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Stability analysis of the modular multilevel converter (MMC) requires accurate models over wide frequency ranges, accounting for both the time-periodic nature of internal dynamics and non-passive behavior resulting from time delays. Frequency-lifting techniques such as harmonic state-space (HSS) can be used to tackle the time-periodicity and obtain linear time-invariant models on which classical stability analysis techniques can be applied. However, exact time delays are generally approximated by rational polynomial functions to study the eigenvalues of ordinary differential equations (ODEs), which leads to inaccuracies at higher frequencies. Such approximations can be avoided by preserving delays in their exact form and studying eigenvalues of delay differential equations (DDEs). To that end, this paper presents the generalization of the original ODE-based HSS framework to DDEs, accounting for both time-delay and time-periodic behaviors of the MMC. The generalized HSS framework benefits from the insights of eigenvalues and participation factors analysis while at the same time guaranteeing the derivation of accurate frequency-response models. Using the developed HSS-DDE framework, AC and DC-side admittances of the MMC are obtained and validated against frequency scans. Lastly, the stability of a test system is studied in both time and frequency domains to show the advantages of the developed framework. ispartof: Ieee Transactions On Power Delivery vol:37 issue:4 status: Published online

In this work, we provide for the first time the characterization of delay terms in systems of delay differential equations within the frame of the Network Simulation Method, a procedure that sets a formal equivalence between the system of differential equations and an electrical network. The results were not achieved previously, which strengths the formalism of the model and provides new opportunities for the method. Free circuit software LTspice is employed to conduct the simulations, which requires few simulation rules and can be programmed either by electrical symbol code or text file. Very few devices are needed to design the network model, and the delay terms are easily implemented by voltage-controlled voltage sources. A practical example for delayed adsorption/desorption kinetics is employed to test the methodology, being the results compared with software Mathematica. Additionally, the modelling of pulse width in passively mode-locked quantum dot lasers by the application of a reverse bias voltage is addressed, which constitutes a promising application in communications and advanced sensing. The power, versatility and simplicity of the Network Simulation Method enables it as an exceptional alternative to solve complex systems described by delay differential equations, from both researching and educational points of view. The authors want to thank the University Centre of Defence at the Spanish Air Force Academy for financial support.

This paper presents 2-step p-th order (p = 2,3,4) multi-step methods that are based on the combination of both polynomial and exponential functions for the solution of Delay Differential Equations (DDEs). Furthermore, the delay argument is approximated using the Lagrange interpolation. The local truncation errors and stability polynomials for each order are derived. The Local Grid Search Algorithm (LGSA) is used to determine the stability regions of the method. Moreover, applicability and suitability of the method have been demonstrated by some numerical examples of DDEs with constant delay, time dependent and state dependent delays. The numerical results are compared with the theoretical solution as well as the existing Rational Multi-step Method2 (RMM2).

In this paper, we investigate the boundedness and uniformly asymtotically stability of the solutions to a certain third order non-autonomous differential equations with bounded delay. By constructing a Lyapunov functional, sufficient conditions for the stability and boundedness of solutions for equations considered are obtained. We used an example to demonstrate the feasibility of our results. The results obtain essentially improve, include and complement the results in the literature.

Inspired by the experimental and numerical findings, we study the dynamic instabilities of two coupled nonlinear delay differential equations that are used to describe the coherent oscillations between the top and bottom boundary layers in turbulent Rayleigh-Bénard convection. By introducing two sensitivity parameters for the instabilities of the top and bottom boundary layers, we find three different types of solutions, namely in-phase single-period oscillations, multi-period oscillations and chaos. The chaos solution contains rare but large amplitude fluctuations. The statistical properties of these fluctuations are consistent with those observed in the experiment for the massive eruption of thermal plumes, which causes random reversals of the large-scale circulation in turbulent Rayleigh-Bénard convection. Our study thus provides new insights into the origin of rare massive eruptions and sudden changes of large-scale flow pattern that are often observed in convection systems of geophysical and astrophysical scales.

