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Ordinary differential equations (ODE) have long been an important tool for modelling and understanding the dynamics of many real systems. However, mathematical modelling in several areas of the life sciences requires the use of time-delayed differential models (DDEs). The time delays in these models refer to the time required for the manifestation of certain hidden processes, such as the time between the onset of cell infection and the production of new viruses (incubation periods), the infection period, or the immune period. Since real biological systems are always subject to perturbations that are not fully understood or cannot be explicitly modeled, stochastic delay differential systems (SDDEs) provide a more realistic approximation to these models. In this work, we study the predator–prey system considering three time-delay models: one deterministic and two types of stochastic models. Our numerical results allow us to distinguish between different asymptotic behaviours depending on whether the system is deterministic or stochastic, and in particular, when considering stochasticity, we see that both the nature of the stochastic systems and the magnitude of the delay play a crucial role in determining the dynamics of the system.

Delays and stochasticity have both served as crucially valuable ingredients in mathematical descriptions of control, physical and biological systems. In this work, we investigate how explicitly dynamical stochasticity in delays modulates the effect of delayed feedback. To do so, we consider a hybrid model where stochastic delays evolve by a continuous-time Markov chain, and between switching events, the system of interest evolves via a deterministic delay equation. Our main contribution is the calculation of an effective delay equation in the fast switching limit. This effective equation maintains the influence of all subsystem delays and cannot be replaced with a single effective delay. To illustrate the relevance of this calculation, we investigate a simple model of stochastically switching delayed feedback motivated by gene regulation. We show that sufficiently fast switching between two oscillatory subsystems can yield stable dynamics.

A delayed term in a differential equation reflects the fact that information takes significant time to travel from one place to another within a process being studied. Despite de apparent similarity with ordinary differential equations, delay-differential equations (DDE) are known to be fundamentally different and to require a dedicate mathematical apparatus for their analysis. Indeed, when the delay is large, it was found that they can sometimes be related to spatially extended dynamical systems. The purpose of this paper is to explain this fact in the simplest possible DDE by way of a multiple-scale analysis. We show the asymptotic correspondence of that linear DDE with the diffusion equation. This partial differential equations arises from a solvability condition that differs from the ones usually encountered in textbooks on asymptotics: In the limit of large delays, the leading-order problem is a map and secular divergence at subsequent orders stem from forcing terms in that map., info:eu-repo/semantics/published

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In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with an arbitrary number of (forward and backward) delays. When the nonlinearities include nonpolynomial terms we introduce auxiliary variables to first rewrite the problem into an equivalent polynomial one. We then apply a flexible fixed point technique in a space of geometrically decaying Fourier coefficients. We showcase the efficacy of this method by proving periodic solutions in the well-known Mackey–Glass delay differential equation for the classical parameter values.

In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with an arbitrary number of (forward and backward) delays. When the nonlinearities include nonpolynomial terms we introduce auxiliary variables to first rewrite the problem into an equivalent polynomial one. We then apply a flexible fixed point technique in a space of geometrically decaying Fourier coefficients. We showcase the efficacy of this method by proving periodic solutions in the well-known Mackey–Glass delay differential equation for the classical parameter values.

This thesis concerns the study of the global dynamics of delay differential equations of the so-called production and destruction type, which find applications to the modelling of several phenomena in areas such as population growth dynamics, economics, cell production, etc. For instance, by applying tools coming from discrete dynamics, we provide sufficient conditions for the existence of globally attracting equilibria for families of scalar or multidimensional equations. Moreover, we extend some known results in the scalar non-autonomous case by the use of integral inequalities. Finally, the existence of periodic solutions is analysed in the general context of infinite delay, impulses and periodic coefficients.

The version of record is available online at: http://dx.doi.org/10.1007/s00285-021-01699-8, Cells and tissues exhibit sustained oscillatory deformations during remodelling, migration or embryogenesis. Although it has been shown that these oscillations correlate with intracellular biochemical signalling, the role of these oscillations is as yet unclear, and whether they may trigger drastic cell reorganisation events or instabilities remains unknown. Here, we present a rheological model that incorporates elastic, viscous and frictional components, and that is able to generate oscillatory response through a delay adaptive process of the rest-length. We analyse its stability as a function of the model parameters and deduce analytical bounds of the stable domain. While increasing values of the delay and remodelling rate render the model unstable, we also show that increasing friction with the substrate destabilises the oscillatory response. This fact was unexpected and still needs to be verified experimentally. Furthermore, we numerically verify that the extension of the model with non-linear deformation measures is able to generate sustained oscillations converging towards a limit cycle. We interpret this sustained regime in terms of non-linear time varying stiffness parameters that alternate between stable and unstable regions of the linear model. We also note that this limit cycle is not present in the linear model. We study the phase diagram and the bifurcations of the non-linear model, based on our conclusions on the linear one. Such dynamic analysis of the delay visco-elastic model in the presence of friction is absent in the literature for both linear and non-linear rheologies. Our work also shows how increasing values of some parameters such as delay and friction decrease its stability, while other parameters such as stiffness stabilise the oscillatory response., JJM and MD have been financially supported by the Spanish Ministry of Science, Innovation and Universities (MICINN) with grant DPI2016-74929-R and by the local government Generalitat de Catalunya with grant 2017 SGR 1278., Peer Reviewed, Postprint (author's final draft)

