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11 pages, 496 KiB

Open AccessArticle

Kuramoto Model with Delay: The Role of the Frequency Distribution

by Vladimir V. Klinshov and Alexander A. Zlobin

Mathematics 2023, 11(10), 2325; https://doi.org/10.3390/math11102325 - 16 May 2023

Cited by 4 | Viewed by 2164

The Kuramoto model is a classical model used for the describing of synchronization in populations of oscillatory units. In the present paper we study the Kuramoto model with delay with a focus on the distribution of the oscillators’ frequencies. We consider a series [...] Read more.

The Kuramoto model is a classical model used for the describing of synchronization in populations of oscillatory units. In the present paper we study the Kuramoto model with delay with a focus on the distribution of the oscillators’ frequencies. We consider a series of rational distributions which allow us to reduce the population dynamics to a set of several delay differential equations. We use the bifurcation analysis of these equations to study the transition from the asynchronous to synchronous state. We demonstrate that the form of the frequency distribution may play a substantial role in synchronization. In particular, for Lorentzian distribution the delay prevents synchronization, while for other distributions the delay can facilitate synchronization. Full article

(This article belongs to the Special Issue Advances in Delay Differential Equations)

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18 pages, 378 KiB

Open AccessArticle

Relaxation Oscillations in the Logistic Equation with Delay and Modified Nonlinearity

by Alexandra Kashchenko and Sergey Kashchenko

Mathematics 2023, 11(7), 1699; https://doi.org/10.3390/math11071699 - 2 Apr 2023

Viewed by 1252

Abstract

We consider the dynamics of a logistic equation with delays and modified nonlinearity, the role of which is to bound the values of solutions from above. First, the local dynamics in the neighborhood of the equilibrium state are studied using standard bifurcation methods. [...] Read more.

We consider the dynamics of a logistic equation with delays and modified nonlinearity, the role of which is to bound the values of solutions from above. First, the local dynamics in the neighborhood of the equilibrium state are studied using standard bifurcation methods. Most of the paper is devoted to the study of nonlocal dynamics for sufficiently large values of the ‘Malthusian’ coefficient. In this case, the initial equation is singularly perturbed. The research technique is based on the selection of special sets in the phase space and further study of the asymptotics of all solutions from these sets. We demonstrate that, for sufficiently large values of the Malthusian coefficient, a ‘stepping’ of periodic solutions is observed, and their asymptotics are constructed. In the case of two delays, it is established that there is attractor in the phase space of the initial equation, whose dynamics are described by special nonlinear finite-dimensional mapping. Full article

(This article belongs to the Special Issue Advances in Delay Differential Equations)

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14 pages, 301 KiB

Open AccessArticle

Improved Properties of Positive Solutions of Higher Order Differential Equations and Their Applications in Oscillation Theory

by Barakah Almarri and Osama Moaaz

Mathematics 2023, 11(4), 924; https://doi.org/10.3390/math11040924 - 11 Feb 2023

Cited by 1 | Viewed by 1066

Abstract

In this article, we present new criteria for testing the oscillation of solutions of higher-order neutral delay differential equation. By deriving new monotonic properties of a class of the positive solutions of the studied equation, we establish better criteria for oscillation. Furthermore, we [...] Read more.

In this article, we present new criteria for testing the oscillation of solutions of higher-order neutral delay differential equation. By deriving new monotonic properties of a class of the positive solutions of the studied equation, we establish better criteria for oscillation. Furthermore, we improve these properties by giving them an iterative character, allowing us to apply the criteria more than once. The results obtained in this paper are characterized by the fact that they do not require the existence of unknown functions and do not need the commutation condition to composition of the delay functions, which are necessary conditions for the previous related results. Full article

(This article belongs to the Special Issue Advances in Delay Differential Equations)

24 pages, 720 KiB

Open AccessArticle

Forecasting the Effect of Pre-Exposure Prophylaxis (PrEP) on HIV Propagation with a System of Differential–Difference Equations with Delay

by Mostafa Adimy, Julien Molina, Laurent Pujo-Menjouet, Grégoire Ranson and Jianhong Wu

Mathematics 2022, 10(21), 4093; https://doi.org/10.3390/math10214093 - 2 Nov 2022

Cited by 2 | Viewed by 1868

Abstract

The HIV/AIDS epidemic is still active worldwide with no existing definitive cure. Based on the WHO recommendations stated in 2014, a treatment, called Pre-Exposure Prophylaxis (PrEP), has been used in the world, and more particularly in France since 2016, to prevent HIV infections. [...] Read more.

