Delay Differential Systems Research Articles

Stability and bifurcation of a delayed Aron–May model for malaria transmission

In this paper, the dynamical behavior in a delayed Aron–May model for malaria transmission is investigated. The basic reproduction number is defined. The global stability of the malaria-free equilibrium (MFE) is established. By using the Bendixson theorem, a sufficient condition for the global stability of the delay-free equilibrium (DFE) is also established. Furthermore, to deal with the local stability of endemic equilibrium (EE), by means of the stability switches analysis method proposed in [E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33(5) (2002) 1144–1165.] the related characteristic equation at EE is investigated, and the occurrence of Hopf bifurcation is discussed by using the incubation period in mosquito as a bifurcation parameter. Last, the simulation analysis is also performed to verify the dynamical behavior of the model.

Design of Nonlinear Delay Differential System for Analyzing Vulnerabilities in Nanoscale Hardware Implants: A Deep Dive into Intelligent Computing Networks

Design of Nonlinear Delay Differential System for Analyzing Vulnerabilities in Nanoscale Hardware Implants: A Deep Dive into Intelligent Computing Networks

Stability and convergence rate analysis of continuous-time delay-difference systems with both point and distributed delays

This paper studies the exponential stability and convergence rate analysis of continuous-time delay-difference systems. Firstly, stability and convergence rate analysis of delay-difference systems with both point delays and distributed delays having exponential integral kernels are studied by using the weighted Lyapunov–Krasovskii functionals (LKFs) approach and the state transformation approach, respectively. Different sufficient stability conditions expressed by linear matrix inequalities (LMIs) are presented. Secondly, as a particular case, LMIs based conditions and spectral radius based conditions are given to ensure the exponential stability with a guaranteed convergence rate for delay-difference systems with both point delays and distributed delays having constant integral kernels, respectively. Finally, numerical examples illustrate the effectiveness of the obtained results.

On exponential and L2-exponential stability of continuous-time delay-difference systems

In this paper, the exponential stability (ES) and L2-exponential stability (L2-ES) of continuous-time delay-difference systems are studied. Firstly, the relationship between ES and L2-ES of the studied system is systematically presented. Secondly, a novel Lyapunov stability theorem is proposed to test both the ES and L2-ES with a guaranteed convergence rate of the system. Then, for a particular class of delay-difference systems with both point delays and distributed delays having exponential integral kernels, some stability criteria based on linear matrix inequalities (LMIs) are established by selecting suitable Lyapunov-Krasovskii functionals (LKFs) and using a delay decomposition technique. Finally, a numerical example is worked out to illustrate the effectiveness of the theoretical results.

Persistency and stability of a class of nonlinear forced positive discrete-time systems with delays

Persistence, excitability and stability properties are considered for a class of nonlinear, forced, positive discrete-time systems with delays. As will be illustrated, these equations arise in a number of biological and ecological contexts. Novel sufficient conditions for persistence, excitability and stability are presented. Further, similarities and differences between the delayed equations considered presently and their corresponding undelayed versions are explored, and some striking differences are noted. It is shown that recent results for a corresponding class of positive, nonlinear delay-differential (continuous-time) systems do not carry over to the discrete-time setting. Detailed discussion of three examples from population dynamics is provided.

Advanced stability analysis of a fractional delay differential system with stochastic phenomena using spectral collocation method.

In recent years, there has been a growing interest in incorporating fractional calculus into stochastic delay systems due to its ability to model complex phenomena with uncertainties and memory effects. The fractional stochastic delay differential equations are conventional in modeling such complex dynamical systems around various applied fields. The present study addresses a novel spectral approach to demonstrate the stability behavior and numerical solution of the systems characterized by stochasticity along with fractional derivatives and time delay. By bridging the gap between fractional calculus, stochastic processes, and spectral analysis, this work contributes to the field of fractional dynamics and enriches the toolbox of analytical tools available for investigating the stability of systems with delays and uncertainties. To illustrate the practical implications and validate the theoretical findings of our approach, some numerical simulations are presented.

Boosting reservoir computer performance with multiple delays.