Rabies is a fatal zoonotic disease caused by a virus through bites or saliva of an infected animal. Dogs are the main reservoir of rabies and responsible for most cases in humans worldwide. In this article, a delay differential equations model for assessing the effects of controls and time delay as incubation period on the transmission dynamics of rabies in human and dog populations is formulated and analyzed. Analysis from the model show that there is a locally and globally asymptotic stable disease-free equilibrium whenever a certain epidemiological threshold, the control reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_v$$\end{document}Rv, is less than unity. Furthermore, the model has a unique endemic equilibrium when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_v$$\end{document}Rv exceed unity which is also locally and globally asymptotically stable for all delays. Time delay is found to have influence on the endemicity of rabies. Vaccination of humans and dogs coupled with annual crop of puppies are imposed to curtail the spread of rabies in the populations. Sensitivity analysis on the number of infected humans and dogs revealed that increasing dog vaccination rate and decreasing annual birth of puppies are more effective in human populations. However in dog populations, the vaccination and birth control of puppies, have equal effective measures for rabies control. Numerical experiments are conducted to illustrate the theoretical results and control strategies.

The oscillation of nonlinear differential equations is used in many applications of mathematical physics, biological and medical physics, engineering, aviation, complex networks, sociophysics and econophysics. The goal of this study is to create some new oscillation criteria for fourth-order differential equations with delay and advanced terms ( a 1 ( x ) ( w ‴ ( x ) ) n ) ′ + ∑ j = 1 r β j ( x ) w k ( γ j ( x ) ) = 0 , {({a}_{1}(x){({w}^{\prime\prime\prime }(x))}^{n})}^{^{\prime} }+\mathop{\sum }\limits_{j=1}^{r}{\beta }_{j}(x){w}^{k}({\gamma }_{j}(x))=0, and ( a 1 ( x ) ( w ‴ ( x ) ) n ) ′ + a 2 ( x ) h ( w ‴ ( x ) ) + β ( x ) f ( w ( γ ( x ) ) ) = 0 . {({a}_{1}(x){({w}^{\prime\prime\prime }(x))}^{n})}^{^{\prime} }+{a}_{2}(x)h({w}^{\prime\prime\prime }(x))+\beta (x)f(w(\gamma (x)))=0. The method is based on the use of the comparison technique and Riccati method to obtain these criteria. These conditions complement and extend some of the results published on this topic. Two examples are provided to prove the efficiency of the main results.

The main purpose of this paper is to study the oscillatory properties of solutions of the third-order quasi-linear delay differential equation ℒ y ( t ) + f ( t ) y β ( σ ( t ) ) = 0 {\cal L}y(t) + f(t){y^\beta }(\sigma (t)) = 0 where ℒy(t) = (b(t)(a(t)(y 0(t)) )0)0 is a semi-canonical differential operator. The main idea is to transform the semi-canonical operator into canonical form and then obtain new oscillation results for the studied equation. Examples are provided to illustrate the importance of the main results.

In this work, we study the problem of $ p- $th moment global exponential stability for functional differential equations and scalar chaotic delayed equations under random impulsive effects. Meanwhile, the $ p- $th moment global exponential synchronization for the proposed equations is also discussed, whereas the main results are proved by using Lyapunov function and Razumikhin technique. Furthermore, the impact of fixed and random time impulses are presented by applying the results to Mackey Glass blood cell production model and Ikeda bistable resonator model. Finally, the effectiveness of fixed and random impulses are depicted via graphical representations.

Somitogenesis is the process by means of which a tissue known as presomitic mesoderm (PSM) is segmented in blocks of cells, called somites, along the anterior-posterior axis of the developing embryo in segmented animals. In vertebrates, somites give rise to axial skeleton, cartilage, tendons, skeletal muscle, and dermis. Somite formation occurs periodically, and this periodicity is driven by a genetic oscillator that operates within PSM cells and is known as the segmentation clock. The correct synchronization of the segmentation clock among PSM cells is essential for somitogenesis to develop normally. When synchronization is disrupted, somites form irregularly and, in consequence, the tissues that originate from them show clear malformations. In this work, based in a model for zebrafish segmentation clock, we investigate by means of a mathematical modeling approach, how PSM-cell synchronization is affected by factors like: the size of PSM-cell networks, the amount of cell-to-cell interactions per PSM cell, the strength of these interactions, and the inherent variability among PSM cells. Interestingly we found that very small PSM-cell networks are unable to synchronize. Moreover, the effect of decreasing the strength of interactions among PSM cells is corrected by increasing the network connectivity-level, and a moderated level of variability among cells can have a positive effect on synchronization, specially in large networks.

