abstract: The Kuramoto model is an archetypal model for studying synchronization in groups of nonidentical oscillators where oscillators are imbued with their own frequency and coupled with other oscillators though a network of interactions. As the coupling strength increases, there is a bifurcation to complete synchronization where all oscillators move with the same frequency and show a collective rhythm. Kuramoto-like dynamics are considered a relevant model for instabilities of the AC-power grid which operates in synchrony under standard conditions but exhibits, in a state of failure, segmentation of the grid into desynchronized clusters. In this dissertation the minimum coupling strength required to ensure total frequency synchronization in a Kuramoto system, called the critical coupling, is investigated. For coupling strength below the critical coupling, clusters of oscillators form where oscillators within a cluster are on average oscillating with the same long-term frequency. A unified order parameter based approach is developed to create approximations of the critical coupling. Some of the new approximations provide strict lower bounds for the critical coupling. In addition, these approximations allow for predictions of the partially synchronized clusters that emerge in the bifurcation from the synchronized state. Merging the order parameter approach with graph theoretical concepts leads to a characterization of this bifurcation as a weighted graph partitioning problem on an arbitrary networks which then leads to an optimization problem that can efficiently estimate the partially synchronized clusters. Numerical experiments on random Kuramoto systems show the high accuracy of these methods. An interpretation of the methods in the context of power systems is provided. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2018

In this work, we investigate the performance limitations characterizing certain common closure models for nonlinear models of fluid flow. The need for closures arises when for computational reasons first-principles models, such as the Navier-Stokes equations, are replaced with their simplified (filtered) versions such as the Large-Eddy Simulation (LES). In the present work, we focus on a simple model problem based on the 1D Kuramoto-Sivashinsky equation with a Smagorinsky-type eddy-viscosity closure model. The eddy viscosity is assumed to be a function of the state (flow) variable whose optimal functional form is determined in a very general form in the continuous setting. It is found by solving a PDE-constrained optimization problem in which the least-squares error between the output of the LES and the true flow evolution is minimized with respect to the functional form of the eddy viscosity. This problem is solved using a gradient-based technique utilizing a suitable adjoint-based variational data-assimilation approach implemented in the optimize-then-discretize setting using state-of-the-art techniques. The numerical computations are thoroughly validated. The obtained results indicate how the standard Smagorinsky closure model can be refined such that the corresponding LES evolution approximates more accurately the evolution of the original (unfiltered) flow. / Thesis / Master of Science (MSc)