Oscillatory response of cellular tissues has been observed in multipleembryogenetic developmental stages. The source of these oscillations hasbeen attributed to imbalance of instabilities in the chemo-mechanical sig-nalling and delayed cell activity. Although the relation of these oscillationswith further drastic tissue deformation remains uncertain, it is apparentthat intracellular remodelling events seem to drive the viscoelastic prop-erties and the measured pulsatile deformations.We here resort to a viscoelastic model that is based on a variable rest-length of the cell. We include a delay between the measured elastic strainand the evolution of the rest-length which dynamically adapts to the cellstrain. This law is not only able to reproduce the relaxation phenomenaobserved in embryonic tissues in vitro and in vivo, but also to give riseto oscillatory cell responses. We analyse the stability of the resultingoscillations on minimal systems with two cells, and also on planar or outof plane deformation modes of monolayers. We conclude that in all cases,the stability decreases with an increasing delay or with the ability to adaptin a faster manner., Peer Reviewed, Postprint (author's final draft)

This paper is devoted to analyze the stability of the economy according to an extended version of Kaldor's economic growth model. We consider the role of the government and its simultaneous monetary and scal policies and we study whether or not a time delay between the recognition and the implementation of its scal policy can a ect the economic stability. Numerical simulations provide further conclusions about the long-term behavior of the four variables modeled - namely, national income, capacity of production, bonds value and money supply.

This thesis considers the problem of robust geometric regulation for tracking and disturbance rejection of systems governed by delay differential equations. It is well known that geometric regulation can be highly sensitive to system parameters and hence such designs are not always robust. In particular, when employing numerical approximations to delay systems, the resulting finite dimensional models inherit natural approximation errors that can impact robustness. This demonstrates this lack of robustness and then addresses robustness by employing versions of robust regulation that have been developed for infinite dimensional systems. Numerical examples are given to illustrate the ideas and to test the robustness of the regulator.

Sufficient conditions for the existence of at least one positive periodic solution are established for a family of scalar periodic differential equations with infinite delay and nonlinear impulses. Our criteria, obtained by applying a fixed‐point argument to an original operator constructed here, allow to treat equations incorporating a rather general nonlinearity and impulses whose signs may vary. Applications to some classes of Volterra integro‐differential equations with unbounded or periodic delay and nonlinear impulses are given, extending and improving results in the literature.

This paper is devoted to analyze the stability of the economy according to an extended version of Kaldor's economic growth model. We consider the role of the government and its simultaneous monetary and scal policies and we study whether or not a time delay between the recognition and the implementation of its scal policy can a ect the economic stability. Numerical simulations provide further conclusions about the long-term behavior of the four variables modeled - namely, national income, capacity of production, bonds value and money supply.

This paper is devoted to analyze the stability of the economy according to an extended version of Kaldor's economic growth model. We consider the role of the government and its simultaneous monetary and scal policies and we study whether or not a time delay between the recognition and the implementation of its scal policy can a ect the economic stability. Numerical simulations provide further conclusions about the long-term behavior of the four variables modeled - namely, national income, capacity of production, bonds value and money supply.

This paper is devoted to analyze the stability of the economy according to an extended version of Kaldor's economic growth model. We consider the role of the government and its simultaneous monetary and scal policies and we study whether or not a time delay between the recognition and the implementation of its scal policy can a ect the economic stability. Numerical simulations provide further conclusions about the long-term behavior of the four variables modeled - namely, national income, capacity of production, bonds value and money supply.

In the present work, the existence of Uniformly Bound Solutions of a SI Mathematical Model with vital dynamics, with logistic growth for the Susceptibles, developed by Delay Differential Equations is constructed, and the behavior of the solutions will be studied (qualitative analysis) for the Infection-Free Point where the necessary conditions for its asymptotic stability will be determined; and furthermore, that the Uniformly Bounded Solution of the Model tends to the steady state of the Infection-Free Point. In addition, it will be simulated computationally (approximate solutions) with initial populations and epidemiological rates of the model. The simulation will complement the qualitative analysis (behavior of solutions) to conclude trends of behaviors of the transmission of the disease overtime., En el presente trabajo, se construye la existencia de Soluciones Uniformemente Acotadas de un Modelo Matemático SI con dinámica vital, con crecimiento logístico para los Susceptibles, desarrollado mediante las Ecuaciones Diferenciales con Retardo, y se estudiará el comportamiento de las soluciones (análisis cualitativo) para el Punto Libre de Infección donde se determinará las condiciones necesarias para su estabilidad asintótica; y más aún, la Solución Uniformemente Acotada del Modelo tiende al estado estacionario del Punto Libre de Infección. Además, se simulará computacionalmente (soluciones aproximadas) con poblaciones iniciales y tasas epidemiológicas del modelo. La simulación complementará el análisis cualitativo (comportamiento de soluciones) para concluir tendencias de comportamientos de la transmisión de la enfermedad en el tiempo.

