In this article we investigate a pitchfork bifurcation of the local attractor of a simple capsizing model proposed by Thompson. Although this is a very simple system it has a very complicate dynamic. We try to reveal some properties of this dynamic with modern numerical methods. For this reason we approximate stable and unstable manifolds which connect the steady states to obtain a complete understanding of the topology in the phase space. We also consider approximations of the Lyapunov Exponents (resp. Floquet Exponents) which indicates the pitchfork bifurcation.

Based on the generating matrix to the Tribonacci sequence, the Tribonacci cat map is a discrete chaotic dynamical system, similar to Arnold's discrete cat map, but on three dimensional space. In this thesis, this new mapping is introduced and the properties of its matrix are presented. The main results of the investigation prove how the size of the domain of the map affects its period and explore the orbit lengths of non-trivial points. Different upper bounds to the map are studied and proved, and a conjecture based on numerical calculations is proposed. The Tribonacci cat map is used for applications such as 3D image encryption and colour encryption. In the latter case, the results provided by the mapping are compared to those from a generalised form of the map.

Afin de prévoir les rendements ou réduire la consommation d’intrants nous pouvons, en exploitant les données expérimentales, créer des modèles mathématiques afin de simuler la croissance des cultures en fonction des caractéristiques de l’environnement. Dans cette optique, cette thèse s’intéresse particulièrement aux modèles dits ”mécanistes”.Des premières tentatives, dans les années 70, à nos jours, il y a eu pléthore de nouveaux modèles créés, à différentes échelles, afin d’étudier certains phénomènes dans les cultures ou au sein des plantes. On peut par exemple citer : CERES, STICS, APSIM, LNAS pour les modèles dits de culture ou LIGNUM, ADEL, GreenLab, MAppleT, pour les modèles dits structure-fonction.Ces modèles nécessitent d’être créés et évalués en conduisant une analyse rigoureuse possédant de nombreuses étapes et dont chacune est composée de plusieurs algorithmes complexes. Cette étude devrait s’inscrire dans une démarche dite de bonnes pratiques de modélisation, ”Good Modelling Practices”. On peut citer comme fonctionnalités : l’analyse de sensibilité, l’estimation paramétrique, l’analyse d’incertitude, l’assimilation de données, la sélection de modèles, le contrôle optimal ... En fonction de la configuration du cas, chacune de ces fonctionnalités peut faire appel à un grand nombre d’algorithmes avec chacun des caractéristiques propres. On retrouve dans l’état de l’art des plateformes qui s’occupent souvent d’une fonctionnalité mais très rarement qui s’attaquent à l’ensemble de la chaîne de travail.Cette thèse propose une formalisation des modèles dynamiques stochastiques (cadre adapté à la modélisation des plantes), de méthodes et algorithmes statistiques dédiés à leur étude et de l’interfaçage entre les modèles et les algorithmes dans cette chaîne de travail. Nous en déduisons la conception d’un système informatique (ou plateforme logicielle) permettant d’aider les modélisateurs, ou plutôt les équipes de modélisation tant l’activité est complexe et transverse, afin de créer et valider des modèles agronomiques par le truchement d’un langage dédié et d’outils statistiques associés. Le système facilite ainsi l’écriture des modèles, leur analyse de sensibilité, leur identification paramétrique et leur évaluation à partir de données expérimentales, leur optimisation. Notre domaine d’étude est au coeur de ”l’agronomie quantitative”, qui combine à la fois agronomie, modélisation, statistiques et informatique. Nous décrirons les types de modèles mathématiques pris en compte et comment nous les traduisons sur machine afin de permettre des simulations. Puis nous passerons en revue le flux de travail général ainsi que les algorithmes utilisés afin de montrer la conduite générale des études qui sont désormais plus facilement et rapidement faisables. Ce flux sera testé sur plusieurs cas d’étude, en particulier pour les modèles LNAS et STICS. Finalement, nous ouvrirons sur la possibilité d’injecter ces études dans une base de connaissance générale, ou ontologie, avec un langage dédié avant de conclure sur les perspectives du travail développé pour la communauté et notamment celles en termes de plateformes à destination des modélisateurs en général et des utilisateurs des modèles agronomiques en particulier. / In agriculture, in order to predict crop yield or to reduce inputs, mathematical models of plant growth open new perspectives by simulating crop growth in interaction with the environment. In this thesis we will particularly focus on ”mechanistic” models based on the description of ecophysiological and archictectural processes in plants.Since the first attempts, in the seventies, the scientific community has created a large number of models with va- rious objectives : for instance, CERES, STICS, APSIM, LNAS as crop models and LIGNUM, ADEL, GreenLab, MAppleT as functional-structural models.These models have to be designed and evaluated with a rigourous process in several steps, according to what is usually described as ”good modelling practices”. The methods involved in the different steps are : sensitivity and uncertainty analysis, parameter estimation, model selection, data assimilation, optimal control ... According to the configuration of the study case, various algorithms can be used at each of these steps. The state-of-the-art software systems generally focus on one aspect of the global workflow, but very few focus on the workflow itself and propose the whole chain of mathematical methodologies adapted to the type of models and configurations faced