The HIV/AIDS epidemic is still active worldwide with no existing definitive cure. Based on the WHO recommendations stated in 2014, a treatment, called Pre-Exposure Prophylaxis (PrEP), has been used in the world, and more particularly in France since 2016, to prevent HIV infections. In this paper, we propose a new compartmental epidemiological model with a limited protection time offered by this new treatment. We describe the PrEP compartment with an age-structure hyperbolic equation and introduce a differential equation on the parameter that governs the PrEP starting process. This leads us to a nonlinear differential–difference system with discrete delay. After a local stability analysis, we prove the global behavior of the system. Finally, we illustrate the solutions with numerical simulations based on the data of the French Men who have Sex with Men (MSM) population. We show that the choice of a logistic time dynamics combined with our Hill-function-like model leads to a perfect data fit. These results enable us to forecast the evolution of the HIV epidemics in France if the populations keep using PrEP. Full article

(This article belongs to the Special Issue Advances in Delay Differential Equations)

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16 pages, 759 KiB

Open AccessArticle

Asymptotics of Solutions to a Differential Equation with Delay and Nonlinearity Having Simple Behaviour at Infinity

by Alexandra Kashchenko

Mathematics 2022, 10(18), 3360; https://doi.org/10.3390/math10183360 - 16 Sep 2022

Cited by 4 | Viewed by 1363

Abstract

In this paper, we study nonlocal dynamics of a nonlinear delay differential equation. This equation with different types of nonlinearities appears in medical, physical, biological, and ecological applications. The type of nonlinearity in this paper is a generalization of two important for applications [...] Read more.

In this paper, we study nonlocal dynamics of a nonlinear delay differential equation. This equation with different types of nonlinearities appears in medical, physical, biological, and ecological applications. The type of nonlinearity in this paper is a generalization of two important for applications types of nonlinearities: piecewise constant and compactly supported functions. We study asymptotics of solutions under the condition that nonlinearity is multiplied by a large parameter. We construct all solutions of the equation with initial conditions from a wide subset of the phase space and find conditions on the parameters of equations for having periodic solutions. Full article

(This article belongs to the Special Issue Advances in Delay Differential Equations)

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21 pages, 399 KiB

Open AccessArticle

Mathematical Modeling and Stability Analysis of a Delayed Carbon Absorption-Emission Model Associated with China’s Adjustment of Industrial Structure

by Leilei Han, Haokun Sui and Yuting Ding

Mathematics 2022, 10(17), 3089; https://doi.org/10.3390/math10173089 - 27 Aug 2022

Cited by 1 | Viewed by 2228

Abstract

Global warming has brought about enormous damage, therefore, some scholars have begun to conduct in-depth research on peak carbon dioxide emissions and carbon neutrality. In this paper, based on the background of China’s upgrading industrial structure and energy structure, we establish a delayed [...] Read more.

Global warming has brought about enormous damage, therefore, some scholars have begun to conduct in-depth research on peak carbon dioxide emissions and carbon neutrality. In this paper, based on the background of China’s upgrading industrial structure and energy structure, we establish a delayed two-dimensional differential equation model associated with China’s adjustment of industrial structure. Firstly, we analyze the existence of the equilibrium for the model. We also analyze the characteristic roots of the characteristic equation at each equilibrium point for the model, then, we analyze the stability of the equilibrium point for the model according to the characteristic root, and discuss the existence of Hopf bifurcation of the system by using bifurcation theory. Secondly, we derive the normal form of Hopf bifurcation by using the multiple time scales method. Then, through the official real data, we present the range of some parameters in the model, and determine a set of parameters by reasonable analysis. The validity of the theoretical results is verified by numerical simulations. Finally, we use the real data to forecast the time of peak carbon dioxide emissions and carbon neutralization. Eventually, we put forward some suggestions based on the current situation of carbon emission and absorption in China, such as planting trees to increase the growth rate of carbon absorption, deepening industrial reform and optimizing energy structure to reduce carbon emissions. Full article

(This article belongs to the Special Issue Advances in Delay Differential Equations)

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32 pages, 446 KiB

Open AccessArticle

Quasinormal Forms for Chains of Coupled Logistic Equations with Delay

by Sergey Kashchenko

Mathematics 2022, 10(15), 2648; https://doi.org/10.3390/math10152648 - 28 Jul 2022

Cited by 5 | Viewed by 948

Abstract

In this paper, chains of coupled logistic equations with delay are considered, and the local dynamics of these chains is investigated. A basic assumption is that the number of elements in the chain is large enough. This implies that the study of the [...] Read more.