In the present work, the existence of Uniformly Bound Solutions of a SI Mathematical Model with vital dynamics, with logistic growth for the Susceptibles, developed by Delay Differential Equations is constructed, and the behavior of the solutions will be studied (qualitative analysis) for the Infection-Free Point where the necessary conditions for its asymptotic stability will be determined; and furthermore, that the Uniformly Bounded Solution of the Model tends to the steady state of the Infection-Free Point. In addition, it will be simulated computationally (approximate solutions) with initial populations and epidemiological rates of the model. The simulation will complement the qualitative analysis (behavior of solutions) to conclude trends of behaviors of the transmission of the disease overtime., En el presente trabajo, se construye la existencia de Soluciones Uniformemente Acotadas de un Modelo Matemático SI con dinámica vital, con crecimiento logístico para los Susceptibles, desarrollado mediante las Ecuaciones Diferenciales con Retardo, y se estudiará el comportamiento de las soluciones (análisis cualitativo) para el Punto Libre de Infección donde se determinará las condiciones necesarias para su estabilidad asintótica; y más aún, la Solución Uniformemente Acotada del Modelo tiende al estado estacionario del Punto Libre de Infección. Además, se simulará computacionalmente (soluciones aproximadas) con poblaciones iniciales y tasas epidemiológicas del modelo. La simulación complementará el análisis cualitativo (comportamiento de soluciones) para concluir tendencias de comportamientos de la transmisión de la enfermedad en el tiempo.

We review a series of key travelling front problems in reaction-diffusion systems with a time-delayed feedback, appearing in ecology, nonlinear optics and neurobiology. For each problem, we determine asymptotic approximations for the wave shape and its speed. Particular attention is devoted to their validity and all analytical solutions are compared to solutions obtained numerically. We also extend the work by Erneux et al. (Erneux et al. 2010 Phil. Trans. R. Soc. A 368, 483-493 (doi:10.1098/rsta.2009.0228)) by considering the case of a slowly propagating front subject to a weak delayed feedback. The delay may either speed up the front in the same direction or reverse its direction., SCOPUS: ar.j, info:eu-repo/semantics/published

The long-run aim of this thesis is to solve delay differential equations with infinite delay of the type d dt u(t) = Au(t)+Zt −∞ u(s)k(t−s)ds+ ft,u(t) , on Fréchet spaces under an extended theory of groups of linear operators; where A is a linear operator, k(s) ⩾ 0 satisfiesR∞ 0 k(s)ds = 1 and f is a nonlinear map. In order to pursue such a goal we study a discrete delay model which explains the natural economic fluctuations considering how economic stability is affected by the role of the fiscal and monetary policies and a possible government inefficiency concerning its fiscal policy decision-making. On the other hand, we start to develop such an extended theory by considering linear Cauchy problems associated to a continuous linear operator on Fréchet spaces, for which we establish necessary and sufficient conditions for generation of a uniformly continuous group which provides the unique solution. Further consequences arises by considering pseudodifferential operators with constant coefficients defined on a particular Fréchet space of distributions, namely FL2loc, and special attention is given to the distributional solution of the heat equation on FL2loc for all time, which extends the standard solution on Hilbert spaces for positive time., El objetivo a largo plazo de esta tesis doctoral es resolver ecuaciones diferenciales de la forma d dt u(t) = Au(t)+Zt −∞ u(s)k(t−s)ds+ ft,u(t) ,en espacios de Fréchet extendiendo la teoria de grupos de operadores lineales; siendo A un operador linear, k(s)⩾0 satisfaceR∞ 0 k(s)ds = 1 y f una función no linear. Persiguiendo tal fin, estudiamos un modelo con retraso que enseña las fluctuaciones de la economía considerandocomolaestabilidadeconómicaesafectadaporlaactuacióndelgobierno,sus políticasfiscalymonetariayunaposibleineficienciadelgobiernoenloqueserefierealasu tomadedecisión.Porotrolado,damosinicioalareferidaextensióndelateoríadegrupos puesto que consideramos problemas de Cauchy lineales asociados a operadores lineales contínuos en espacios de Fréchet, para los cuales establecimos condiciones necessarias y suficientes para la generación de un grupo uniformemente contínuo en tal espacio y que proporcionelaúnicasolucióndelproblema.Consequenciasadicionalessurgencuando se considera operadores pseudodiferenciales con coeficientes constantes definidos en un particular espacio de Fréchet de distribuciones, a saber FL2loc, y una atención especial es dada a la solución distribucional de la ecuación del calor en FL2loc para todo el tiempo, la cual extiende la solución usual en espacios de Hilbert para tiempo positivo.

