Search Results (633)

In this study, we establish new oscillation criteria for solutions of the fourth-order differential equation $(a$$\varphi \left(u\right)$$u^{‴})$+q$(u\circ h)=0$, which is of a functional type with a delay. The oscillation behavior of solutions of fourth-order delay equations has been studied using many techniques, but previous results did not take into account the existence of the function $\varphi$ except in second-order studies. The existence of $\varphi$ increases the difficulty of obtaining monotonic and asymptotic properties of the solutions and also increases the possibility of applying the results to a larger area of special cases. We present two criteria to ensure the oscillation of the solutions of the studied equation for two different cases of $\varphi$. Our approach is based on using the comparison principle with equations of the first or second order to benefit from recent developments in studying the oscillation of these orders. We also provide several examples and compare our results with previous ones to illustrate the novelty and effectiveness. Full article

This paper investigates the existence and uniqueness of bounded nonoscillatory solutions for two classes of m-th-order nonlinear neutral differential equations that incorporate both discrete and distributed delays. By applying Banach’s fixed-point theorem, we establish sufficient conditions under which such solutions exist. The results extend and generalize previous works by relaxing assumptions on the nonlinear terms and accommodating a wider range of feedback structures, including positive, negative, bounded, and unbounded cases. The mathematical framework is unified and applicable to a broad class of problems, providing a comprehensive treatment of neutral equations beyond the first or second order. To demonstrate the practical relevance of the theoretical findings, we analyze a delayed temperature control system as an application and provide numerical simulations to illustrate nonoscillatory behavior. This paper concludes with a discussion of analytical challenges, limitations of the numerical scope, and possible future directions involving stochastic effects and more complex delay structures. Full article

For systems such as the van der Pol and van der Pol–Duffing oscillators, the study of their oscillation is currently a very active area of research. Many authors have used the bifurcation method to try to determine oscillatory behavior. But when the system involves n separate delays, the equations for bifurcation become quite complex and difficult to deal with. In this paper, the existence of periodic oscillatory behavior was studied for a system consisting of n coupled equations with multiple delays. The method begins by rewriting the second-order system of differential equations as a larger first-order system. Then, the nonlinear system of first-order equations is linearized by disregarding higher-degree terms that are locally small. The instability of the trivial solution to the linearized equations implies the instability of the nonlinear equations. Periodic behavior often occurs when the system is unstable and bounded, so this paper also studied the boundedness here. It follows from previous work on the subject that the conditions here did result in periodic oscillatory behavior, and this is illustrated in the graphs of computer simulations. Full article

A well-known model of infectious diseases, described by a nonlinear system of delay differential equations, is investigated under the influence of stochastic perturbations. Using the general method of Lyapunov functional construction combined with the linear matrix inequality (LMI) approach, we derive sufficient conditions for the stability of the equilibria of the considered system. Numerical simulations illustrating the system’s behavior under stochastic perturbations are provided to support the thoretical findings. The proposed method for stability analysis is broadly applicable to other systems of nonlinear stochastic differential equations across various fields. Full article

The stability of hybrid stochastic differential equations (SDEs in short) depends on multiple factors, such as the structures and parameters of subsystems, switching rules, delay, etc. Regarding stability analysis for hybrid stochastic systems incorporating subsystems with diverse structures, existing research results require the system to possess either Markovian properties or finite memory characteristics. However, the stability problem remains unresolved for hybrid stochastic differential equations with infinite memory (hybrid IMSDEs in short), as no systematic theoretical framework currently exists for such systems. To bridge this gap, this paper develops a rigorous stability analysis for a class of hybrid IMSDEs by introducing a suitably chosen phase space and leveraging the theory of fading memory spaces. We establish sufficient conditions for exponential stability, extending the existing results to systems with unbounded memory effects. Finally, a numerical example is provided to illustrate the effectiveness of the proposed criteria. Full article