This study aims to investigate the asymptotic behavior of a class of third-order delay differential equations. Here, we consider an equation with a middle term and several delays. We obtain an iterative relationship between the positive solution of the studied equation and the corresponding function. Using this new relationship, we derive new criteria that ensure that all non-oscillatory solutions converge to zero. The new findings are an extension and expansion of relevant findings in the literature. We apply our results to a special case of the equation under study to clarify the importance of the new criteria.

In this paper, we study the dynamics of an infectious disease in the presence of a continuous-imperfect vaccine and latent period. We consider a general incidence rate function with a non-monotonicity property to interpret the psychological effect in the susceptible population. After we propose the model, we provide the well-posedness property and calculate the effective reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_E$$\end{document}RE. Then, we obtain the threshold dynamics of the system with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_E$$\end{document}RE by studying the global stability of the disease-free equilibrium when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_E1$$\end{document}RE>1. For the endemic equilibrium, we use the semi-discretization method to analyze its linear stability. Then, we discuss the critical vaccination coverage rate that is required to eliminate the disease. Numerical simulations are provided to implement a case study regarding data of influenza patients, study the local and global sensitivity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_E

This is the accepted version of the following article: Buedo‐Fernández, S, Faria, T. Positive periodic solutions for impulsive differential equations with infinite delay and applications to integro‐differential equations. Math Meth Appl Sci. 2020; 43: 3052–3075, which has been published in final form at https://doi.org/10.1002/mma.6100. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy http://www.wileyauthors.com/self-archiving 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 work was supported by Ministerio de Educacion, Cultura y Deporte (Spain) under grant FPU16/04416 (Sebastián Buedo-Fernández) and by Fundação para a Ciência e a Tecnologia (Portugal) under project UID/MAT/04561/2019 (Teresa Faria) SI

Recently, many realistic models of structured populations are described through delay equations which involve quantities defined by the solutions of external problems. For instance, the size or survival probability of individuals may be described by ordinary differential equations, and their maturation age may be determined by a nonlinear condition. When treating these complex models with existing continuation approaches in view of analyzing stability and bifurcations, the external quantities are computed from scratch at every continuation step. As a result, the requirements from the computational point of view are often demanding. In this work we propose to improve the overall performance by investigating a suitable numerical treatment of the external problems in order to include the relevant variables into the continuation framework, thus exploiting their values computed at each previous step. We explore and test this internal continuation with prototype problems first. Then we apply it to a representative class of realistic models, demonstrating the superiority of the new approach in terms of computational time for a given accuracy threshold.

Self-mode-locked semiconductor disk lasers possess compact resonant cavity and stable construction. These devices have a wide application prospect because of their picosecond to sub-picosecond pulse width, excellent beam quality and tailorable emission wavelength. In this paper, dynamics simulations of self-mode-locking in a semiconductor disk laser are performed by using delay differential equations for the first time. The corresponding conditions of different modality of mode-locking, including Q-switched mode-locking, continuous-wave mode-locking and harmonic mode-locking are calculated, and their dynamics evolution processes are presented. We also analyze the characteristics of the three different mode-locking modalities and summarize their overall dynamics evolution tendency. This kind of numerical simulation and analysis provides an understanding of the dynamics process of self-mode-locking, and may be referenced for related experiments.

Most plant viral infections are vector-borne. There is a latent period of disease inside the vector after obtaining the virus from the infected plant. Thus, after interacting with an infected vector, the plant demonstrates an incubation time before becoming diseased. This paper analyzes a mathematical model for persistent vector-borne viral plant disease dynamics. The backpropagated neural network based on the Levenberg—Marquardt algorithm (NN-BLMA) is used to study approximate solutions for fluctuations in natural plant mortality and vector mortality rates. A state-of-the-art numerical technique is utilized to generate reference data for obtaining surrogate solutions for multiple cases through NN-BLMA. Curve fitting, regression analysis, error histograms, and convergence analysis are used to assess accuracy of the calculated solutions. It is evident from our simulations that NN-BLMA is accurate and reliable.