in plant growth modelling : stochastic and nonlinear dynamical models involving a lot of processes and parameters, heterogeneous and irregular system observations.This thesis considers the formalization of stochastic dynamical models, of statistical methods and algorithms dedicated to their study and of the interface between models and algorithms to generate the analysis workflow. We deduce the conception of a software platform which allows modelers (or more exactly modelling teams, since the activity is quite complex) to create and validate crop/plant models by using a single language and dedicated statistical tools. Our system facilitates model design, sensitivity and uncertainty analysis, parameter estimation and evaluation from experimental data and optimization.Our research is at the heart of ”quantitative agronomy” which combines agronomy, modeling, statistics and computer science. We describe and formalize the type of models faced in agronomy and plant sciences and how we simulate them. We detail the good modelling practices workflow and which algorithms are available at all steps. Thanks to this formalization and tools, model studies can be conducted in an easier and more efficient way. It is illustrated on several test cases, particularly for the LNAS and STICS models. Based on this conception and results, we also discuss the possibility to deduce an ontology and a domain-specific language in order to improve the communication between experts. Finally, we conclude about the perspectives in terms of community platforms, first generally for modellers, and second more specifically in quantitative agronomy.

In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We substantiate that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.:1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Preliminaries and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Preliminaries related to topology and measure theory . . . . . . . . 4 2.2 Basic notions from category theory . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Classical dynamical systems theory . . . . . . . . . . . . . . . . . . . . . . 23 3 Dynamical Systems in Abstract Categories . . . . . . . . . . . . . . . . . . 30 3.1 Monoids and monoid actions in abstract categories . . . . . . . . . . 31 3.2 Abstract dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Nonautonomous dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Dynamical Systems as Algebras and Coalgebras . . . . . . . . . . . . . .38 4.1 From monoids to monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 From abstract dynamical systems to monadic algebras . . . . . . . 48 4.3 Connections to coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4 Exponential objects in Top for locally compact Hausdorff spaces . . 52 4.5 (Co)Monadic (co)algebras and adjoint functors . . . . . . . . . . . . . .56

In this dissertation, we address the problem of understanding human activities in videos by developing a two-pronged approach: coarse level modeling of scene activities and fine level modeling of individual activities. At the coarse level, where the resolution of the video is low, we rely on person tracks. At the fine level, richer features are available to identify different parts of the human body, therefore we rely on the body joint tracks. There are three main goals of this dissertation: (1) identify unusual activities at the coarse level, (2) recognize different activities at the fine level, and (3) predict the behavior for synthesizing and tracking activities at the fine level. The first goal is addressed by modeling activities at the coarse level through two novel and complementing approaches. The first approach learns the behavior of individuals by capturing the patterns of motion and size of objects in a compact model. Probability density function (pdf) at each pixel is modeled as a multivariate Gaussian Mixture Model (GMM), which is learnt using unsupervised expectation maximization (EM). In contrast, the second approach learns the interaction of object pairs concurrently present in the scene. This can be useful in detecting more complex activities than those modeled by the first approach. We use a 14-dimensional Kernel Density Estimation (KDE) that captures motion and size of concurrently tracked objects. The proposed models have been successfully used to automatically detect activities like unusual person drop-off and pickup, jaywalking, etc. The second and third goals of modeling human activities at the fine level are addressed by employing concepts from theory of chaos and non-linear dynamical systems. We show that the proposed model is useful for recognition and prediction of the underlying dynamics of human activities. We treat the trajectories of human body joints as the observed time series generated from an underlying dynamical system. The observed data is used to reconstruct a phase (or state) space of appropriate dimension by employing the delay-embedding technique. This transformation is performed without assuming an exact model of the underlying dynamics and provides a characteristic representation that will prove to be vital for recognition and prediction tasks. For recognition, properties of phase space are captured in terms of dynamical and metric invariants, which include the Lyapunov exponent, correlation integral, and correlation dimension. A composite feature vector containing these invariants represents the action and will be used for classification. For prediction, kernel regression is used in the phase space to compute predictions with a specified initial condition. This approach has the advantage of modeling dynamics without making any assumptions about the exact form (polynomial, radial basis, etc.) of the mapping function. We demonstrate the utility of these predictions for human activity synthesis and tracking.