The long-run aim of this thesis is to solve delay differential equations with infinite delay of the type d dt u(t) = Au(t)+Zt −∞ u(s)k(t−s)ds+ ft,u(t) , on Fréchet spaces under an extended theory of groups of linear operators; where A is a linear operator, k(s) ⩾ 0 satisfiesR∞ 0 k(s)ds = 1 and f is a nonlinear map. In order to pursue such a goal we study a discrete delay model which explains the natural economic fluctuations considering how economic stability is affected by the role of the fiscal and monetary policies and a possible government inefficiency concerning its fiscal policy decision-making. On the other hand, we start to develop such an extended theory by considering linear Cauchy problems associated to a continuous linear operator on Fréchet spaces, for which we establish necessary and sufficient conditions for generation of a uniformly continuous group which provides the unique solution. Further consequences arises by considering pseudodifferential operators with constant coefficients defined on a particular Fréchet space of distributions, namely FL2loc, and special attention is given to the distributional solution of the heat equation on FL2loc for all time, which extends the standard solution on Hilbert spaces for positive time., El objetivo a largo plazo de esta tesis doctoral es resolver ecuaciones diferenciales de la forma d dt u(t) = Au(t)+Zt −∞ u(s)k(t−s)ds+ ft,u(t) ,en espacios de Fréchet extendiendo la teoria de grupos de operadores lineales; siendo A un operador linear, k(s)⩾0 satisfaceR∞ 0 k(s)ds = 1 y f una función no linear. Persiguiendo tal fin, estudiamos un modelo con retraso que enseña las fluctuaciones de la economía considerandocomolaestabilidadeconómicaesafectadaporlaactuacióndelgobierno,sus políticasfiscalymonetariayunaposibleineficienciadelgobiernoenloqueserefierealasu tomadedecisión.Porotrolado,damosinicioalareferidaextensióndelateoríadegrupos puesto que consideramos problemas de Cauchy lineales asociados a operadores lineales contínuos en espacios de Fréchet, para los cuales establecimos condiciones necessarias y suficientes para la generación de un grupo uniformemente contínuo en tal espacio y que proporcionelaúnicasolucióndelproblema.Consequenciasadicionalessurgencuando se considera operadores pseudodiferenciales con coeficientes constantes definidos en un particular espacio de Fréchet de distribuciones, a saber FL2loc, y una atención especial es dada a la solución distribucional de la ecuación del calor en FL2loc para todo el tiempo, la cual extiende la solución usual en espacios de Hilbert para tiempo positivo.

This project consists on a study of the displacement oscillations observed in biological tissues during tissue deformation processes. The final aim of this project is to determine the stability behaviour of such oscillations by defining the expressions of the stability boundaries. By using a cell model based on a delayed visco-elastic response to displacement of the cell walls, we will define two different deformation models and study their respective stability boundaries analytically and numerically in order to corroborate the analytic results. For each deformation model, we will also study different cases under different hypothesis, such as uniform delay or multiple delay.

We study a continuous Hénon system obtained by considering the discrete original model in continuous time. While the dynamics of the continuous model is trivial, we are able to recover the complexity of the discrete model by the introduction of time delays. In particular high period limit cycles and chaotic attractors are observed. We illustrate the results with some numerical simulations.

Delay differential equations (DDEs) are often used to model systems with time-delayed effects, and they have found applications in fields such as climate dynamics, biosciences, engineering, and control theory. In contrast to ordinary differential equations (ODEs), the phase space associated even with a scalar DDE is infinite-dimensional. Oftentimes, it is desirable to have low-dimensional ODE systems that capture qualitative features as well as approximate certain quantitative aspects of the DDE dynamics. In this thesis, we present a Galerkin scheme for a broad class of DDEs and derive convergence results for this scheme. In contrast to other Galerkin schemes devised in the DDE literature, the main new ingredient here is the use of the so called Koornwinder polynomials, which are orthogonal polynomials under an inner product with a point mass. A main advantage of using such polynomials is that they live in the domain of the underlying linear operator, which arguably simplifies the related numerical treatments. The obtained results generalize a previous work to the case of DDEs with multiply delays in the linear terms, either discrete or distributed, or both. We also consider the more challenging case of discrete delays in the nonlinearity and obtain a convergence result by assuming additional assumptions about the Galerkin approximations of the linearized systems.

We study a continuous Hénon system obtained by considering the discrete original model in continuous time. While the dynamics of the continuous model is trivial, we are able to recover the complexity of the discrete model by the introduction of time delays. In particular high period limit cycles and chaotic attractors are observed. We illustrate the results with some numerical simulations.

Delay differential models present characteristic dynamical properties that should ideally be preserved when computing numerical approximate solutions. In this work, exact numerical schemes for a general linear delay differential model, as well as for the special case of a pure delay model, are obtained. Based on these exact schemes, a family of nonstandard methods, of increasing order of accuracy and simple computational properties, is proposed. Dynamic consistency of the new nonstandard methods are proved, and illustrated with numerical examples, for asymptotic stability, positive preserving properties, and oscillation behaviour.

We study a continuous Hénon system obtained by considering the discrete original model in continuous time. While the dynamics of the continuous model is trivial, we are able to recover the complexity of the discrete model by the introduction of time delays. In particular high period limit cycles and chaotic attractors are observed. We illustrate the results with some numerical simulations.