The calculation of the probability of a minor outbreak is crucial in analyzing a stochastic epidemic model. For stochastic epidemic models with fixed delays, the linear chain trick is applied to transform the delayed models into a family of ODE models with increasing shape parameters. We then prove that the well-established results on the probability of a minor outbreak for continuous-time Markov chain (CTMC) epidemic models also hold for the stochastic epidemic models with fixed delays. All theoretical results are verified by numerical simulations implemented by the delay stochastic simulation algorithm (DSSA) in Python. It is shown that DSSA is able to generate exact realizations for underlying delayed models in the context of mathematical epidemiology, and therefore, provides insights into the effect of delays during the outbreak phases of epidemics.

This article discusses several forms of Ulam stability of nonlinear fractional delay differential equations. Our investigation is based on a generalised Gronwall’s inequality and Picard operator theory. Implementations are provided to demonstrate the stability results obtained for finite intervals.

In this paper, we obtain new monotonic properties for positive solutions of even-order delay differential equations in the non-canonical case. Using these properties, we establish a new oscillation criterion for solutions by comparison with an equation of the first order. The approach adopted is based on the use of symmetry between positive and negative solutions.

In this paper, we propose and derive a new system called pure hybrid fuzzy neutral delay differential equations. We apply the classical fourth-order Runge–Kutta method (RK-4) to solve the proposed system of ordinary differential equations. First, we define the RK-4 method for hybrid fuzzy neutral delay differential equations and then establish the efficiency of this method by utilizing it to solve a particular type of fuzzy neutral delay differential equation. We provide a numerical example to verify the theoretical results. In addition, we compare the RK-4 and Euler solutions with the exact solutions. An error analysis is conducted to assess how much deviation from exactness is found in the two numerical methods. We arrive at the same conclusion for our hybrid fuzzy neutral delay differential system since the RK-4 method outperforms the classical Euler method.

Nonhomogeneous systems governed by second-order linear differential equations with pure delay are considered. As an application, the exact solutions of these systems and their delayed matrix functions are used to obtain the finite-time stability results. Our results extend and improve some previous results by removing some restrictive conditions. Finally, an example is provided to illustrate our theoretical results.

Orientador: João Frederico da Costa Azevedo Meyer Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica Resumo: O modelo com retardamento da respiração humana é revisitado segundo as referências [5,13,14] com o objetivo de rever resultados sobre estabilidade, a fim de obter parâmetros para certos problemas, como a "Respiração de Cheyne-Stokes". Após uma análise qualitativa retirada de [5], são apresentadas simulações numéricas em que esses parâmetros são escolhidos de modo a obter, através de uma análise gráfica, algumas informações importantes para a conclusão sobre certos problemas respiratórios Abstract: The human breathing model with delay is revisited according to the references [5,13,14] in order to review results on stability and to obtain parameters for certain problems, such as "Cheyne-Stokes Respiration". After a qualitative analysis taken from [5], numerical simulations are presented in which these parameters are chosen to make it possible, through a graphical analysis, to obtain some important information for the conclusion on certain respiratory problems Mestrado Matemática Aplicada Mestre em Matemática Aplicada CNPQ 132100/2018-5

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.

In this paper, we study two mathematical models, involving delay differential equations, which describe the processes of erythropoiesis and leukopoiesis in the case of maintenance therapy for acute lymphoblastic leukemia. All types of possible equilibrium points were determined, and their stability was analyzed. For some of the equilibrium points, conditions for parameters that imply stability were obtained. When this was not feasible, due to the complexity of the characteristic equation, we discuss the stability through numerical simulations. An important part of the stability study for each model is the examination of the critical case of a zero root of the characteristic equation. The mathematical results are accompanied by biological interpretations.