Synchonisation ist ein universelles Phänomen welches in den Natur- und Ingenieurwissenschaften, aber auch in Sozialsystemen vorkommt. Verschiedene Modellsysteme wurden zur Beschreibung von Synchronisation vorgeschlagen, wobei das Kuramoto-Modell das am weitesten verbreitete ist. Das Kuramoto-Modell zweiter Ordnung beschreibt eigenständige Phasenoszillatoren mit heterogenen Eigenfrequenzen, die durch den Sinus ihrer Phasendifferenzen gekoppelt sind, und wird benutzt um nichtlineare Dynamiken in Stromnetzen, Josephson-Kontakten und vielen anderen Systemen zu analysieren. Im Laufe der letzten Jahre wurden insbesondere Netzwerke von Kuramoto-Oszillatoren studiert, da sie einfach genug für eine analytische Beschreibung und denoch reich an vielfältigen Phänomenen sind. Eines dieser Phänomene, explosive synchronization, entsteht in skalenfreien Netzwerken wenn eine Korrelation zwischen den Eigenfrequenzen der Oszillatoren und der Netzwerktopolgie besteht. Im ersten Teil dieser Dissertation wird ein Kuramoto-Netzwerk zweiter Ordnung mit einer Korrelation zwischen den Eigenfrequenzen der Oszillatoren und dem Netzwerkgrad untersucht. Die Theorie im Kontinuumslimit und für unkorrelierte Netzwerke wird für das Modell mit asymmetrischer Eigenfrequenzverteilung entwickelt. Dabei zeigt sich, dass Cluster von Knoten mit demselben Grad nacheinander synchronisieren, beginnend mit dem kleinsten Grad. Dieses neue Phänomen wird als cluster explosive synchronization bezeichnet. Numerische Untersuchungen zeigen, dass dieses Phänomen auch durch die Zusammensetzung der Netzwerkgrade beeinflusst wird. Zum Beispiel entstehen unstetige Übergänge nicht nur in disassortativen, sondern auch in stark assortativen Netzwerken, im Gegensatz zum Kuramoto-Modell erster Ordnung.Unstetige Phasenübergänge lassen sich anhand eines Ordnungsparameters und der Hysterese auf unterschiedliche Anfangsbedingungen zurückführen. Unter starken Störungen kann das System von wünschenswerten in nicht gewünschte Zustände übergehen. Diese Art der Stabilität unter starken Störungen kann mit dem Konzept der basin stability quantifiziert werden. Im zweiten Teil dieser Dissertation wird die basin stability der Synchronisation im Kuramoto-Modell zweiter Ordnung untersucht, wobei die Knoten separat gestört werden. Dabei wurde ein neues Phänomen mit zwei nacheinander auftretenden Übergängen erster Art entdeckt: Eine \emph{onset transition} von einer globalen Stabilität zu einer lokalen Instabilität, und eine suffusing transition von lokaler zu globaler Stabilität. Diese Abfolge wird als onset and suffusing transition bezeichnet.Die Stabilität von Netzwerknoten kann durch die lokale Netzwerktopologie beeinflusst werden, zum Beispiel haben Knoten neben Netzwerk-Endpunkten eine geringe basin stability. Daraus folgend wird ein neues Konzept der partiellen basin stability vorgeschlagen, insbesondere für cluster synchronization, um die wechselseitigen Stabilitätseinflüsse von Clustern zu quantifizieren.Dieses Konzept wird auf zwei wichtige reale Beispiele angewandt: Neuronale Netzwerke und das nordeuropäische Stromnetzwerk. Die neue Methode erlaubt es instabile und stabile Cluster in neuronalen Netzwerken zu identifizieren und erklärt wie Netzwerk-Endpunkte die Stabilität gefährden. / Synchronization phenomena are ubiquitous in the natural sciences and engineering, but also in social systems. Among the many models that have been proposed for a description of synchronization, the Kuramoto model is most popular. It describes self-sustained phase oscillators rotating at heterogeneous intrinsic frequencies that are coupled through the sine of their phase differences. The second-order Kuramoto model has been used to investigate power grids, Josephson junctions, and other systems.The study of Kuramoto models on networks has recently been boosted because it is simple enough to allow for a mathematical treatment and yet complex enough to exhibit rich phenomena. In particular, explosive synchronization emerges in scale-free networks in the presence of a correlation between the natural frequencies and the network topology. The first main part of this thesis is devoted to study the networked second-order Kuramoto model in the presence of a correlation between the oscillators'' natural frequencies and the network''s degree. The theoretical framework in the continuum limit and for uncorrelated networks is provided for the model with an asymmetrical natural frequency distribution. It is observed that clusters of nodes with the same degree join the synchronous component successively, starting with small degrees. This novel phenomenon is named cluster explosive synchronization. Moreover, this phenomenon is also influenced by the degree mixing in the network connection as shown numerically. In particular, discontinuous transitions emerge not just in disassortative but also in strong assortative networks, in contrast to the first-order model. Discontinuous phase transitions indicated by the order parameter and hysteresis emerge due to different initial conditions. For very large perturbations, the system could move from a desirable state to an undesirable state. Basin stability was proposed to quantify the stability of a system to stay in the desirable state after being subjected to strong perturbations. In the second main part of this thesis, the basin stability of the synchronization of the second-order Kuramoto model is investigated via perturbing nodes separately. As a novel phenomenon uncovered by basin stability it is demonstrated that two first-order transitions occur successively in complex networks: an onset transition from a global instability to a local stability and a suffusing transition from a local to a global stability. This sequence is called onset and suffusing transition.Different nodes could have a different stability influence from or to other nodes. For example, nodes adjacent to dead ends have a low basin stability. To quantify the stability influence between clusters, in particular for cluster synchronization, a new concept of partial basin stability is proposed. The concept is implemented on two important real examples: neural networks and the northern European power grid. The new concept allows to identify unstable and stable clusters in neural networks and also explains how dead ends undermine the network stability of power grids.

On s’intéresse dans cette thèse à des systèmes couplés de type champ moyen en étudiant l’existence de l’état de synchronisation qui se caractérise par une distance uniformément bornée dans le temps entre chaque paire de composantes d’une solution. L’étude se base sur une méthode perturbative. Néanmoins les résultats obtenus ne sont pas évidents dans le cas non-perturbé. En outre dans le cas où le système couplé est périodique et grâce au Théorème du point fixe on montre l’existence d’une solution périodique sur le tore. L’étude de stabilité et de stabilité exponentielle est établie dans le cas linéaire et appliquée à ce type de systèmes couplés / We study in this thesis a class of a perturbed interconnected mean-field system, also known as a coupled systems. Under some assumptions we prove the existence of an invariant open set by the flow of the perturbed system ; in other word, we prove that the distance between the components of an orbit is uniformly bounded, this property is also called synchronization. We use the perturbation method to obtain the result. However the result is not trivial for the not perturbed system. We use the fixed point theorem to prove the existence of a periodic orbit in the torus. We study in addition the stability and the exponential stability of such systems by studying the stability of a linear systems.