Time delays play a significant role in dynamical systems, as they affect their transient behavior and the dimensionality of their attractors. The number, values, and spacing of these time delays influences the eigenvalues of a nonlinear delay-differential system at its fixed point. Here we explore a multidelay system as the core computational element of a reservoir computer making predictions on its input in the usual regime close to fixed point instability. Variations in the number and separation of time delays are first examined to determine the effect of such parameters of the delay distribution on the effectiveness of time-delay reservoirs for nonlinear time series prediction. We demonstrate computationally that an optoelectronic device with multiple different delays can improve the mapping of scalar input into higher-dimensional dynamics, and thus its memory and prediction capabilities for input time series generated by low- and high-dimensional dynamical systems. In particular, this enhances the suitability of such reservoir computers for predicting input data with temporal correlations. Additionally, we highlight the pronounced harmful resonance condition for reservoir computing when using an electro-optic oscillator model with multiple delays. We illustrate that the resonance point may shift depending on the task at hand, such as cross prediction or multistep ahead prediction, in both single delay and multiple delay cases.

Relative Controllability and Hyers–Ulam Stability of Riemann–Liouville Fractional Delay Differential System

Relative Controllability and Hyers–Ulam Stability of Riemann–Liouville Fractional Delay Differential System

Existence and exponential stability of a periodic solution of an infinite delay differential system with applications to Cohen–Grossberg neural networks

In the present paper, we investigate both the global exponential stability and the existence of a periodic solution of a general differential equation with unbounded distributed delays. The main stability criterion depends on the dominance of the non-delay terms over the delay terms. The criterion for the existence of a periodic solution is obtained with the application of the coincidence degree theorem. We use the main results to get criteria for the existence and global exponential stability of periodic solutions of a generalized higher-order periodic Cohen–Grossberg neural network model with discrete-time varying delays and infinite distributed delays. Additionally, we provide a comparison with the results in the literature and two numerical examples to illustrate the effectiveness of some of our results.

A stochastic scale conjugate neural network procedure for the SIRC epidemic delay differential system

In this study, a stochastic computing structure is provided for the numerical solutions of the SIRC epidemic delay differential model, i.e. SIRC-EDDM using the dynamics of the COVID-19. The design of the scale conjugate gradient (CG) neural networks (SCGNNs) is presented for the numerical treatment of SIRC-EDDM. The mathematical model is divided into susceptible S ( ρ ) , recovered R ( ρ ) , infected I ( ρ ) , and cross-immune C ( ρ ) , while the numerical performances have been provided into three different cases. The exactitude of the SCGNNs is perceived through the comparison of the accomplished and reference outcomes (Runge-Kutta scheme) and the negligible absolute error (AE) that are performed around 10−06 to 10−08 for each case of the SIRC-EDDM. The obtained results have been presented to reduce the mean square error (MSE) using the performances of train, validation, and test data. The neuron analysis is also performed that shows the AE by taking 14 neurons provide more accurateness as compared to 4 numbers of neurons. To check the proficiency of SCGNNs, the comprehensive studies are accessible using the error histograms (EHs) investigations, state transitions (STs) values, MSE performances, regression measures, and correlation.

Boundedness and stability of nonlinear hybrid neutral stochastic delay differential equation with Lévy jumps under different structures

This paper investigates the boundedness and stability of a class of nonlinear hybrid neutral stochastic differential delay systems with Lévy jumps and different structures. The coefficients in this system satisfy the local Lipschitz condition and a suitable Khasminskii-type condition, and the state space of the system is separated into two subsets, the existence uniqueness, asymptotic boundedness, and exponential stability of the system are obtained by designing a new Lyapunov function and applying the M-matrix technique as well as dealing with the non-differentiable delay function. Different with the existing work, we not only consider the neutral term, but also the case of the delay function being bounded and non-differentiable. At last, numerical examples are performed to manifest the obtained results.

Hyers–Ulam Stability of Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

In this paper, we explore the stability of a new class of Caputo-type fractional stochastic delay differential systems with Poisson jumps. We prove the Hyers–Ulam stability of the solution by utilizing a version of fixed point theorem, fractional calculus, Cauchy–Schwartz inequality, Jensen inequality, and some stochastic analysis techniques. Finally, an example is provided to illustrate the effectiveness of the results.

Delay‐independent conditions of exponential ISS for linear time‐varying delay differential systems

AbstractThis work focuses on linear continuous‐time systems with time‐varying delays, in the general case of time‐varying matrices. Initially, we explore the case of positive systems within this class and introduce two distinct delay‐independent conditions of exponential input‐to‐state stability. Moreover, we provide guaranteed exponential convergence rates for such conditions, explicitly stating the ISS gains. Then, we extend our analysis to systems without sign constraints, adopting a state‐bounding approach that takes advantage of the properties of positive systems. Due to the time‐varying nature of the systems, all proposed conditions are in terms of an infinite number of inequalities. Hence, implementation issues are discussed and significant special cases in which the conditions can be cast into a linear programming problem of finite dimension are presented.