Capelin (Mallotus villosus) is a short-lived (1-4 years) fish species, that plays a crucial role by dominating the intermediate trophic level in the Barents Sea. Several episodes of extreme biomass decline (collapse) have been observed during the last three decades. We postulate that these collapses might be regulated by food availability (bottom-up effect) and/or by time discrepancy between capelin feeding and abundance of its prey (match-mismatch hypothesis). This paper investigates our postulate using a model consisting of a set of coupled differential equations to describe the predator-prey system, with a single delay term, τ , in description of the predator dynamics. We derive theoretical conditions on τ , as well as determine how changes in these conditions define different stability regimes of the system. Unconstrained optimization is used to calculate optimal model parameters by fitting the predator-prey model to empirical data. The optimization results are combined with those from the theoretical analysis, to make inference about the empirical system stability. Our results show that Hopf bifurcation occurs in the predatory-prey system when τ exceeds a theoretically derived value τ ∗ > 0 . This value represents the critical time for prey availability in advance of the optimal predator growth period.Set into an ecological context, our findings provide mathematical evidence for validity of the match-mismatch hypothesis and a bottom-up effect for capelin.

BackgroundSimeoni and colleagues introduced a compartmental model for tumor growth that has proved quite successful in modeling experimental therapeutic regimens in oncology. The model is based on a system of ordinary differential equations (ODEs), and accommodates a lag in therapeutic action through delay compartments. There is some ambiguity in the appropriate number of delay compartments, which we examine in this note.MethodsWe devised an explicit delay differential equation model that reflects the main features of the Simeoni ODE model. We evaluated the original Simeoni model and this adaptation with a sample data set of mammary tumor growth in the FVB/N-Tg(MMTVneu)202Mul/J mouse model.ResultsThe experimental data evinced tumor growth heterogeneity and inter-individual diversity in response, which could be accommodated statistically through mixed models. We found little difference in goodness of fit between the original Simeoni model and the delay differential equation model relative to the sample data set.ConclusionsOne should exercise caution if asserting a particular mathematical model uniquely characterizes tumor growth curve data. The Simeoni ODE model of tumor growth is not unique in that alternative models can provide equivalent representations of tumor growth.

[eng] This thesis deals with the jet transport for numerical integrators and the effective invariant object computation of delay differential equations. Firstly we study how automatic differentiation (AD) affects when they are applied to numerical integrators of ordinary differential equations (ODEs). We prove that the use of AD is exactly the same as considering the initial ODE and add new equations to the calculation of the variational flow up to a certain order. With this result we propose to detail the effective computation when these equations are affected by a delay. In particular, the computation of the stability of equilibrium points, the computation of periodic orbits as well as their stability and continuation. Similarly the computation of quasi-orbits periodic and its stability. For such computations, we avoid the explicit generation of the Jacobian matrix and we only require the matrix-vector evaluation. Finally, we cover the existence, uniqueness and numerical computation of the slowest direction of the stable manifold of a limit cycle of a state-dependent delay equation differential. The results are formulated in a posteriori format, which leads to rigorous proofs of numerical experiments. Specifically our result is applicable when you have a delayed perturbation and it depends on the state of an ODE in the plane.

We address the problem of the numerical bifurcation analysis of general nonlinear delay equations, including integral and integro-differential equations, for which no software is currently available. Pseudospectral discretization is applied to the abstract reformulation of equations with infinite delay to obtain a finite dimensional system of ordinary differential equations, whose properties can be numerically studied with well-developed software. We explore the applicability of the method on some test problems and provide some numerical evidence of the convergence of the approximations.

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. Ministerio de Economía y Competitividad Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía)

International audience; During the primary CD8 T cell immune response, CD8 T cells undergo proliferation and continuous differentiation, acquiring cytotoxic abilities to address the infection and generate an immune memory. At the end of the response, the remaining CD8 T cells are antigen-specific memory cells that will respond stronger and faster in case they are presented this very same antigen again. We propose a nonlinear multiscale mathematical model of the CD8 T cell immune response describing dynamics of two inter-connected physical scales. At the intracellular scale, the level of expression of key proteins involved in proliferation, death, and differentiation of CD8 T cells is modeled by a delay differential system whose dynamics define maturation velocities of CD8 T cells. At the population scale, the amount of CD8 T cells is represented by a discrete density and cell fate depends on their intracellular content. We introduce the model, then show essential mathematical properties (existence, uniqueness, positivity) of solutions and analyse their asymptotic behavior based on the behavior of the intracellular regulatory network. We numerically illustrate the model's ability to qualitatively reproduce both primary and secondary responses, providing a preliminary tool for investigating the generation of long-lived CD8 memory T cells and vaccine design.