In this thesis, we develop and investigate numerical methods for solving nonsmooth convex optimization problems in real Hilbert spaces. We construct algorithms, such that they handle the terms in the objective function and constraints of the minimization problems separately, which makes these methods simpler to compute. In the first part of the thesis, we extend the well known AMA method from Tseng to the Proximal AMA algorithm by introducing variable metrics in the subproblems of the primal-dual algorithm. For a special choice of metrics, the subproblems become proximal steps. Thus, for objectives in a lot of important applications, such as signal and image processing, machine learning or statistics, the iteration process consists of expressions in closed form that are easy to calculate. In the further course of the thesis, we intensify the investigation on this algorithm by considering and studying a dynamical system. Through explicit time discretization of this system, we obtain Proximal AMA. We show the existence and uniqueness of strong global solutions of the dynamical system and prove that its trajectories converge to the primal-dual solution of the considered optimization problem. In the last part of this thesis, we minimize a sum of finitely many nonsmooth convex functions (each can be composed by a linear operator) over a nonempty, closed and convex set by smoothing these functions. We consider a stochastic algorithm in which we take gradient steps of the smoothed functions (which are proximal steps if we smooth by Moreau envelope), and use a mirror map to 'mirror'' the iterates onto the feasible set. In applications, we compare them to similar methods and discuss the advantages and practical usability of these new algorithms.

Numerous prediction models of SARS-CoV-2 pandemic were proposed in the past. Unknown parameters of these models are often estimated based on observational data. However, lag in case-reporting, changing testing policy or incompleteness of data lead to biased estimates. Moreover, parametrization is time-dependent due to changing age-structures, emerging virus variants, non-pharmaceutical interventions, and vaccination programs. To cover these aspects, we propose a principled approach to parametrize a SIR-type epidemiologic model by embedding it as a hidden layer into an input-output non-linear dynamical system (IO-NLDS). Observable data are coupled to hidden states of the model by appropriate data models considering possible biases of the data. This includes data issues such as known delays or biases in reporting. We estimate model parameters including their time-dependence by a Bayesian knowledge synthesis process considering parameter ranges derived from external studies as prior information. We applied this approach on a specific SIR-type model and data of Germany and Saxony demonstrating good prediction performances. Our approach can estimate and compare the relative effectiveness of non-pharmaceutical interventions and provide scenarios of the future course of the epidemic under specified conditions. It can be translated to other data sets, i.e., other countries and other SIR-type models.