Symmetries of differential delay systems with applications to observer design

AbstractIn this paper we construct a general observer theory for differential delay systems based on different types of symmetries (exact symmetry or asymptotic symmetry), ending up with a certain number of semi‐global and global observers, with bounded or unbounded system's solutions. We introduce the notions of symmetry for a differential delay system, being inspired by well‐known definitions of symmetry for an ordinary or partial differential system, and variational symmetry for the associated variational differential delay system. We illustrate observer design procedures in details, by proving that the existence of a (variational asymptotic) symmetry with system's detectability in the first approximation have a central role in the design of a state observer. The symmetry is a one‐parameter group of transformations which maps the system into itself (exact symmetry) or into a different system (asymptotic symmetry), approximating the original one with better and better accuracy as the parameter of the symmetry is larger. The types of symmetries we consider here show an important contractive action on the state and input spaces for which the system's solutions are mapped into arbitrarily small neighbourhoods of the origin in which the transformed system can be well‐approximated by its linearization. The parameter of the symmetry may be constant (semiglobal observers) or updated on‐line by a state norm estimator (global observers).

Novel neuro-stochastic adaptive supervised learning for numerical treatment of nonlinear epidemic delay differential system with impact of double diseases

ABSTRACT This paper presents a novel neuro-stochastic adaptive processing to investigate the dynamic behavior of the nonlinear SIS epidemic delayed model with the impact of double disease (SIS-EDDM) using the knack of prediction and modeling of artificial neural networks (ANNs) optimized through Levenberg-Marquardt technique (LMT), i.e. ANNs-LMT. The model captures the dynamic progression of individuals in the population, differentiating between those susceptible to infection and affected by two unique viruses. The inclusion of time delays in the differential equations introduces a critical temporal aspect to the model, enhancing its precision in portraying the transmutation process. The reference dataset for ANNs-LMT is produced using the explicit Runge-Kutta method, with variations in the transmission coefficients between susceptible and infective populations, the death rate of infective population caused by diseases, the cure rate of treatment for infected population, vaccination proportional coefficient for the susceptible population, white noise of the environment, and time delay. The designed computing framework of ANNs-LMT is used to determine the numerical solutions of the variants of SIS-EDDM by incorporating the training, testing, and validation samples based adaptive learning processing. The neuro-stochastic adaptive processing of ANNs-LMT is validated by minimal absolute and mean squared errors, coupled with the attainment of nearly optimal regression measures, demonstrating its accuracy and effectiveness.

The Existence and Averaging Principle for Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo-type fractional stochastic delay differential systems(FSDDSs) with Poisson jumps by utilizing the delayed perturbation of the Mittag–Leffler function. Moreover, by using the Burkholder–Davis–Gundy inequality, Doob’s martingale inequality, and Hölder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of Lp. Some known results in the literature are extended.

Uniform Ultimate Boundedness Results for some System of Third Order Nonlinear Delay Differential Equations

Lyapunov functional, third-order delay differential system, Boundedness.

A Study of Multi-Term Time-Fractional Delay Differential System with Monotonic Conditions

In this paper, the existence and uniqueness of mild solution for a class of multi-term time-fractional delay differential system have been discussed in ordered Banach space by enforcing monotone iterative technique. The generalized semigroup theory, fractional calculus and measure of noncompactness have been implemented to obtain the required results. A new set of sufficient conditions with the coefficients in the equations satisfying some monotonic properties has been obtained. Finally, an application is given to illustrate the obtained results.

Analysis of Functional and Neutral Differential Equations via Lyapunov Functionals

We employ Lyapunov types functions and functionals and obtain sufficient conditions that guarantee the boundedness and the exponential decay of solutions, stability and exponential stability of the zero solution in nonlinear delay and neutral differential systems. The theory is illustrated with several examples. 2000 Mathematics Subject Classification: Primary 39A13, 39A23; Secondary 34K42

Stability of conformable fractional delay differential systems with impulses

This paper is concerned with the stability of conformable fractional delay differential systems with impulses. By establishing a conformable fractional Halanay inequality, some sufficient criteria on the conformable exponential stability are presented for the systems.