This paper analyzes the stability of trajectory tracking in fixed-wing UAV swarms subject to time-delayed feedback control. A delay-dependent stability criterion is established using a combination of Routh–Hurwitz analysis and a transcendental characteristic equation method. The study identifies a critical delay threshold beyond which the tracking objective becomes unstable. The influence of delayed feedback on the system dynamics is analyzed, showing how time delays affect the swarm’s ability to maintain formation. Numerical simulations confirm the theoretical predictions and illustrate the loss of stability as the delay increases. The findings underline the importance of accounting for delays when evaluating control performance in UAV swarm coordination. Full article

In this paper, we investigate the nonlinear dynamic behavior of a cart–double pendulum system with single time delay feedback control. Based on the center manifold theorem and stochastic averaging method, a reduced-order dynamic model of the system is established, with a focus on analyzing the influence of time delay parameters and stochastic excitation on the system’s Hopf bifurcation characteristics. By introducing stochastic differential equation theory, the system is transformed into the form of an Itô equation, revealing bifurcation phenomena in the parameter space. Numerical simulation results demonstrate that decreasing the time delay and increasing the time delay feedback gain can effectively enhance system stability, whereas increasing the time delay and decreasing the time delay feedback gain significantly improves dynamic performance. Additionally, it is observed that Gaussian white noise intensity modulates the bifurcation threshold. This study provides a novel theoretical framework for the stochastic stability analysis of time delay-controlled multibody systems and offers a theoretical basis for subsequent research. Full article

The dynamic vibration absorber (DVA) based on delayed fractional PID (DFOPID) can achieve a more superior vibration suppression effect. However, the strong nonlinear characteristics of the system and the computational burden resulting from its high dimensionality make solving and optimizing more challenging. This paper presents a coupled model of DFOPID and DVA, exploring its parameter tuning mechanism and optimization problem. First, using the averaging method and Lyapunov stability theory, the amplitude-frequency equation and the stability condition of the steady-state solution of the primary system are derived. Numerical simulations validate the accuracy of the analytical result. Next, based on the mechanics of vibration, the approximate expressions of the controller under different differential conditions are calculated, and their equivalent action mechanisms are analyzed. Finally, by minimizing the maximum amplitude of the primary system as the objective function, the Particle Swarm Optimization (PSO) algorithm is applied to optimize the parameters of the passive DVA and the DVA models controlled by PID, FOPID, and DFOPID, successfully addressing the parameter optimization challenges posed by traditional fixed-point theory. The vibration reduction performance is compared across different loading environments. The results demonstrate that the model presented in this paper performs the best, exhibiting excellent vibration suppression and robustness. Full article

Phase stability at low radio frequencies is severely impacted by ionospheric propagation delays. Radio interferometers such as the giant metrewave radio telescope (GMRT) are capable of detecting changes in the ionosphere’s total electron content (TEC) over larger spatial scales and with greater sensitivity compared to conventional tools like the global navigation satellite system (GNSS). Thanks to its unique design, featuring both a dense central array and long outer arms, and its strategic location, the GMRT is particularly well-suited for studying the sensitive ionospheric region located between the northern peak of the equatorial ionization anomaly (EIA) and the magnetic equator. In this study, we observe the bright flux calibrator 3C48 for ten hours to characterize and study the low-latitude ionosphere with the upgraded GMRT (uGMRT). We outline the methods used for wideband data reduction and processing to accurately measure differential TEC ($\delta \mathrm{TEC}$) between antenna pairs, achieving a precision of< mTECU (1 mTECU = ${10}^{-3}$ TECU) for central square antennas and approximately mTECU for arm antennas. The measured $\delta \mathrm{TEC}$ values are used to estimate the TEC gradient across GMRT arm antennas. We measure the ionospheric phase structure function and find a power-law slope of $\beta =1.72\pm 0.07$, indicating deviations from pure Kolmogorov turbulence. The inferred diffractive scale, the spatial separation over which the phase variance reaches $1{\mathrm{rad}}^2$, is ∼6.66 km. The small diffractive scale implies high phase variability across the field of view and reduced temporal coherence, which poses challenges for calibration and imaging. Full article