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<p dir="ltr">The kinetic-based lattice Boltzmann method (LBM) is a specialized computational fluid dynamics (CFD) technique that resolves intricate flow phenomena at the mesoscale level. The LBM is particularly suited for large-scale parallel computing on Graphic Processing Units (GPU) and simulating multi-phase flows. By incorporating a volume fraction parameter, LBM becomes a volumetric lattice Boltzmann method (VLBM), leading to advantages such as easy handling of complex geometries with/without movement. These capabilities render VLBM an effective tool for modeling various complex flows. In this study, we investigated the computational modeling of complex flows using VLBM, focusing particularly on pulsatile flows, the transition to turbulent flows, and pore-scale porous media flows. Furthermore, a new discrete dynamical system (DDS) is derived and validated for potential integration into large eddy simulations (LES) aimed at enhancing modeling for turbulent and pulsatile flows. Pulsatile flows are prevalent in nature, engineering, and the human body. Understanding these flows is crucial in research areas such as biomedical engineering and cardiovascular studies. However, the characteristics of oscillatory, variability in Reynolds number (Re), and shear stress bring difficulties in the numerical modeling of pulsatile flows. To analyze and understand the shear stress variability in pulsatile flows, we first developed a unique computational method using VLBM to quantify four-dimensional (4-D) wall stresses in image-based pulsatile flows. The method is validated against analytical solutions and experimental data, showing good agreement. Additionally, an application study is presented for the non-invasive quantification of 4-D hemodynamics in human carotid and vertebral arteries. Secondly, the transition to turbulent flows is studied as it plays an important role in the understanding of pulsatile flows since the flow can shift from laminar to transient and then to turbulent within a single flow cycle. We conducted direct numerical simulations (DNS) using VLBM in a three-dimensional (3-D) pipe and investigated the flow at Re ranging from 226 to 14066 in the Lagrangian description. Results demonstrate good agreement with analytical solutions for laminar flows and with open data for turbulent flows. Key observations include the disappearance of parabolic velocity profiles when Re>2300, the fluctuation of turbulent kinetic energy (TKE) between laminar and turbulent states within the range 2300</p>

This thesis focuses on analyzing the synchronization of time delays between mutually coupled phase-locked loops (PLLs) in a flat hierarchy. Mutual synchronization refers to decentralized synchronization where there is no primary or secondary unit or control source. Consequently, it is an inherently self-organizing system in which each unit has equal rights, making it a democratic system. In this research, a dynamic nonlinear time-domain model is used to describe mutually delayed coupled oscillators. The predictions of this model are evaluated against experimental measurements. The time-domain model is based on the Kuramoto model. The Kuramoto model describes a network of coupled oscillators. As a first impression, this Kuramoto model is first analyzed for understanding of the effects of time delays between oscillators. The time domain model is based on a conventional PLL architecture modified to allow mutual coupling. The modifications include a circuit section that sums and weights all incoming phase differences. Overall, the measured results of this research study are in good agreement with the theoretical predictions of the time-domain model. The analysis allows the identification of the transient dynamics and the mechanisms that lead to mutual coupling and the formation of synchronized states through self-organized synchronization. The results show that the mutual coupling can self-organize its dynamics to synchronize even at time delays where the phenomenon of multistability of synchronized states occurs. A critical time delay beyond which a stable synchronized state cannot be achieved has been identified. The work also analyzes the dynamics and noise of synchronized states and finds that the dynamics near a synchronized state are correlated due to mutual coupling, leading to a reduction in noise. The noise correlation