Nonlinear Delay Differential Equations Research Articles

Synchronization and limit cycles in a simple contagion model with time delays

We examine a system of N=2 coupled non-linear delay-differential equations representing financial market dynamics. In such time delay systems, coupled oscillations have been derived. We linearize the system for small time delays and study its collective dynamics. Using analytical and numerical solutions, we obtain the bifurcation diagrams and analyze the corresponding regions of amplitude death, phase locking, limit cycles and market synchronization in terms of the system frequency-like parameters and time delays. We further numerically explore higher order systems with N>2, and demonstrate that limit cycles can be maintained for coupled N−asset models with appropriate parameterization.

Mathematical modeling of Ebola using delay differential equations

Nonlinear delay differential equations (NDDEs) are essential in mathematical epidemiology, computational mathematics, sciences, etc. In this research paper, we have presented a delayed mathematical model of the Ebola virus to analyze its transmission dynamics in the human population. The delayed Ebola model is based on the four human compartments susceptible, exposed, infected, and recovered (SEIR). A time-delayed technique is used to slow down the dynamics of the host population. Two significant stages are analyzed in the said model: Ebola-free equilibrium (EFE) and Ebola-existing equilibrium (EEE). Also, the reproduction number of a model with the sensitivity of parameters is studied. Furthermore, the local asymptotical stability (LAS) and global asymptotical stability (GAS) around the two stages are studied rigorously using the Jacobian matrix Routh–Hurwitz criterion strategies for stability and Lyapunov function stability. The delay effect has been observed in the model in inverse relation of susceptible and infected humans (it means the increase of delay tactics that the susceptibility of humans increases and the infectivity of humans decreases eventually approaches zero which means that Ebola has been controlled into the population). For the numerical results, the Euler method is designed for the system of delay differential equations (DDEs) to verify the results with an analytical model analysis.

Linearized Stability Analysis of Nonlinear Delay Differential Equations with Impulses

Linearized Stability Analysis of Nonlinear Delay Differential Equations with Impulses

Modeling Rugose Spiraling Whitefly Infestation on Coconut Trees Using Delay Differential Equations: Analysis via HPM

In this paper, a delay - induced pest control model is proposed. We have introduced a time delay in healthy trees and whitefly population in the infected tree density of the proposed system of equations to reduce the probability of healthy trees becoming infected, as well as the level of infection. We have analyzed the impact of time delay on the stability of the equilibrium and establish requirements to verify its asymptotic stability over all delays. The solutions of this system of non-linear ordinary differential equations(ODEs) and delay differential equations(DDEs) are presented by using homotopy perturbation method(HPM). Numerical simulation is also obtained for the same model of both ODE and DDE by using MATLAB software. The proposed method works very well and is easy to use, as shown by these findings. Our aim is to control the spread of whitefly such that harvest is not affected.

Ultimate boundedness and stability of highly nonlinear neutral stochastic delay differential equations with semi-Markovian switching signals

The ultimate boundedness and stability of highly nonlinear neutral stochastic delay differential equations (NSDDEs) are investigated in this paper. Different from many previous works, the highly nonlinear NSDDEs with semi-Markov switching signals are considered for the first time in this paper. Meanwhile, the time delay function in this paper is only required to meet much more relaxed restrictions, which can invalidate plentiful methods with requirements on its derivatives for the stability of NSDDEs. To overcome this difficulty, several novel techniques to tackle NSDDEs with such a delay are developed. Furthermore, a crucial property on the ergodic semi-Markov process is established, which plays a key role in the proof on the existence and boundedness of global solution. The generalized Khasminskii-type theorems are established for the existence and uniqueness of global solution by applying new methods and this property, and the criteria for boundedness and stability are also provided. In particular, feasible measures are provided to deal with the neutral term in our stability criteria. Finally, the effectiveness is verified by a numerical example.

Novel criteria for exponential stability in mean square of stochastic delay differential equations with Markovian switching

This paper addresses the exponential stability in mean square of nonlinear stochastic delay differential equations with Markovian switching. Using a novel approach, we present new explicit criteria for exponential stability in mean square. A discussion and two examples are given to illustrate the effectiveness of our criteria.