Coupled oscillators, such as groups of fireflies or clusters of neurons, are found throughout nature and are frequently modeled in the applied mathematics literature. Earlier work by Kuramoto, Strogatz, and others has led to a deep understanding of the emergent behavior of systems of such oscillators using traditional dynamical systems methods. In this project we outline the application of techniques from topological data analysis to understanding the dynamics of systems of coupled oscillators. This includes the examination of partitions, partial synchronization, and attractors. By looking for clustering in a data space consisting of the phase change of oscillators over a set of time delays we hope to reconstruct attractors and identify members of these clusters.

This dissertation focuses on the further development of creating accurate numerical models of complex dynamical systems using the holistic discretisation technique [Roberts, Appl. Num. Model., 37:371-396, 2001]. I extend the application from second to fourth order systems and from only one spatial dimension in all previous work to two dimensions (2D). We see that the holistic technique provides useful and accurate numerical discretisations on coarse grids. We explore techniques to model the evolution of spatial patterns governed by pdes such as the Kuramoto-Sivashinsky equation and the real-valued Ginzburg-Landau equation. We aim towards the simulation of fluid flow and convection in three spatial dimensions. I show that significant steps have been taken in this dissertation towards achieving this aim. Holistic discretisation is based upon centre manifold theory [Carr, Applications of centre manifold theory, 1981] so we are assured that the numerical discretisation accurately models the dynamical system and may be constructed systematically. To apply centre manifold theory the domain is divided into elements and using a homotopy in the coupling parameter, subgrid scale fields are constructed consisting of actual solutions of the governing partial differential equation(pde). These subgrid scale fields interact through the introduction of artificial internal boundary conditions. View the centre manifold (macroscale) as the union of all states of the collection of subgrid fields (microscale) over the physical domain. Here we explore how to extend holistic discretisation to the fourth order Kuramoto-Sivashinsky pde. I show that the holistic models give impressive accuracy for reproducing the steady states and time dependent phenomena of the Kuramoto-Sivashinsky equation on coarse grids. The holistic method based on local dynamics compares favourably to the global methods of approximate inertial manifolds. The excellent performance of the holistic models shown here is strong evidence in support of the holistic discretisation technique. For shear dispersion in a 2D channel a one-dimensional numerical approximation is generated directly from the two-dimensional advection-diffusion dynamics. We find that a low order holistic model contains the shear dispersion term of the Taylor model [Taylor, IMA J. Appl. Math., 225:473-477, 1954]. This new approach does not require the assumption of large x scales, formerly absolutely crucial in deriving the Taylor model. I develop holistic discretisation for two spatial dimensions by applying the technique to the real-valued Ginzburg-Landau equation as a representative example of second order pdes. The techniques will apply quite generally to second order reaction-diffusion equations in 2D. This is the first study implementing holistic discretisation in more than one spatial dimension. The previous applications of holistic discretisation have developed algebraic forms of the subgrid field and its evolution. I develop an algorithm for numerical construction of the subgrid field and its evolution for 1D and 2D pdes and explore various alternatives. This new development greatly extends the class of problems that may be discretised by the holistic technique. This is a vital step for the application of the holistic technique to higher spatial dimensions and towards discretising the Navier-Stokes equations.

La actividad cerebral exhibe complejos fenómenos oscilatorios similares a los que se observan en modelos de redes artificiales con osciladores acoplados. Por un lado, estudios sobre la actividad cerebral durante el reposo han demostrado la presencia de fluctuaciones lentas estructuradas y modulaciones de potencia a distintas frecuencias. Simultáneamente, estudios teóricos en el ámbito de la física muestran dinámicas similares usando osciladores acoplados. En este trabajo, por primera vez, se usan modelos de osciladores de fase en redes inspiradas en la arquitectura real del cerebro. Los resultados muestran la aparición espontánea de una dinámica similar a la observada experimentalmente. Además, esta correspondencia es comparable cuantitativamente con datos de neuroimagen, lo que sugiere procesos generales de integración subyacentes a la cognición. Por otra parte, se propone que la actividad cerebral alterada observada en algunas enfermedades psiquiátricas podría tener su origen en desconexiones estructurales que afectarían el comportamiento cooperativo de regiones corticales. / Neural activity in the brain exhibits complex oscillatory phenomena that can be compared with the ones observed in artificial network models of coupled oscillators. In particular, neuroimaging studies of brain activity during rest have reported slow spatiotemporally organized fluctuations and correlated band-limited power modulations. Simultaneously, theoretical works on the area of physics have reported similar dynamic behaviours using simple models of coupled oscillators with intermittent modular synchronization. In this work, for the first time, we use models of phase oscillators in networks inspired in the brain’s wiring architecture. Results show the spontaneous emergence of a dynamics similar to the one observed experimentally. In addition, this correspondence is quantitatively comparable to neuroimaging data, which is suggestive of general integrative processes underlying cognition. Furthermore, we propose that altered brain activity observed in some psychiatric diseases might originate from structural disconnections, which affect the cooperative behaviour of coupled cortical regions.

