Synchronization Analysis of Complex Networks of Nonlinear Oscillators Ali El Ati

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Synchronization Analysis of Complex Networks of Nonlinear Oscillators Ali El Ati Synchronization analysis of complex networks of nonlinear oscillators Ali El Ati To cite this version: Ali El Ati. Synchronization analysis of complex networks of nonlinear oscillators. Other [cond- mat.other]. Université Paris Sud - Paris XI, 2014. English. NNT : 2014PA112362. tel-01241761 HAL Id: tel-01241761 https://tel.archives-ouvertes.fr/tel-01241761 Submitted on 11 Dec 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE´ PARIS-SUD ECOLE´ DOCTORALE : Sciences et Technologie de l'Information, des T´el´ecommunications et des Syst`emes Laboratoire des signaux et syst`emes DISCIPLINE : G´enie informatique, automatique et traitement du signal THESE` DE DOCTORAT Pr´esent´ee le 04/12/2014 par Ali EL ATI Synchronization analysis of complex networks of nonlinear oscillators Directeur de th`ese : Antonio Loria Directeur de Recherche CNRS (LSS) Encadrant : Elena Panteley Charg´ede recherche CNRS (LSS) Composition du jury : Rapporteurs : Jean-Pierre Barbot Professeur (ENSEA ) Alexander Pogromskiy Professeur (Universit´ede Eindhoven) Examinateurs : Luca Zaccarian Directeur de recherche CNRS (LAAS) William Pasillas-lepine Charg´ede recherche CNRS (LSS) ii R´esum´e Cette th`eseporte sur l'analyse de la synchronisation des grands r´eseaux d'oscillateurs non lin´eaires et h´et´erog`enes `al'aide d'outils et de m´ethodes issues de la th´eorie du contr^ole. Nous consid´erons deux mod`eles de r´eseaux ; `asavoir, le mod`ele de Kuramoto qui consid`ere seulement les coordonn´ees de phase des oscillateurs et des r´eseaux com- pos´esd'oscillateurs non lin´eaires de Stuart-Landau connect´espar un couplage lin´eaire. Pour le mod`ele de Kuramoto nous construisons un syst`emelin´eaire qui conserve les informations sur les fr´equences naturelles et sur les gains d'interconnexion du mod`ele original de Kuramoto. Nous montrons en suite que l'existence de solutions `averrouillage de phase du mod`ele de Kuramoto est ´equivalente `al'existence d'un tel syst`eme lin´eaire avec certaines propri´et´es. Ce syst`eme est utilis´epour formuler les conditions d'existence de solutions `averrouillage de phase et de leur stabilit´epour des structures particuli`eres de l'interconnexion. Ensuite, cette analyse s'est ´etendue au cas o`udes interactions at- tractives et r´epulsives sont pr´esentes dans le r´eseau. Nous consid´erons cette situation lorsque les gains d'interconnexion peuvent ^etre `ala fois positif et n´egatif. Dans le cadre de r´eseaux d'oscillateurs de Stuart-Landau, nous pr´esentons une nouvelle transformation de coordonn´eesdu r´eseau qui permet de r´e´ecrire le mod`ele du r´eseau en deux parties : une d´ecrivant le comportement de l'oscillateur « moyenne » du r´eseau et la seconde partie pr´esentant les dynamiques des erreurs de synchronisation par rapport `acet oscillateur « moyenne ». Cette transformation nous permet de caract´eriser les propri´et´esdu r´eseau en termes de la stabilit´edes erreurs de synchronisation et du cycle limite de l'oscillateur « moyenne ». Pour ce faire, nous reformulons ce probl`emeen un probl`emede stabilit´ede deux ensembles compacts et nous utilisons des outils issus de la stabilit´ede Lyapunov pour montrer la stabilit´epratique de ces derniers pour des valeurs suffisamment grandes du gain d'interconnexion. Mots cl´es: synchronisation, r´eseau d'oscillateurs non-lin´eaires, oscillateur `acycle li- mite, graphe orient´e, graphe sign´e,mod`ele de Kuramoto, ´equation de Stuart-Landau, interactions attractives et r´epulsives, stabilit´easymptotique, stabilit´epratique. iv Abstract This thesis is devoted to the analysis of synchronization in large networks of hetero- geneous nonlinear oscillators using tools and methods issued from control theory. We consider two models of networks ; namely, the Kuramoto model which takes into ac- count only phase coordinates of the oscillators and networks composed of nonlinear Stuart-Landau oscillators interconnected by linear coupling. For the Kuramoto model we construct an auxiliary linear system that preserves information on the natural fre- quencies and interconnection gains of the original Kuramoto model. We show next that existence of phase locked solutions of the Kuramoto model is equivalent to the existence of such a linear system with certain properties. This system is used to formulate condi- tions that ensure existence of phase-locked solutions and their stability for particular structures of network interconnections. Next, this analysis is extended to the case where both attractive and repulsive interactions are present in the network that is we consider the