is affected by the direction of coupling, the number of nodes in the network, and the network topology. An improvement in phase noise of up to 14.42 dBc/Hz at 100 kHz offset from the carrier and 49.47ns delay was achieved using all-to-all coupling with four nodes. Furthermore, the hybrid approach, the combination of hierarchical and self-organizing synchronization architectures, is investigated. The dissertation presents an experimental study to understand how this affects a network of mutually delayed delay-coupled oscillators and whether the network of mutually coupled nodes can be abstracted as a secondary oscillator. A range in which the mutually delay-coupled network can be successfully synchronized by an external reference oscillator, depending on the synchronized state, is identified. In summary, this thesis provides valuable insights into the properties of mutually delay-coupled PLLs and their synchronization in flat hierarchies, and contributes to the understanding, design, and optimization of more practical networks of mutually delayed PLLs.:Abstract/Zusammenfassung Symbols and Abbreviations Previous Publications 1 Introduction 1.1 Classifications of Synchronization 1.2 A Historical Perspective on Mutual Synchronization 1.3 Extending the Understanding of Mutual Synchronization 1.4 Definitions and Methodologies 2 Model of Networks of Mutually Coupled PLLs 2.1 Coupled Oscillators – Kuramoto Model 2.1.1 Consequences of a Time Delay between Oscillators 2.1.2 Arbitrary Time Delays between Oscillators 2.2 Time-Domain Model of Delay-Coupled PLLs 2.2.1 Phase Detection 2.2.2 Loop-Filter 2.2.3 Voltage Controlled Oscillator 2.3 Prediction and Stability Analysis of Synchronized States 2.3.1 Assessing the Linear Stability of Synchronized States 2.3.2 Stability Consideration for Two Identical PLL Nodes 2.3.3 The Notion of Mode Locking 2.3.4 Effects of Heterogeneity on Synchronized States 2.4 Dynamics and Noise in Synchronized States 2.4.1 Gain and Phase Margin of a PLL Node 2.4.2 Phase Noise 2.5 Key Findings of the Theoretical Model 3 Design of Phase-Locked-Loops for Mutual Synchronization 3.1 PLL Nodes Dedicated for Mutual Synchronization 3.1.1 Phase Detection Circuitry 3.1.2 Adder Chain 3.2 Additional Circuitry for Implementing a Time Delay 4 Experimental Analysis of Mutually Time-Delayed Coupled PLLs 4.1 Synchronized States Including Oscillator Nonlinearity 4.2 Stability of Multistable Synchronized States 4.3 Critical Time Delay Between Two Coupled Nodes 4.4 Combining Hierarchical and Flat Synchronization Concepts 4.4.1 Entrainment of a Chain Network Topology 4.4.2 Entrainment of a Ring Network Topology 4.5 Heterogeneous Time Delays between Coupled PLLs 4.6 Phase Noise Analysis of Time Delay Coupled PLLs 4.6.1 Phase Noise for Two Mutually Coupled Nodes 4.6.2 The Impact of Coupling Directionality 4.6.3 Long Term Frequency Stability 4.6.4 Effect of Time Delay on Phase Noise 4.6.5 Network Topology Dependency on Phase Noise 5 Conclusion and Future Prospects Bibliography Own Publications – Periodicals Own Publications – Conference Proceedings Co-Authored Publications Other References List of Figures List of Tables Acknowledgement Curriculum Vitae / Diese Arbeit befasst sich mit der Analyse der Auswirkungen von Zeitverzögerungen auf die Synchronisation von gegenseitig gekoppelten Phasenregelschleifen (engl. phase-locked loop (PLL)) in einer flachen Hierarchie. Gegenseitige Synchronisation bezieht sich auf eine dezentrale Synchronisation, bei der es keine primäre oder sekundäre Einheit oder Steuerquelle gibt. Folglich ist es ein inhärent selbstorganisierendes System, in dem jede Einheit gleichberechtigt ist, was es zu einem demokratischen System macht. Für die Untersuchung wird ein dynamisches nichtlineares Zeitbereichsmodell verwendet, um gegenseitig verzögert gekoppelte Oszillatoren zu modellieren und die Vorhersagen dieses Modells anhand experimenteller Messungen zu bewerten. Dieses Zeitbereichsmodell basiert auf dem sogenannten Kuramoto-Modell, das ein Netzwerk gekoppelter Oszillatoren beschreibt. Um einen ersten Eindruck zu erhalten, wird zunächst dieses Kuramoto-Modell analysiert, um die Auswirkungen von Zeitverzögerungen zwischen den Oszillatoren zu verstehen. Das Zeitbereichsmodell basiert auf einer konventionellen PLL-Architektur, die modifiziert wurde, um eine gegenseitige Kopplung zu ermöglichen. Die Modifikationen beinhalten einen Schaltungsteil, der alle