Classifications of Solutions of Second Order Nonlinear Neutral Delay Difference Equations With Positive and Negative Coefficients

Classifications of Solutions of Second Order Nonlinear Neutral Delay Difference Equations With Positive and Negative Coefficients

Existence and Hyers–Ulam Stability of Stochastic Delay Systems Governed by the Rosenblatt Process

Under the effect of the Rosenblatt process, time-delay systems of nonlinear stochastic delay differential equations are considered. Utilizing the delayed matrix functions and exact solutions for these systems, the existence and Hyers–Ulam stability results are derived. First, depending on the fixed point theory, the existence and uniqueness of solutions are proven. Next, sufficient criteria for the Hyers–Ulam stability are established. Ultimately, to illustrate the importance of the results, an example is provided.

Meromorphic Solutions to Higher Order Nonlinear Delay Differential Equations

Meromorphic Solutions to Higher Order Nonlinear Delay Differential Equations

An improved oscillation theorem for nonlinear delay differential equations

In this article, we establish a new oscillation criterion for all solutions of first order nonlinear delay differential equations. The main result improves well-known oscillation conditions in the literature. Besides, an example is presented to prove the importance of the main theorem.

Hyers–Ulam–Rassias stability of fractional delay differential equations with Caputo derivative

This paper is devoted to the study of Hyers–Ulam–Rassias (HUR) stability of a nonlinear Caputo fractional delay differential equation (CFrDDE) with multiple variable time delays. We obtain two new theorems with regard to HUR stability of the CFrDDE on bounded and unbounded intervals. The method of the proofs is based on the fixed point approach. The HUR stability results of this paper have indispensable contributions to theory of Ulam stabilities of CFrDDEs and some earlier results in the literature.

A novel approaches to 6th-order delay differential equations in toxic plant interactions and soil impact: beyond newton-raphson

This paper focuses on investigating a 6th-order delay differential equation root within the context of toxic interactions between competing plant populations and their impact on soil dynamics. The study introduces a novel approach for approximating solutions to nonlinear delay differential equations, drawing inspiration from the fundamental principles of Newton-Raphson’s method. This technique leverages the complex root theorem to ensure stability, enabling it to effectively handle widely dispersed roots within dynamic systems. Consequently, this approach holds considerable potential for a diverse array of applications. The analysis introduces time delay into a nonlinear dynamical system and explores the system’s threshold value. At this threshold, the dynamical system’s stability undergoes fluctuations, and observations of hopf bifurcation phenomena are made. The study also successfully identifies both real and complex roots of the dynamical system. Visualization of the dynamic system is accomplished using MATLAB-generated graphical representations. Moreover, this research’s implications extend to the realm of climate action and terrestrial ecosystems, underscoring its significance for promoting a sustainable environment and fostering healthy life on land.

Stabilization and destabilisation of non-autonomous stochastic nonlinear delay differential equations

This paper focuses on a class of non-autonomous stochastic nonlinear delay differential equations, where the time delay functions are no longer required to be bounded or differentiable. The key aim is to study the stabilisation and destabilisation of the non-autonomous nonlinear stochastic delay differential equations by using the function ln ⁡ | x ( t ) | 2 , the improved LaSalle-type theorem, and non-negative semimartingale convergence theorem. In comparison with recent works on stabilisation for stochastic time-varying delay nonlinear systems (see, e.g. [Dong, H., & Mao, X. (2022). Advances in stabilization of highly nonlinear hybrid delay systems. Automatica, 136, 110086. https://doi.org/10.1016/j.automatica.2021.110086; Hu, J., Mao,W., & Mao, X. (2023). Advances in nonlinear hybrid stochastic differential delay equations: Existence, boundness and stability. Automatica, 147, 110682. https://doi.org/10.1016/j.automatica.2022.110682; Xu, H., & Mao, X. (2023). Improved delay-dependent stability of superlinear hybrid stochastic systems with general time-varying delays. Nonlinear Analysis: Hybrid Systems, 50, 101413. https://doi.org/10.1016/j.nahs.2023.101413]), the proposed results exhibit significant improvements and can be applied to a broader class of non-autonomous equations.