In this paper, chains of coupled logistic equations with delay are considered, and the local dynamics of these chains is investigated. A basic assumption is that the number of elements in the chain is large enough. This implies that the study of the original systems can be reduced to the study of a distributed integro–differential boundary value problem that is continuous with respect to the spatial variable. Three types of couplings of greatest interest are considered: diffusion, unidirectional, and fully connected. It is shown that the critical cases in the stability of the equilibrium state have an infinite dimension: infinitely many roots of the characteristic equation tend to the imaginary axis as the small parameter tends to zero, which characterizes the inverse of the number of elements of the chain. In the study of local dynamics in cases close to critical, analogues of normal forms are constructed, namely quasinormal forms, which are boundary value problems of Ginzburg–Landau type or, as in the case of fully connected systems, special nonlinear integro–differential equations. It is shown that the nonlocal solutions of the obtained quasinormal forms determine the principal terms of the asymptotics of solutions to the original problem from a small neighborhood of the equilibrium state. Full article

(This article belongs to the Special Issue Advances in Delay Differential Equations)

19 pages, 1193 KiB

Open AccessArticle

An Epidemic Model with Time Delay Determined by the Disease Duration

by Samiran Ghosh, Vitaly Volpert and Malay Banerjee

Mathematics 2022, 10(15), 2561; https://doi.org/10.3390/math10152561 - 22 Jul 2022

Cited by 14 | Viewed by 3467

Abstract

Immuno-epidemiological models with distributed recovery and death rates can describe the epidemic progression more precisely than conventional compartmental models. However, the required immunological data to estimate the distributed recovery and death rates are not easily available. An epidemic model with time delay is [...] Read more.

Immuno-epidemiological models with distributed recovery and death rates can describe the epidemic progression more precisely than conventional compartmental models. However, the required immunological data to estimate the distributed recovery and death rates are not easily available. An epidemic model with time delay is derived from the previously developed model with distributed recovery and death rates, which does not require precise immunological data. The resulting generic model describes epidemic progression using two parameters, disease transmission rate and disease duration. The disease duration is incorporated as a delay parameter. Various epidemic characteristics of the delay model, namely the basic reproduction number, the maximal number of infected, and the final size of the epidemic are derived. The estimation of disease duration is studied with the help of real data for COVID-19. The delay model gives a good approximation of the COVID-19 data and of the more detailed model with distributed parameters. Full article

(This article belongs to the Special Issue Advances in Delay Differential Equations)

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15 pages, 288 KiB

Open AccessArticle

Laplace Transform and Semi-Hyers–Ulam–Rassias Stability of Some Delay Differential Equations

by Daniela Marian

Mathematics 2021, 9(24), 3260; https://doi.org/10.3390/math9243260 - 15 Dec 2021

Cited by 5 | Viewed by 1839

Abstract

In this paper, we study semi-Hyers–Ulam–Rassias stability and generalized semi-Hyers–Ulam–Rassias stability of differential equations

x^{'} (t) + x (t - 1) = f (t)

and [...] Read more.

In this paper, we study semi-Hyers–Ulam–Rassias stability and generalized semi-Hyers–Ulam–Rassias stability of differential equations

x^{'} (t) + x (t - 1) = f (t)

and

x^{″} (t) + x^{'} (t - 1) = f (t), x (t) = 0 if t \leq 0,

using the Laplace transform. Our results complete those obtained by S. M. Jung and J. Brzdek for the equation

x^{'} (t) + x (t - 1) = 0

. Full article

(This article belongs to the Special Issue Advances in Delay Differential Equations)

17 pages, 282 KiB

Open AccessArticle

Ulam Stability of n-th Order Delay Integro-Differential Equations

by Shuyi Wang and Fanwei Meng

Mathematics 2021, 9(23), 3029; https://doi.org/10.3390/math9233029 - 26 Nov 2021

Cited by 1 | Viewed by 1532

Abstract

In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the [...] Read more.

In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the expression of the solution for delay integro-differential equation is derived by mathematical induction. On this basis, we obtain the Ulam stability of the delay integro-differential equation via Gronwall–Bellman inequality. Finally, two examples of delay integro-differential equations are given to explain our main results. Full article

(This article belongs to the Special Issue Advances in Delay Differential Equations)

13 pages, 251 KiB

Open AccessArticle

On a Coupled System of Random and Stochastic Nonlinear Differential Equations with Coupled Nonlocal Random and Stochastic Nonlinear Integral Conditions

by Ahmed M. A. El-Sayed and Hoda A. Fouad

Mathematics 2021, 9(17), 2111; https://doi.org/10.3390/math9172111 - 1 Sep 2021

Cited by 5 | Viewed by 1583

Abstract

It is well known that Stochastic equations had many useful applications in describing numerous events and problems of real world, and the nonlocal integral condition is important in physics, finance and engineering. Here we are concerned with two problems of a coupled system [...] Read more.

It is well known that Stochastic equations had many useful applications in describing numerous events and problems of real world, and the nonlocal integral condition is important in physics, finance and engineering. Here we are concerned with two problems of a coupled system of random and stochastic nonlinear differential equations with two coupled systems of nonlinear nonlocal random and stochastic integral conditions. The existence of solutions will be studied. The sufficient condition for the uniqueness of the solution will be given. The continuous dependence of the unique solution on the nonlocal conditions will be proved. Full article

(This article belongs to the Special Issue Advances in Delay Differential Equations)

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