situation where some of the interconnection gains are allowed to be negative. In the context of networks of Stuart-Landau oscillators, we present a new coordinate transformation of the network which allows to split the network model into two parts, one describing behaviour of an "averaged" network oscillator and the second one, descri- bing dynamics of the synchronization errors relative to this "averaged" oscillator. This transformation allows us to characterize properties of the network in terms of stability of synchronization errors and limit cycle of the "averaged" oscillator. To do so, we re- cast this problem as a problem of stability of compact sets and use Lyapunov stability tools to ensure practical stability of both sets for sufficiently large values of the coupling strength. Keywords : synchronization, network of nonlinear systems, limit-cycle oscillator, di- graph, signed graphs, Kuramoto model, phase-locked solution, Stuart-Landau equation, attractive and repulsive interactions, asymptotic stability, practical stability. vi A` la m´emoire d'un homme courageux et g´en´ereux, `ala m´emoire de mon p`ere, Mohamed El-Ati paix `ason ^ame. "Pendant le vivant de votre p`ere, observez avec soin sa volont´e; Apr`es sa mort, ayez toujours les yeux fix´es sur ses actions." Citation de Confucius ; Les entretiens - VIe s. av. J.-C. vii A` la personne la plus ch`ere `amon coeur, `aune m`ere d'exception sans qui je n'aurai sans doute jamais tant accompli, `ama tr`es ch`ere maman. "Le paradis est sous les pieds des m`eres." Proph`ete Mohamed. ix Remerciements La pr´esente ´etude n'aurait pas ´et´epossible sans le bienveillant soutien de certaines personnes. Je voudrais les prier d'accueillir tous mes sentiments de gratitude en acceptant mes remerciements. Tout d'abord, mes remerciements s'adressent aux Elena Panteley et Antonio Loria mes directeurs de th`ese. Je leurs exprime toute ma reconnaissance et ma gratitude pour leur soutien et la confiance qu'ils m'ont donn´es. Je tiens `aexprimer ma gratitude `aElena pour ses conseils judicieux et son suivi r´egulier. Elle a toujours ´et´edisponible pour d'intenses discussions. Le manuscrit lui doit aussi beaucoup, gr^ace `ases innombrables relectures, `ala fois promptes et minutieuses. Je remercie vivement mes rapporteurs, Jean-Pierre Barbot et Alexander Pogromskiy pour avoir accept´ede juger la qualit´ede ce travail. Je remercie Luca Zaccarian et William Pasillas-lepine de m'avoir fait l'honneur de faire partie de mon jury de th`ese. Durant ces trois ann´ees de th`ese, j'ai eu la chance de cotoyer `aSupelec de nombreuses personnes attachantes, que toutes soient remerci´ees pour les bons moments partag´es. Je souhaite ´egalement exprimer mes remerciements aux mes soeurs et mes fr`eres pour leurs encouragements et soutien. Ce moment est aussi pour moi l'occasion d'exprimer ma reconnaissance envers les pro- fesseurs qui m'ont enseign´eet qui sont pour beaucoup dans mon parcours universitaire. R´esum´efran¸cais 1. Introduction Les travaux de cette th`ese sont consacr´es`al'analyse de la synchronisation des r´eseaux `agrand nombre d'oscillateurs non lin´eaires et h´et´erog`enes, en utilisant des outils et des m´ethodes issues de la th´eorie du contr^ole. Deux mod`eles sont utilis´es pour d´ecrire les r´eseaux d'oscillateurs : le mod`ele de Kuramoto et le mod`ele de Stuart-Landau. Notre analyse se concentre sur l'effet des interactions entre les ´el´ements du r´eseau sur la syn- chronisation du syst`eme. La th`esecontient des r´esultats d'analyse et leurs confirmations `atravers des exemples de simulation d´emonstratifs. La premi`ere partie du travail porte sur l'´etude du mod`ele de Kuramoto. Tout d'abord, nous consid´erons un r´eseau complexe arbitraire des oscillateurs de Kuramoto. Pour ce dernier, nous donnons les conditions d'existence de solutions `averrouillage de phase ainsi que l'expression analytique de la fr´equence de synchronisation. Ensuite, nous proposons une nouvelle g´en´eralisation du mod`ele de Kuromoto avec des interactions plus h´et´ero- g`enes : nous supposons que chaque oscillateur est caract´eris´epar ses poids d'entr´ee et de sortie outre sa fr´equence naturelle. Pour ce mod`ele, nous pr´esentons les solutions `a verrouillage de phase ainsi que leur analyse de stabilit´e.Notre ´etude s'est ´etendue au cas o`ule r´eseau contient des pond´erations n´egatives. Ceci permet de mod´eliser `ala fois les interactions attractives et r´epulsives entre les oscillateurs. Dans ce cas, les crit`eres d'existence de solutions `averrouillage de phase sont formul´eset leur stabilit´elocale est analys´ee. Dans la seconde partie, nous ´etudions le probl`eme de synchronisation d'un r´eseau h´e- t´erog`ene d'oscillateurs de Stuart-Landau.
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