This paper introduces a robust observer-based output feedback control strategy that enables the generation of complex three-dimensional borehole trajectories created by directional drilling systems, while avoiding undesired transient behavior. The model-based controller relies on a set of nonlinear delay differential equations describing the borehole evolution. Herein, only local orientation measurements of the bottom hole assembly of the drilling system are employed. Controller and observer gains are synthesized by optimizing the location of the rightmost pole of the closed-loop dynamics, using a spectral approach for delay differential equations. Moreover, the strategy is extended to cope with the uncertainty of key system parameters in the directional drilling process. The effectiveness of the designed controller is tested in an illustrative benchmark study.

We study a continuous Hénon system obtained by considering the discrete original model in continuous time. While the dynamics of the continuous model is trivial, we are able to recover the complexity of the discrete model by the introduction of time delays. In particular high period limit cycles and chaotic attractors are observed. We illustrate the results with some numerical simulations.

Delay differential models present characteristic dynamical properties that should ideally be preserved when computing numerical approximate solutions. In this work, exact numerical schemes for a general linear delay differential model, as well as for the special case of a pure delay model, are obtained. Based on these exact schemes, a family of nonstandard methods, of increasing order of accuracy and simple computational properties, is proposed. Dynamic consistency of the new nonstandard methods are proved, and illustrated with numerical examples, for asymptotic stability, positive preserving properties, and oscillation behaviour.

During the embryonic development, it has been observed the prior to drastic shape changes of the organism, tissues deformations oscillate in a manner that resembles a fluid. Strikingly though, inertial terms are negligible in such systems, which renders the governing mechanical PDEs as elliptical or parabolic, with no apparent oscillatory solution. This work aims to investigate the coupling between mechanical equilibrium and biochemical coupling as a epossible source of this oscillations.

In the present work, a Basic Model SI with Vital Dynamics Structured by Gender developed by the Ordinary Differential Equations (Transmission of contagion is instantaneous), and also developed in the DelayDifferential Equations (Transmission of contagion occurs after a certain period of time), where the Local and Asymptotic Stability Theorem is proposed The Free of Infection point for both models, respectively., En el presente trabajo de investigación, se analiza cualitativamente un Modelo Matemático SI con Dinámica vital Estrucurado por Sexo desarrollado mediante las ecuaciones diferenciales ordinarias (transmisión de contagio es instantáneo), y también desarrollado por las ecuaciones diferenciales con retardo (transmisión de contagio se da después de un cierto periodo de tiempo), donde se proponen teoremas de Estabilidad Local y Asintótica para el punto libre de infección de ambos modelos, respectivamente. Los modelos permiten una concepción matemática de la dinámica de la enfermedad, y permitirían realizar una mejor previsión a las instituciones nacionales, de manera particular a la Dirección General de Epidemiología debido que es el encargado de realizar las actividades de previsión y de control de la enfermedad que serán considerados como políticas públicas por el Ministerio de Salud.

Two different practical approaches to the application of mathematical modelling in oncology are investigated. The first approach involved a comparative model investigation, where the focus was on determining whether a fractional-order Gompertz model or the ordinary integer-order Gompertz model would be the best descriptor of growth for a particular set of tumour growth volumes. Herein, fractional calculus is used with the ordinary Gompertz model to propose a fractional-order version of the model, with orders between 0 and 1. After obtaining the solution of the fractional-order model through the application of fractional calculus, calibration of the solution together with the solution of the ordinary Gompertz model was carried out with the use of a set of Rhabdomyosarcoma volumes in a mouse. This generated the approximate fractional-order for the proposed model that would best describe the data. Using the approximate fractional-order obtained for the proposed model during calibration, as well as the ordinary Gompertz model, a further comparative investigation was conducted into the ability of both models to predict a data point of the dataset not included in the calibration process. At conclusion of these investigations it was found that the fractional-order Gompertz model of order 0.68 could capture the growth behaviour represented in the data better than that of the ordinary Gompertz model. The proposed model not only fit the data better, but also had better prediction capability over the period of observation versus its integer-order counterpart. The second approach involved a data-driven investigation leading to the construction of a mathe- matical model of acute lymphoblastic leukaemia. The aim of this investigation was to determine whether a model of reasonable suitability could be proposed and whether such a model could gen- erate meaningful and relevant insights not only into the system under investigation, but also into the model. The latter would form the f

This paper continues research initiated in Solodushkin et al. (2015). We develop a finite difference scheme for a first order multidimensional partial differential equation including a time delay. This class of equations is used to model different time lapse phenomena, e.g. birds migration, proliferation of viruses or bacteria and transfer of nuclear particles. For the constructed difference schemes the order of approximation, stability and convergence order are substantiated. To conclude we support the obtained results with some test examples. © 2016 Elsevier B.V.