A recent pseudospectral collocation is used to reduce a nonlinear delay differential equation to a system of ordinary differential equations. Standard methods are then applied to compute Lyapunov exponents. The validity of this simple approach is shown experimentally. Matlab codes are also included.

In the paper we consider a system of delay differential equations describing the interaction between predator and prey populations living on the same territory. The system consists of three equations, herewith the components of solutions characterize the number of individuals of prey population, the number of adult predators, and the number of juvenile predators. It is assumed that only adult predators can attack the individuals of prey population and reproduce. The delay parameter is assumed to be constant and denotes the time that the predators need to become adult. For the system we consider the initial value problem, for which it is discussed the existence, uniqueness, nonnegativity, and boundedness of solutions. It is also discussed the stability of stationary solutions (equilibrium points) corresponding to complete extinction of populations, extinction of only predator populations and coexistence of predator and prey populations. The main attention in the paper is paid to obtaining estimates of solutions characterizing the rate of convergence to the equilibrium point corresponding to the coexistence of populations and the establishment of estimates for the attraction set, i.e. the admissible conditions for the initial data under which the convergence takes place. When obtaining the re- sults, we use the method of Lyapunov–Krasovskii functionals, which is an analogue of the method of Lyapunov functions for ordinary differential equations. Herewith in the paper it is significantly used the modified Lyapunov–Krasovskii functional proposed by G.V. Demidenko and I.I. Matveeva. It is important to note that this functional allows to obtain estimates of solutions to delay systems, which are analogues of the Krein’s estimate for ordinary differential equations, and the construction of such functional is reduced to solving well-conditioned problems.

In this paper we consider a class of delay differential equations of the form $\dot{x}(t)=\alpha (t) h(x(t-\tau), x(t-\sigma))-\beta(t)f(x(t))$, where $h$ is a mixed monotone function. Sufficient conditions are presented for the permanence of the positive solutions. Our results give also lower and upper estimates of the limit inferior and the limit superior of the solutions via a special solution of an associated nonlinear system of algebraic equations. The results are generated to a more general class of delay differential equations with mixed monotonicity.

We are concerned with a class of $n$-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of delay differential equations. Sufficient conditions for the exponential asymptotic stability of the linear system are established. By using this stability, the permanence of the perturbed nonlinear system is studied. Under more restrictive constraints on the coefficients, the system has a cooperative type behaviour, in which case explicit uniform lower and upper bounds for the solutions are obtained. As an illustration, the asymptotic behaviour of a non-autonomous Nicholson system with distributed delays is analysed.

It is noteworthy to observe that a first-order linear ordinary differential equation without delay does not possess oscillatory solutions. Therefore the investigation of oscillatory solutions is of interest for equations with delays or for the discrete analogue difference equations. Furthermore, the mathematical modelling of several real-world problems leads to differential equations that depend on the past history rather than only the current state. In this article conditions are presented such that all solutions of delay and difference equations are oscillatory while all solutions of the corresponding ordinary differential equations without delay are, for example, decreasing and tend to zero. Equations with constant and variable arguments are investigated. Several examples of delay and difference equations with applications to many sectors of life are presented.

In this paper, new oscillatory properties for fourth-order delay differential equations with p-Laplacian-like operators are established, using the Riccati transformation and comparison method. Moreover, our results are an extension and complement to previous results in the literature. We provide some examples to examine the applicability of our results.

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.

The aim of this work is to present new oscillation results for a class of second-order delay differential equations with damping term. The new criterion of oscillation depends on improving the asymptotic properties of the positive solutions of the studied equation by using an iterative technique. Our results extend some of the results recently published in the literature.

