Riddling : Chimera’s dilemma

We wish to acknowledge the support: Sao Paulo Research Foundation (FAPESP) under Grants 2011/19296-1, 2015/05186-0, 2015/07311-7, 2015/50122-0, and 2017/20920-8, Conselho Nacional de Desenvolvimento Cientıfico e Tecnologico (CNPq), and Coordena¸cao de Aperfei¸coamento de Pessoal de Nıvel Superior (CAPES).Peer reviewedPublisher PD

Anticipation of Oligocene's climate heartbeat by simplified eigenvalue estimation

The Eocene-Oligocene transition marks a watershed point of earth's climate history. The climate shifts from a greenhouse state to an icehouse state in which Antarctica glaciated for the first time and periodic dynamics arise which are still relevant for our current climate. We analyse a $CaCO_3$ concentration time series which covers the Eocene-Oligocene transition and which is obtained from a Pacific sediment core at site DSDP1218. Therefore, we introduce a simplified autoregression-based variant of the dominant eigenvalue (DEV) estimation procedure. The DEV works as leading indicator of bifurcation-induced transitions and enables us to identify the bifurcation type. We confirm its reliability in a methodological study and demonstrate the crucial importance of proper detrending to obtain unbiased results. As a remark, we discuss also possible pathways to estimate the stability of limit cycles based on the DEV and the alternative drift slope as a proof of principle. Finally, we present the DEV analysis results of the $CaCO_3$ concentration time series which are reproducible in a wide parameter range. Our findings demonstrate that the onset of Oligocene's periodic dynamics might be announced by a Neimark-Sacker/Hopf bifurcation in course of the Eocene-Oligocene transition 34 mya. (We follow the convention and use mya$\widehat{=}$"million years ago" and Ma$\widehat{=}$"million years" throughout the article.)Comment: 14 pages, 6 figures. Appendix included with 14 pages, 13 figures and 2 tables. Total pages: 31. Data and code available onlin

Mathematical modeling with applications in biological systems, physiology, and neuroscience

Doctor of PhilosophyDepartment of MathematicsBacim AlaliDynamical systems modeling is used to describe different biological and physical systems as well as to predict the interactions between multiple components of a system over time. A dynamical system describes the evolution of a given system over time using a set of mathematical laws, typically described by differential equations. There are two main methods to model the dynamical behaviors of a system: continuous time modeling and discrete-time modeling. When the time between two measurements is negligible, the continuous time modeling governs the evolution of the system, however, when there is a gap between any two consecutive measurements, discrete-time system modeling comes into play. Differential equations are used to model continuous systems and iterated maps represent the generations in discrete-time systems. In this dissertation, we study some dynamical systems and present their applications to different problems in biological systems, physiology, and neuroscience. In chapter one, we study the local dynamics of some interesting systems and show the local stable behavior of the system around its critical points. Moreover, we investigate the local dynamical behavior of different systems including the Hénon-Heiles system, the Duffing oscillator, and the Van der Pol equation. Furthermore, we discuss about the chaotic behavior of Hamiltonian systems using two different and new examples. In chapter two, we consider some models in computational neuroscience. Due to the complexity of nerve systems, linear modeling methods are not sufficient to understand the various phenomena in neuroscience. We use nonlinear methods and models, which aim at capturing certain properties of the neurons and their complex dynamics. Specifically, we explore the interesting phenomenon of firing spikes and complex dynamics of the Morris-Lecar model. We consider a set of parameters such that the model exhibits a wide range of phenomenon. We investigate the influences of injected current and temperature on the spiking dynamics of Morris-Lecar model. In addition, we study bifurcations, and computational properties of this neuron model. Moreover, we provide a bound for the membrane potential and a certain voltage value or threshold for firing the spikes. Studying the two co-dimension bifurcations demonstrates more complicated behaviors for this single neuron model. Furthermore, we describe the phenomenon of neural bursting and investigate the dynamics of Morris-Lecar model as a square-wave burster, elliptic burster and parabolic burster. Pharmacokinetic models are mathematical models, which provide insights into the interaction of chemicals with certain biological processes. In chapter three, we consider the process of drug and nanoparticle (NPs) distribution throughout the body. We use a tricompartmental model to study the perfusion of NPs in tissues and a six-compartmental model to study drug distribution in different body organs. We perform global sensitivity analysis by LHS Monte Carlo method using Partial Rank Correlation Coefficient (PRCC). We identify the key parameters that contribute most significantly to the absorption and distribution of drugs and NPs in different organs in the body. In chapter four, we study two infectious disease models and use nonlinear optimization and optimal control theory to help in identifying strategies for transmission control and forecasting the spread of infectious diseases. We analyze the effect of vaccination on the disease transmission in these models. Moreover, we perform global sensitivity analysis to investigate the key parameters in these models. In chapter five, we investigate the complex dynamics of two-species Ricker-type discrete-time competitive model. We perform local stability analysis for the fixed points of the system and discuss about its persistence for boundary fixed points. This system inherits properties of the dynamics of a one-dimensional Ricker model such as the cascade of period-doubling bifurcation, periodic windows, and chaos. We explore the existence of chaos for the equilibrium points for a specific case of this system using Marotto theorem and show the existence of snap-back repeller. In chapter six, we study the problem of chaos synchronization in certain discrete-time dynamical systems. We introduce a drive-response discrete-time dynamical system, which is coupled using convex link function. We investigate a synchronization threshold, after which, the drive-response system uncouples and loses its synchronized behaviors. We apply this method to the synchronized cycles of the Ricker model and show that this model displays a rich cascade of complex dynamics from a stable fixed point and cascade of period-doubling bifurcation to chaos. We numerically verify the effectiveness of the proposed scheme and demonstrate how this type of coupling affects the synchronization of the system. In chapter seven, we study the synchronized cycles of a generalized Nicholson-Bailey model. This model demonstrates a rich cascade of complex dynamics from a stable fixed point to periodic orbits, quasi periodic orbits and chaos. We introduce a coupling of these two chaotic systems with different initial conditions and show how they synchronize over a short time. We investigate the qualitative behavior of Generalized Nicholson-Bailey model and its synchronized model using time series analysis and its long-time dynamics by using its bifurcation diagram