Dynamik auf Netzwerken ist ein mathematisches Feld, das in den letzten Jahrzehnten schnell gewachsen ist und Anwendungen in zahlreichen Disziplinen wie z.B. Physik, Biologie und Soziologie findet. Die Funktion vieler Netzwerke hängt von der Fähigkeit ab, die Elemente des Netzwerkes zu synchronisieren. Mit anderen Worten, die Existenz und die transversale Stabilität der synchronen Mannigfaltigkeit sind zentrale Eigenschaften. Erst seit einigen Jahren wird versucht, den verwickelten Zusammenhang zwischen der Kopplungsstruktur und den Stabilitätseigenschaften synchroner Zustände zu verstehen. Genau das ist das zentrale Thema dieser Arbeit. Zunächst präsentiere ich erste Ergebnisse zur Klassifizierung der Kanten eines gerichteten Netzwerks bezüglich ihrer Bedeutung für die Stabilität des synchronen Zustands. Folgend untersuche ich ein komplexes Verzweigungsszenario in einem gerichteten Ring von Stuart-Landau Oszillatoren und zeige, dass das Szenario persistent ist, wenn dem Netzwerk eine schwach gewichtete Kante hinzugefügt wird. Daraufhin untersuche ich synchrone Zustände in Ringen von Phasenoszillatoren die mit Zeitverzögerung gekoppelt sind. Ich bespreche die Koexistenz synchroner Lösungen und analysiere deren Stabilität und Verzweigungen. Weiter zeige ich, dass eine Zeitverschiebung genutzt werden kann, um Muster im Ring zu speichern und wiederzuerkennen. Diese Zeitverschiebung untersuche ich daraufhin für beliebige Kopplungsstrukturen. Ich zeige, dass invariante Mannigfaltigkeiten des Flusses sowie ihre Stabilität unter der Zeitverschiebung erhalten bleiben. Darüber hinaus bestimme ich die minimale Anzahl von Zeitverzögerungen, die gebraucht werden, um das System äquivalent zu beschreiben. Schließlich untersuche ich das auffällige Phänomen eines nichtstetigen Übergangs zu Synchronizität in Klassen großer Zufallsnetzwerke indem ich einen kürzlich eingeführten Zugang zur Beschreibung großer Zufallsnetzwerke auf den Fall zeitverzögerter Kopplungen verallgemeinere., Since a couple of decades, dynamics on networks is a rapidly growing branch of mathematics with applications in various disciplines such as physics, biology or sociology. The functioning of many networks heavily relies on the ability to synchronize the network’s nodes. More precisely, the existence and the transverse stability of the synchronous manifold are essential properties. It was only in the last few years that people tried to understand the entangled relation between the coupling structure of a network, given by a (di-)graph, and the stability properties of synchronous states. This is the central theme of this dissertation. I first present results towards a classification of the links in a directed, diffusive network according to their impact on the stability of synchronization. Then I investigate a complex bifurcation scenario observed in a directed ring of Stuart-Landau oscillators. I show that under the addition of a single weak link, this scenario is persistent. Subsequently, I investigate synchronous patterns in a directed ring of phase oscillators coupled with time delay. I discuss the coexistence of multiple of synchronous solutions and investigate their stability and bifurcations. I apply these results by showing that a certain time-shift transformation can be used in order to employ the ring as a pattern recognition device. Next, I investigate the same time-shift transformation for arbitrary coupling structures in a very general setting. I show that invariant manifolds of the flow together with their stability properties are conserved under the time-shift transformation. Furthermore, I determine the minimal number of delays needed to equivalently describe the system’s dynamics. Finally, I investigate a peculiar phenomenon of non-continuous transition to synchrony observed in certain classes of large random networks, generalizing a recently introduced approach for the description of large random networks to the case of delayed couplings.

Continuous-time random walks combining diffusive scattering and ballistic propagation on lattices model a class of Lévy walks. The assumption that transitions in the scattering phase occur with exponentially-distributed waiting times leads to a description of the process in terms of multiple states, whose distributions evolve according to a set of delay differential equations, amenable to analytic treatment. We obtain an exact expression of the mean squared displacement associated with such processes and discuss the emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of initial conditions on the transport coefficients. Of particular interest is the case of rare ballistic propagation, in which a regime of superdiffusion may lurk underneath one of normal diffusion.

The paper deals with a dynamical system described by delay differential equations which is controlled under conditions of unknown disturbances. This system is approximated by a system of ordinary differential equations which is used as a leader. An aiming procedure between the delay and approximating systems is elaborated. The procedure is organized in such a way that the necessary proximity of motions of these systems is guaranteed by means of the delay system control and a certain part of the approximating system controls. Another part of the approximating system controls can be used to compensate disturbances in the initial delay system and to ensure the required quality of the control process. Thus, by using the elaborated aiming procedure, results of control theory for ordinary differential systems can be applied to delay differential systems. © 2015, IFAC Hosting by Elsevier Ltd.

We construct a finite difference scheme for the numerical solution of a first order partial differential equation with a time delay and retardation of a state variable. Such equations are used to model the dynamics of structured cell populations when age and maturity level are taken into account. For the supplied difference schemes the order of approximation, stability and convergence order are studied. We illustrate the obtained results with a test example. © 2015 Elsevier B.V. All rights reserved.