Application of generalized Haar wavelet technique on simultaneous delay differential equations

In this paper, the Haar wavelet technique is applied to a system of simultaneous linear and nonlinear proportional delay differential equations. By implementing this method, the delay differential equations are transformed into a series of linear and nonlinear algebraic equations, respectively. These equations are then solved using numerical code constructed in MATLAB. The obtained approximate solutions are compared with exact solutions, demonstrating the accuracy and efficacy of the Haar wavelet method. Using the fixed point theorem, the existence and uniqueness of the solutions is established. The reliability and efficiency of the proposed technique are illustrated through two numerical examples. Additionally, a comprehensive error analysis is presented and the convergence result is discussed, providing a thorough examination of the robustness of the method. This study not only shows that the Haar wavelet technique is a powerful tool for solving delay differential equations but also sets the groundwork for future research in this area.

New Results on the Ulam–Hyers–Mittag–Leffler Stability of Caputo Fractional-Order Delay Differential Equations

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work.

Numerical simulation of nonlinear fractional delay differential equations with Mittag-Leffler kernels

This study is concerned with finding numerical solutions of nonlinear delay differential equations involving extended Mittag-Leffler fractional derivatives of the Caputo-type. The main benefit of the used extension is to address the complexity resulting from the limitations of using fractional derivatives with non-singular Mittag-Leffler kernels. We discussed the existence and uniqueness of solutions for the studied delay models. Next, we modified an Adams-type method to numerically solve fractional delay differential equations combined with Mittag-Leffler kernels. A new type of solution belonging to the L1 space is presented for the studied models using the proposed scheme.

Advancing real-time drilling trajectory prediction with an efficient nonlinear DDE model and online parameter estimation

This study introduces a streamlined approach to predict planar drilling trajectory by employing an efficient nonlinear Delay Differential Equation (DDE) model with high computational speed and predictive accuracy. The efficient model is validated and compared with other models by utilizing not just data from existing literature but also two sets of experimental data with two distinct rigs. Following the validation, the model is integrated with an online estimation technique—calibrating one parameter in real-time using historical data—empowering it to forecast drill bit paths with remarkable fidelity. The validation of the model and its estimation strategy underscores their potential for application in real-time drilling control systems, setting a new benchmark for future advancements in the field.

Asymptotic stability of nonlinear fractional delay differential equations with α ∈ (1, 2): An application to fractional delay neural networks.

We introduce a theorem on linearized asymptotic stability for nonlinear fractional delay differential equations (FDDEs) with a Caputo order α∈(1,2), which can be directly used for fractional delay neural networks. It relies on three technical tools: a detailed root analysis for the characteristic equation, estimation for the generalized Mittag-Leffler function, and Lyapunov's first method. We propose coefficient-type criteria to ensure the stability of linear FDDEs through a detailed root analysis for the characteristic equation obtained by the Laplace transform. Further, under the criteria, we provide a wise expression of the generalized Mittag-Leffler functions and prove their polynomial long-time decay rates. Utilizing the well-established Lyapunov's first method, we establish that an equilibrium of a nonlinear Caputo FDDE attains asymptotically stability if its linearization system around the equilibrium solution is asymptotically stable. Finally, as a by-product of our results, we explicitly describe the asymptotic properties of fractional delay neural networks. To illustrate the effectiveness of our theoretical results, numerical simulations are also presented.

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.

Subdomain collocation method based on successive integration technique for solving delay differential equations

The main objective of this work is to propose the polynomial based Subdomain collocation method using successive integration technique for solving delay differential equations (DDEs). In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomial, the Chebyshev polynomial, the Hermite polynomial, and the Fibonacci polynomial are considered. Numerical examples of linear and nonlinear DDEs have been considered to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N (order of the polynomial) increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear DDEs in real-world problems.