In this paper, we propose an implicit finite difference scheme for a wave equation with strong damping and a discrete delay term. Although the scheme is implicit, the use of second-order finite difference approximations for the strong damping term in both space and time prevents it from being unconditionally stable. A sufficient condition for the asymptotic stability of the scheme is established by applying the Jury stability criterion to show that all roots of the characteristic polynomial associated with the resulting linear recurrence lie strictly inside the unit disk. This stability condition is derived under an appropriate constraint that links the time and space discretization steps with the damping and delay parameters. A numerical example is provided to illustrate the decay behavior of the scheme and confirm the theoretical findings. Full article

In this work, we address a nonlinear $\psi$-Hilfer fractional-order Volterra integro-differential equation that incorporates n-multiple-variable time delays. Employing the $\psi$-Hilfer fractional derivative operator, we investigate the existence of a unique solution, as well as the Ulam–Hyers–Rassias stability, semi-Ulam–Hyers–Rassias stability, and Ulam–Hyers stability of the proposed $\psi$-Hilfer fractional-order Volterra integro-differential equation through the fixed-point approach. In this study, we enhance and generalize existing results in the literature on $\psi$-Hilfer fractional-order Volterra integro-differential equations, both including and excluding single delay, by establishing new findings for nonlinear $\psi$-Hilfer fractional-order Volterra integro-differential equations involving n-multiple-variable time delays. This study provides novel theoretical insights that deepen the qualitative understanding of fractional calculus. Full article

The purpose of this article is to analyze a hybrid-type iterative algorithm for a class of generalized non-expansive mappings satisfying the Garcia-Falset property in uniformly convex Banach spaces. Some existing results for such mappings have been obtained using the given algorithm. The ${\omega }^2$-stability of the iterative process is also studied. Using some examples, numerical experiments are conducted by comparing this iterative algorithm with different well-known iterative schemes. It is concluded that this iterative algorithm converges faster to the fixed point and is preferable over the previously known iterative schemes using the Garcia-Falset property. A weak solution of the Volterra–Stieltjes-type delay functional differential equation is presented to demonstrate the significance of the proposed results. Full article

The objective of this study is to develop a kinetic model that incorporates time-delay mechanisms to describe the gradual release behavior of pectin. Three models were used: modified Peleg, logistic, and second-order linear with time delay (SOPDT), to represent the extraction kinetics of grapefruit peels at 60, 70, and 90 °C. Model fitting was performed using a hybrid methodology combining Monte Carlo simulations with genetic algorithms. The SOPDT model provided the best fit, with the lowest squared error values (J = 7.06 at 60 °C, 5.63 at 70 °C, and 8.71 at 90 °C), requiring only two parameters to represent the entire process. In contrast, the other models required six. The maximum pectin yield reached over 120% at 90 °C, compared to <80% at lower temperatures. These results demonstrate the potential of the SOPDT model for efficient process description and control. Full article

This paper mainly investigates a class of third-order semilinear delay differential equations with a nonhomogeneous term ${\left({\left[x^{″}\left(t\right)\right]}^{\alpha }\right)}^{\prime }+q\left(t\right)x^{\alpha }\left(\sigma \left(t\right)\right)+f\left(t\right)=0,t\ge t_0.$ Under the oscillation criteria, we propose a sufficient condition to ensure that all solutions for the equation exhibit oscillatory behavior when $\alpha$ is the quotient of two positive odd integers, supported by concrete examples to verify the accuracy of these conditions. Furthermore, for the case $\alpha =1,$ a sufficient condition is established to guarantee that the solutions either oscillate or asymptotically converge to zero. Moreover, under these criteria, we demonstrate that the global oscillatory behavior of solutions remains unaffected by time-delay functions, nonhomogeneous terms, or nonlinear perturbations when $\alpha =1.$ Finally, numerical simulations are provided to validate the effectiveness of the derived conclusions. Full article

This paper continues a series of papers by the author devoted to unsolved problems in the theory of stability and optimal control for stochastic systems. A delay differential equation with stochastic perturbations of the white noise and Poisson’s jump types is considered. In contrast with the known stability condition, in which it is assumed that stochastic perturbations fade on the infinity quickly enough, a new situation is studied, in which stochastic perturbations can either fade on the infinity slowly or not fade at all. Some unsolved problem in this connection is brought to readers’ attention. Additionally, some unsolved problems of stabilization for one stochastic delay differential equation and one stochastic difference equation are also proposed. Full article