In this paper, a structure-preserving model reduction approach for a class of delay differential equations is proposed. Benefits of this approach are, firstly, the fact that the delay nature of the system is preserved after reduction, secondly, that input-output stability properties are preserved and, thirdly, that a computable error bound reflecting the accuracy of the reduction is provided. These results are applicable to large-scale linear delay differential equations with constant delays, but also extensions to a class of nonlinear delay differential equations with time-varying delays are presented. The effectiveness of the results is evidenced by means of an illustrative example., QC 20150414

In this paper a parameter class of Linear multistep method are applied to solve delay differential equations of the form y’(t) = f(t; y(t); y(t- t(t))), (t >0) subject to the initial condition y(t) = j (t) for tmin = 0 . The stability properties when the methods were applied to the test equation with a fixed delay t ; y’(t) = λy(t) + µy(t - t ); t ³ 0; are studied λ; µ are complex constants and Ф(t) is a continuous complex-valued function. The stability regions are plotted and numerical results are introduced.

Dynamik auf Netzwerken ist ein mathematisches Feld, das in den letzten Jahrzehnten schnell gewachsen ist und Anwendungen in zahlreichen Disziplinen wie z.B. Physik, Biologie und Soziologie findet. Die Funktion vieler Netzwerke hängt von der Fähigkeit ab, die Elemente des Netzwerkes zu synchronisieren. Mit anderen Worten, die Existenz und die transversale Stabilität der synchronen Mannigfaltigkeit sind zentrale Eigenschaften. Erst seit einigen Jahren wird versucht, den verwickelten Zusammenhang zwischen der Kopplungsstruktur und den Stabilitätseigenschaften synchroner Zustände zu verstehen. Genau das ist das zentrale Thema dieser Arbeit. Zunächst präsentiere ich erste Ergebnisse zur Klassifizierung der Kanten eines gerichteten Netzwerks bezüglich ihrer Bedeutung für die Stabilität des synchronen Zustands. Folgend untersuche ich ein komplexes Verzweigungsszenario in einem gerichteten Ring von Stuart-Landau Oszillatoren und zeige, dass das Szenario persistent ist, wenn dem Netzwerk eine schwach gewichtete Kante hinzugefügt wird. Daraufhin untersuche ich synchrone Zustände in Ringen von Phasenoszillatoren die mit Zeitverzögerung gekoppelt sind. Ich bespreche die Koexistenz synchroner Lösungen und analysiere deren Stabilität und Verzweigungen. Weiter zeige ich, dass eine Zeitverschiebung genutzt werden kann, um Muster im Ring zu speichern und wiederzuerkennen. Diese Zeitverschiebung untersuche ich daraufhin für beliebige Kopplungsstrukturen. Ich zeige, dass invariante Mannigfaltigkeiten des Flusses sowie ihre Stabilität unter der Zeitverschiebung erhalten bleiben. Darüber hinaus bestimme ich die minimale Anzahl von Zeitverzögerungen, die gebraucht werden, um das System äquivalent zu beschreiben. Schließlich untersuche ich das auffällige Phänomen eines nichtstetigen Übergangs zu Synchronizität in Klassen großer Zufallsnetzwerke indem ich einen kürzlich eingeführten Zugang zur Beschreibung großer Zufallsnetzwerke auf den Fall zeitverzögerter Kopplungen verallgemeinere., Since a couple of decades, dynamics on networks is a rapidly growing branch of mathematics with applications in various disciplines such as physics, biology or sociology. The functioning of many networks heavily relies on the ability to synchronize the network’s nodes. More precisely, the existence and the transverse stability of the synchronous manifold are essential properties. It was only in the last few years that people tried to understand the entangled relation between the coupling structure of a network, given by a (di-)graph, and the stability properties of synchronous states. This is the central theme of this dissertation. I first present results towards a classification of the links in a directed, diffusive network according to their impact on the stability of synchronization. Then I investigate a complex bifurcation scenario observed in a directed ring of Stuart-Landau oscillators. I show that under the addition of a single weak link, this scenario is persistent. Subsequently, I investigate synchronous patterns in a directed ring of phase oscillators coupled with time delay. I discuss the coexistence of multiple of synchronous solutions and investigate their stability and bifurcations. I apply these results by showing that a certain time-shift transformation can be used in order to employ the ring as a pattern recognition device. Next, I investigate the same time-shift transformation for arbitrary coupling structures in a very general setting. I show that invariant manifolds of the flow together with their stability properties are conserved under the time-shift transformation. Furthermore, I determine the minimal number of delays needed to equivalently describe the system’s dynamics. Finally, I investigate a peculiar phenomenon of non-continuous transition to synchrony observed in certain classes of large random networks, generalizing a recently introduced approach for the description of large random networks to the case of delayed couplings.