Bifurcation and Chaos in Fractional-Order Systems

This book presents a collection of seven technical papers on fractional-order complex systems, especially chaotic systems with hidden attractors and symmetries, in the research front of the field, which will be beneficial for scientific researchers, graduate students, and technical professionals to study and apply. It is also suitable for teaching lectures and for seminars to use as a reference on related topics

Nonlinear Time Series Analysis of BL Lac Light Curves

In this work the time properties of the BL Lac objects are elaborated in a detailed manner through linear and nonlinear time series analysis methods. In spite of the large amount of available data in the last 20 years, the variability studies have not provided major progress for understanding the behavior of these objects. Vague time series analysis methods, lacking any mathematical foundation, are usually invoked revealing erroneous time properties in the data sets which then act misleadingly for modeling the dynamics of the system under study. The flaws of some of the current time series analysis methods are reviewed thoroughly throughout this work for specific sources (Mrk 421, Mrk 501) and the need of employing higher order time series analysis methods is demonstrated. An extensive description of the modern nonlinear analysis methods is presented together with examples being implemented in a way to be applicable to astronomical time series. Then, these methods are applied to the X-ray data set of Mrk 421, obtained by RXTE, covering a time period of 9 years, giving some hints to answer the question: Is it possible to explain the variability behavior of these sources based on few physical parameters (deterministic system), or is it the result of numerous components yielding from a stochastic system? Finally the results from the longest multiwavelength campaign, conducted during August--September 2004, for the BL Lac object PKS 2155-304 are presented. The source was observed in the very high energy gamma-rays (>100 GeV) by H.E.S.S., in the X-rays (2-10 keV) by RXTE and in the optical (R-band) by three terrestrial observatories

Distance-based analysis of dynamical systems and time series by optimal transport

The concept of distance is a fundamental notion that forms a basis for the orientation in space. It is related to the scientific measurement process: quantitative measurements result in numerical values, and these can be immediately translated into distances. Vice versa, a set of mutual distances defines an abstract Euclidean space. Each system is thereby represented as a point, whose Euclidean distances approximate the original distances as close as possible. If the original distance measures interesting properties, these can be found back as interesting patterns in this space. This idea is applied to complex systems: The act of breathing, the structure and activity of the brain, and dynamical systems and time series in general. In all these situations, optimal transportation distances are used; these measure how much work is needed to transform one probability distribution into another. The reconstructed Euclidean space then permits to apply multivariate statistical methods. In particular, canonical discriminant analysis makes it possible to distinguish between distinct classes of systems, e.g., between healthy and diseased lungs. This offers new diagnostic perspectives in the assessment of lung and brain diseases, and also offers a new approach to numerical bifurcation analysis and to quantify synchronization in dynamical systems.LEI Universiteit LeidenNWO Computational Life Sciences, grant no. 635.100.006Analyse en stochastie

Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue

Physics-based Machine Learning Approaches to Complex Systems and Climate Analysis

Komplexe Systeme wie das Klima der Erde bestehen aus vielen Komponenten, die durch eine komplizierte Kopplungsstruktur miteinander verbunden sind. Für die Analyse solcher Systeme erscheint es daher naheliegend, Methoden aus der Netzwerktheorie, der Theorie dynamischer Systeme und dem maschinellen Lernen zusammenzubringen. Durch die Kombination verschiedener Konzepte aus diesen Bereichen werden in dieser Arbeit drei neuartige Ansätze zur Untersuchung komplexer Systeme betrachtet. Im ersten Teil wird eine Methode zur Konstruktion komplexer Netzwerke vorgestellt, die in der Lage ist, Windpfade des südamerikanischen Monsunsystems zu identifizieren. Diese Analyse weist u.a. auf den Einfluss der Rossby-Wellenzüge auf das Monsunsystem hin. Dies wird weiter untersucht, indem gezeigt wird, dass der Niederschlag mit den Rossby-Wellen phasenkohärent ist. So zeigt der erste Teil dieser Arbeit, wie komplexe Netzwerke verwendet werden können, um räumlich-zeitliche Variabilitätsmuster zu identifizieren, die dann mit Methoden der nichtlinearen Dynamik weiter analysiert werden können. Die meisten komplexen Systeme weisen eine große Anzahl von möglichen asymptotischen Zuständen auf. Um solche Zustände zu beschreiben, wird im zweiten Teil die Monte Carlo Basin Bifurcation Analyse (MCBB), eine neuartige numerische Methode, vorgestellt. Angesiedelt zwischen der klassischen Analyse mit Ordnungsparametern und einer gründlicheren, detaillierteren Bifurkationsanalyse, kombiniert MCBB Zufallsstichproben mit Clustering, um die verschiedenen Zustände und ihre Einzugsgebiete zu identifizieren. Bei von Vorhersagen von komplexen Systemen ist es nicht immer einfach, wie Vorwissen in datengetriebenen Methoden integriert werden kann. Eine Möglichkeit hierzu ist die Verwendung von Neuronalen Partiellen Differentialgleichungen. Hier wird im letzten Teil der Arbeit gezeigt, wie hochdimensionale räumlich-zeitlich chaotische Systeme mit einem solchen Ansatz modelliert und vorhergesagt werden können.Complex systems such as the Earth's climate are comprised of many constituents that are interlinked through an intricate coupling structure. For the analysis of such systems it therefore seems natural to bring together methods from network theory, dynamical systems theory and machine learning. By combining different concepts from these fields three novel approaches for the study of complex systems are considered throughout this thesis. In the first part, a novel complex network construction method is introduced that is able to identify the most important wind paths of the South American Monsoon system. Aside from the importance of cross-equatorial flows, this analysis points to the impact Rossby Wave trains have both on the precipitation and low-level circulation. This connection is then further explored by showing that the precipitation is phase coherent to the Rossby Wave. As such, the first part of this thesis demonstrates how complex networks can be used to identify spatiotemporal variability patterns within large amounts of data, that are then further analysed with methods from nonlinear dynamics. Most complex systems exhibit a large number of possible asymptotic states. To investigate and track such states, Monte Carlo Basin Bifurcation analysis (MCBB), a novel numerical method is introduced in the second part. Situated between the classical analysis with macroscopic order parameters and a more thorough, detailed bifurcation analysis, MCBB combines random sampling with clustering methods to identify and characterise the different asymptotic states and their basins of attraction. Forecasts of complex system are the next logical step. When doing so, it is not always straightforward how prior knowledge in data-driven methods. One possibility to do is by using Neural Partial Differential Equations. Here, it is demonstrated how high-dimensional spatiotemporally chaotic systems can be modelled and predicted with such an approach in the last part of the thesis