Singularly perturbed boundary value problems with negative shift parameter are special types of differential difference equations whose solution exhibits boundary layer behaviour. A simple but novel numerical method is developed to approximate the numerical solution of the problems of these types. The method gives accurate solutions for h≥ε in the inner region of the boundary layer where other classical numerical methods fail to give smooth solution. The present method is proved to be point-wise uniformly convergent with second order rate of convergence., Finite difference method; Delay differential equations; Singular perturbation problem

The aim of the present study is to present a new model based on the nonlinear singular second order delay differential equation of Lane–Emden type and numerically solved by using the heuristic technique. Four different examples are presented based on the designed model and numerically solved by using artificial neural networks optimized by the global search, local search methods and their hybrid combinations, respectively, named as genetic algorithm (GA), sequential quadratic programming (SQP) and GA-SQP. The numerical results of the designed model are compared for the proposed heuristic technique with the exact/explicit results that demonstrate the performance and correctness. Moreover, statistical investigations/assessments are presented for the accuracy and performance of the designed model implemented with heuristic methodology. This paper has been partially supported by Ministerio de Ciencia, Innovación y Universidades, Spain grant number PGC2018-0971-B-100 and Fundación Séneca de la Región de Murcia, Spain grant number 20783/PI/18.

In this paper, we study the non-monotonic behavior of the solutions of second-order delay differential equations. We establish criteria such that every positive solution is non- monotonic, on a bounded interval. While there has been previous studies on non-monotone solutions, the results are the first to be reported, for functional differential equations. In the final section, examples illustrating the significance of our results are also given.

We study stability and Hopf bifurcation analysis of a model that refers to the competition between the immune system and an aggressive host such as a tumor. The model which describes this competition is governed by a reaction-diffusion system including time delay under the Neumann boundary conditions, and is based on Kuznetsov-Taylor's model. Choosing the delay parameter as a bifurcation parameter, we first show that Hopf bifurcation occurs. Second, we determine two properties of the periodic solution, namely its direction and stability, by applying the normal form theory and the center manifold reduction for partial functional differential equations. Furthermore, we discuss the effects of diffusion on the dynamics by analyzing a model with constant coefficients and perform some numerical simulations to support the analytical results. The results show that diffusion has an important effects on the dynamics of a mathematical model.

White matter pathways form a complex network of myelinated axons that regulate signal transmission in the nervous system and play a key role in behaviour and cognition. Recent evidence reveals that white matter networks are adaptive and that myelin remodels itself in an activity-dependent way, during both developmental stages and later on through behaviour and learning. As a result, axonal conduction delays continuously adjust in order to regulate the timing of neural signals propagating between different brain areas. This delay plasticity mechanism has yet to be integrated in computational neural models, where conduction delays are oftentimes constant or simply ignored. As a first approach to adaptive white matter remodeling, we modified the canonical Kuramoto model by enabling all connections with adaptive, phase-dependent delays. We analyzed the equilibria and stability of this system, and applied our results to two-oscillator and large-dimensional networks. Our joint mathematical and numerical analysis demonstrates that plastic delays act as a stabilizing mechanism promoting the network’s ability to maintain synchronous activity. Our work also shows that global synchronization is more resilient to perturbations and injury towards network architecture. Our results provide key insights about the analysis and potential significance of activity-dependent myelination in large-scale brain synchrony.

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

Using a simple delay differential equation model we study theoretically the dynamics of a unidirectional class-A ring laser with a nonlinear amplifying loop mirror. We perform analytical linear stability analysis of the CW regimes in the large delay limit and demonstrate that these regimes can be destabilized via modulational and Turing-type instabilities, as well as by a bifurcation leading to the appearance of square-waves. We investigate the formation of square-waves and mode-locked pulses in the system. We show that mode-locked pulses are very asymmetric with exponential decay of the trailing and superexponential growth of the leading edge. We discuss asymmetric interaction of these pulses leading to a formation of harmonic mode-locked regimes.

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.

Inthisarticle,wewilldiscusstheapplicationoftheLyapunov’ssecondmethodandfixedpointtheoriestocertaindifferential equations of first and second order. First, we will introduce some basic information about these subjects, and later, we give their applications concerning some specific attitude of Solutions of Delay Differential Equations. We will also do a comparison between them. Include keywords, mathematical subject classification numbers as needed

There are almost no results in mathematical literature on the exponential stability of third-order delay differential equations. One of the main purposes of the paper is to fill this gap. We propose an approach to the study of stability for third-order delay differential equations. On the basis of these results, new possibilities of stabilization by delay feedback input control are proposed.