Bone morphogenic proteins (BMPs) and their inhibitors are increasingly being appreciated for their role in vascular development. Matrix Gla protein (MGP) and crosssveinless 2 (CV2) antagonize BMP-4 and BMP-9, respectively, and the expression of both are stimulated by time-delayed BMP-9/ALK1 signaling, while ALK1 is stimulated by delayed BMP-4 signaling. This double negative feedback system is an important regulator of endothelial cell differentiation, angiogenesis, and maturation. Our collaborators recently discovered oscillatory gene expression of these proteins in human pulmonary artery endothelial cells (HPAECs) undergoing maturation in cell culture. We set out to explore the biological mechanisms guiding this periodic gene expression and to propose a functional role for these oscillations. We used delay differential equations to mathematically model this network and applied dimensionless analysis to study the impact of relative protein degradation rates on oscillations. Monte Carlo parameter sampling and Hopf bifurcation analysis, while adjusting 1 and 2 parameters, helped identify conditions yielding periodic dynamics. We discovered that the presence of the activator, BMP-4, is critical for oscillations and that its inhibitor, MGP, opposes them, making their mutual regulation important for oscillatory gene expression. We also identified other mechanisms for oscillations including the necessary existence of three time-delays, the presence of BMP-9 as a feedback regulator, and the function of CV2 as an indirect inhibitor of MGP. Findings from the cell culture experiments and the mathematical model led us to postulate that oscillations in this network are important for the maintenance of progenitor HPAECs and the lack of oscillations is associated with HPAEC maturation.

Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of Levy walks on lattices. By including exponentially distributed waiting times separating the successive jump events of a walker, we are led to a description of such Levy walks in terms of multistate processes whose time-evolution is shown to obey a set of coupled delay differential equations. Using simple arguments, we obtain asymptotic solutions to these equations and rederive the scaling laws for the mean squared displacement of such processes. Our calculation includes the computation of all relevant transport coefficients in terms of the parameters of the models.

A convergence analysis is presented for the implicit Euler and Lie splitting schemes when applied to nonlinear parabolic equations with delay. More precisely, we consider a vector field which is the sum of an unbounded dissipative operator and a delay term, where both point delays and distributed delays fit into the framework. Such equations are frequently encountered, e.g. in population dynamics. The main theoretical result is that both schemes converge with an order (of at least) q = 1/2, without any artificial regularity assumptions. We discuss implementation details for the methods, and the convergence results are verified by numerical experiments demonstrating both the correct order, as well as the efficiency gain of Lie splitting as compared to the implicit Euler scheme.

Dans ce travail, nous étudions deux types de problèmes à retard. Le premier traite des oscillateurs optoélectroniques (OOEs). Un OOE est un système bouclé permettant de délivrer une onde électromagnétique radio-fréquence de grande pureté spectrale et de faible bruit électronique. Le second problème traite du couplage retardé de neurones. Une nouvelle forme de synchronisation est observée où un régime oscillant est une alternative à un état stationnaire stable. Ces deux problèmes présentent des oscillations de type slow-fast. Une grande partie de ma thèse est dévouée à l’analyse de ces régimes. Etant donné qu’il s’agit d’équations nonlinéaires à retard, les techniques asymptotiques classiques ont dû être revues. En plus d’une étude théorique, des expériences ont été effectuées. Le travail sur les OOEs a été rendu possible grâce aux invitations respectives de L. Larger dans son laboratoire à l’Université de Franche-Comté et de D.J. Gauthier à Duke University. Le travail sur le couplage de neurones a bénéficié d’expériences réalisées par L. Keuninckx du groupe « Applied Physics » de la Vrije Universiteit Brussel.Une contribution importante de cette thèse est à la fois l’analyse mathématique mais aussi l’observation expérimentale d’ondes carrées stables asymétriques présentant des longueurs de plateau différentes mais ayant la même période dans un OOE. Une bifurcation de Hopf primaire d’un état stationnaire est le mécanisme menant à ces régimes. Un deuxième phénomène qui a été à la fois observé pour l’OOE et pour les neurones couplés est la coexistence entre plusieurs ondes carrées ayant des périodes différentes. Pour l’OOE, ces oscillations peuvent être reliées à plusieurs bifurcations de Hopf primaires qui sont proches les unes des autres à cause du grand délai. Le mécanisme de stabilité est similaire à celui de "Eckhaus" pour les systèmes spatialement étendus. Pour le couplage de cellules excitables, nous avons étudié des équations couplées de type FitzHugh-Nagumo, Doctorat en Sciences, info:eu-repo/semantics/nonPublished

As important as the intrinsic properties of an individual nervous cell stands the network of neurons in which it is embedded and by virtue of which it acquires great part of its responsiveness and functionality. In this study we have explored how the topological properties and conduction delays of several classes of neural networks affect the capacity of their constituent cells to establish well-defined temporal relations among firing of their action potentials. This ability of a population of neurons to produce and maintain a millisecond-precise coordinated firing (either evoked by external stimuli or internally generated) is central to neural codes exploiting precise spike timing for the representation and communication of information. Our results, based on extensive simulations of conductance-based type of neurons in an oscillatory regime, indicate that only certain topologies of networks allow for a coordinated firing at a local and long-range scale simultaneously. Besides network architecture, axonal conduction delays are also observed to be another important factor in the generation of coherent spiking. We report that such communication latencies not only set the phase difference between the oscillatory activity of remote neural populations but determine whether the interconnected cells can set in any coherent firing at all. In this context, we have also investigated how the balance between the network synchronizing effects and the dispersive drift caused by inhomogeneities in natural firing frequencies across neurons is resolved. Finally, we show that the observed roles of conduction delays and frequency dispersion are not particular to canonical networks but experimentally measured anatomical networks such as the macaque cortical network can display the same type of behavior.