The PhD thesis considers four topics in dynamical systems and is based on one paper and three manuscripts. In Paper I we apply methods of interval analysis in order to compute the rigorous enclosure of rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points which is used to check rationality of the rotation number. In Manuscript II we provide a numerical algorithm for computing critical points of the multiplier map for the quadratic family (i.e., points where the derivative of the multiplier with respect to the complex parameter vanishes). Manuscript III concerns continued fractions of quadratic irrationals. We show that the generating function corresponding to the sequence of denominators of the best rational approximants of a quadratic irrational is a rational function with integer coefficients. As a corollary we can compute the Lévy constant of any quadratic irrational explicitly in terms of its partial quotients. Finally, in Manuscript IV we develop a method for finding rigorous enclosures of all odd periodic solutions of the stationary Kuramoto-Sivashinsky equation. The problem is reduced to a bounded, finite-dimensional constraint satisfaction problem whose solution gives the desired information about the original problem. Developed approach allows us to exclude the regions in L2, where no solution can exist.

Dans cette thèse, nous étudions le modèle de synchronisation de Kuramoto et plus généralement des systèmes de diffusions interagissant en champ moyen, en présence d'un aléa supplémentaire appelé désordre. La motivation principale en est l'étude du comportement du système en grande population, pour une réalisation fixée du désordre (modèle quenched). Ce document, outre l'introduction, comporte quatre chapitres. Le premier s'intéresse à la convergence de la mesure empirique du système d'oscillateurs vers une mesure déterministe, solution d'un système d'équations aux dérivées partielles non linéaires couplées (équation de McKean-Vlasov). Cette convergence est prouvée indirectement via un principe de grandes déviations dans le cas averaged et directement dans le cas quenched, sous des hypothèses plus faibles sur le désordre. Le deuxième chapitre est issu d'un travail en commun avec Giambattista Giacomin et Christophe Poquet et concerne la régularité des solutions de l'EDP limite ainsi que la stabilité de ses solutions stationnaires synchronisées dans le cas d'un désordre faible. Les deux derniers chapitres étudient l'influence du désordre sur une population d'oscillateurs de taille finie et illustrent des problématiques observées dans la littérature physique. Nous prouvons dans le troisième chapitre un théorème central limite quenched associé à la loi des grands nombres précédente: on montre que le processus de fluctuations quenched converge, en un sens faible, vers la solution d'une EDPS linéaire. Le dernier chapitre étudie le comportement en temps long de cette EDPS, illustrant le fait que les fluctuations dans le modèle de Kuramoto ne sont pas auto-moyennantes.

When statistical assumptions do not hold and coherent structures are present in spatially extended systems such as fluid flows, flame fronts and field theories, a dynamical description of turbulent phenomena becomes necessary. In the dynamical systems approach, theory of turbulence for a given system, with given boundary conditions, is given by (a) the geometry of its infinite-dimensional state space and (b) the associated measure, that is, the likelihood that asymptotic dynamics visits a given state space region. In this thesis this vision is pursued in the context of Kuramoto-Sivashinsky system, one of the simplest physically interesting spatially extended nonlinear systems. With periodic boundary conditions, continuous translational symmetry endows state space with additional structure that often dictates the type of observed solutions. At the same time, the notion of recurrence becomes relative: asymptotic dynamics visits the neighborhood of any equivalent, translated point, infinitely often. Identification of points related by the symmetry group action, termed symmetry reduction, although conceptually simple as the group action is linear, is hard to implement in practice, yet it leads to dramatic simplification of dynamics. Here we propose a scheme, based on the method of moving frames of Cartan, to efficiently project solutions of high-dimensional truncations of partial differential equations computed in the original space to a reduced state space. The procedure simplifies the visualization of high-dimensional flows and provides new insight into the role the unstable manifolds of equilibria and traveling waves play in organizing Kuramoto-Sivashinsky flow. This in turn elucidates the mechanism that creates unstable modulated traveling waves (periodic orbits in reduced space) that provide a skeleton of the dynamics. The compact description of dynamics thus achieved sets the stage for reduction of the dynamics to mappings between a set of Poincare sections.