e)) or |z| > η/(2b). This proves that V < 0, ∀(τ, x, z) ∈ [0, 1] × Ω κ \Ω η .ACKNOWLEDGEMENTThe second author is grateful to Dr. Jin Gyu Lee at University of Cambridge for helpful discussions about Lemma 1.APPENDIX Consider a continuously differentiable functionf : R n ×R m → R l such that f (x, 0) = 0 for all x ∈ R n . For any compact set K ⊂ R n × R m , there are nonnegative constants N i such that |f (x, y)| ≤ N 1 |y 1 | + • • • + N m |y m |, ∀(x, y) ∈ K. Proof:The result follows by applying the mean-value theorem and taking the maximum of the continuous function (∂f /∂y)(x, y) over the compact set K.

Lemma 3 :

 3 Consider two class K functions α 1 and α 2 . Fix s > 0. For every δ > 0, there exists v 12 > 0 such that α 1 (s) ≤ v 12 α 2 (s) + δ, ∀s ∈ [0, s].

  the solution (x(t), y(t)) to (1) does not have finite escape time and satisfies2

	lim sup	|x(t)| A ≤ ρ,	(7)
	t→∞
	lim sup

t→∞ |y(t) -h(x(t))| ≤ ρ.

In equations[START_REF] Lee | A tool for analysis and synthesis of heterogeneous multi-agent systems under rank-deficient coupling[END_REF] and[START_REF] Liberzon | Switching in Systems and Control[END_REF], we provide bounds on forward complete solutions (x(t), y(t)) for t going to +∞. It may happen that, for certain initial conditions, the resulting solution is defined only over a finite time interval, and the aforementioned bounds do not apply for such solutions.

The QSS system[START_REF] Maheshwari | Stabilization under round robin scheduling of control inputs in nonlinear systems[END_REF] is called 'blended dynamics' in the context of heterogeneous multi-agent systems[START_REF] Lee | A tool for analysis and synthesis of heterogeneous multi-agent systems under rank-deficient coupling[END_REF].

This work was supported in part by the National Research Foundation of Korea grant funded by the Korea government (Ministry of Science and ICT), (NRF-2017R1E1A1A03070342), and under the framework of international cooperation program managed by the National Research Foundation of Korea (2019K2A9A1A06093254). The authors also acknowledge the financial support by CNRS for the project DIROBS under the grant PRC-2738.

A. Tanwani is with LAAS-CNRS, France,