eingehenden Phasendifferenzen summiert und gewichtet. Die gemessenen Ergebnisse dieser Untersuchung stimmen insgesamt gut mit den theoretischen Vorhersagen des Zeitbereichsmodells überein. Die Analyse erlaubt es, die transiente Dynamik und die Mechanismen zu identifizieren, die zur gegenseitigen Synchronisation und zur Bildung synchronisierter Zustände durch selbstorganisierte Synchronisation führen. Die Ergebnisse zeigen, dass selbst bei Zeitverzögerungen, bei denen das Phänomen der Multistabilität synchronisierter Zustände auftritt, die gegenseitige Kopplung ihre Dynamik selbst organisieren kann, um sich zu synchronisieren. Die Untersuchung identifizierte eine kritische Zeitverzögerung, bei der kein stabiler synchronisierter Zustand erreicht werden kann. Die Arbeit analysiert auch die Dynamik und das Rauschen von synchronisierten Zuständen und stellt fest, dass die Dynamik in der Nähe eines synchronisierten Zustands aufgrund der gegenseitigen Kopplung korreliert ist, was zu einer Reduktion des Rauschens führt. Die Richtung der Kopplung und die Anzahl der Knoten im Netzwerk sowie die Netzwerktopologie beeinflussen die Korrelation des Rauschens. Eine Verbesserung des Phasenrauschens von bis zu 14.42 dBc/Hz bei einem Versatz von 100 kHz zum Träger und einer Verzögerung von 49.47 ns wurde durch eine globalen oder All-to-All-Kopplung mit vier Knoten erreicht. Des Weiteren wird der hybride Ansatz, die Kombination von hierarchischen und selbstorganisierenden Synchronisationsarchitekturen, untersucht. Die Arbeit präsentiert eine experimentelle Studie, um zu verstehen, wie dies ein Netzwerk von gegenseitig verzögert gekoppelten Oszillatoren beeinflusst und ob das Netzwerk von gegenseitig gekoppelten Knoten als sekundärer Oszillator abstrahiert werden kann. Dabei wird eine vom synchronisierten Zustand abhängige Domäne identifiziert, in der das wechselseitig gekoppelte Netzwerk durch einen externen Referenzoszillator erfolgreich synchronisiert werden kann. Insgesamt liefert diese wissenschaftliche Arbeit wertvolle Erkenntnisse über die Eigenschaften von gegenseitig verzögerungsgekoppelten PLLs und deren Synchronisation in einer flachen Hierarchie und trägt zum Verständnis, zum Entwurf und zur Optimierung von praktisch realisierten Netzwerken gegenseitig verzögerungsgekoppelter PLLs bei.:Abstract/Zusammenfassung Symbols and Abbreviations Previous Publications 1 Introduction 1.1 Classifications of Synchronization 1.2 A Historical Perspective on Mutual Synchronization 1.3 Extending the Understanding of Mutual Synchronization 1.4 Definitions and Methodologies 2 Model of Networks of Mutually Coupled PLLs 2.1 Coupled Oscillators – Kuramoto Model 2.1.1 Consequences of a Time Delay between Oscillators 2.1.2 Arbitrary Time Delays between Oscillators 2.2 Time-Domain Model of Delay-Coupled PLLs 2.2.1 Phase Detection 2.2.2 Loop-Filter 2.2.3 Voltage Controlled Oscillator 2.3 Prediction and Stability Analysis of Synchronized States 2.3.1 Assessing the Linear Stability of Synchronized States 2.3.2 Stability Consideration for Two Identical PLL Nodes 2.3.3 The Notion of Mode Locking 2.3.4 Effects of Heterogeneity on Synchronized States 2.4 Dynamics and Noise in Synchronized States 2.4.1 Gain and Phase Margin of a PLL Node 2.4.2 Phase Noise 2.5 Key Findings of the Theoretical Model 3 Design of Phase-Locked-Loops for Mutual Synchronization 3.1 PLL Nodes Dedicated for Mutual Synchronization 3.1.1 Phase Detection Circuitry 3.1.2 Adder Chain 3.2 Additional Circuitry for Implementing a Time Delay 4 Experimental Analysis of Mutually Time-Delayed Coupled PLLs 4.1 Synchronized States Including Oscillator Nonlinearity 4.2 Stability of Multistable Synchronized States 4.3 Critical Time Delay Between Two Coupled Nodes 4.4 Combining Hierarchical and Flat Synchronization Concepts 4.4.1 Entrainment of a Chain Network Topology 4.4.2 Entrainment of a Ring Network Topology 4.5 Heterogeneous Time Delays between Coupled PLLs 4.6 Phase Noise Analysis of Time Delay Coupled PLLs 4.6.1 Phase Noise for Two Mutually Coupled Nodes 4.6.2 The Impact of Coupling Directionality 4.6.3 Long Term Frequency Stability 4.6.4 Effect of Time Delay on Phase Noise 4.6.5 Network Topology Dependency on Phase Noise 5 Conclusion and Future Prospects Bibliography Own Publications – Periodicals Own Publications – Conference Proceedings Co-Authored Publications Other References List of Figures List of Tables Acknowledgement Curriculum Vitae