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This work investigates the controllability and stability properties of nonlinear piecewise dynamical systems subject to integral boundary conditions studied in arbitrary time domains. The study employs powerful mathematical ideas like fixed point theorems, Gramian-type matrices and the unified approach given by the theory of time scales. This unified approach provides results applicable seamlessly in various settings of continuous-time systems, discrete-time systems, and systems with hybrid behavior of both. This paper expands existing findings in the literature, providing a more complete view. To show the application of the theoretical results, a detailed example is presented and supported by numerical simulations to confirm the efficiency of the proposed methods. Full article

In our research, we developed a Distributed Relaxation Spectrum Delay Differential Equation (DRSDDE) model to simulate viscoelastic responses exhibited by materials with multiple-scale relaxation mechanisms and finite delay times. Our model expanded upon traditional integer-order viscoelastic models to include a continuum relaxation process using a log-time-space Gaussian distribution representing a continuum of relaxation processes, including a direct representation of the effect of delayed feedback via an explicit time delay term. Consequently, the resultant model can be viewed as a generalized Maxwell-type formulation where the viscoelastic behavior exhibits distributed relaxation dynamics and has finite signal propagation characteristics. We then used experimental data obtained from three representative materials: PDMS Sylgard 184, bovine brain white matter, and polyurethane foam to calibrate the model. Calibration was achieved by estimating model parameters through the use of Gauss-Legendre quadrature combined with non-linear optimization of the relaxation spectrum. The results indicate that the coefficients of determination for each of the materials exceeded R² > 0.83. Therefore, the proposed DRSDDE model outperformed the classical Zener model when simulating materials that exhibit a wide relaxation spectrum. The parameter values estimated for each of the examined materials provided additional insight into their physical behaviors. Specifically, the characteristic relaxation times for the studied materials were determined based upon \(\tau\)_c = 10^µ ranging from about 63 s to 158 s. These results illustrate different dominant relaxation regimes for the investigated materials. Additionally, both characteristic equations and frequency domain analyses were utilized to study the stability and bifurcation properties of the DRSDDE model. A significant finding resulted from identifying a delay-insensitive stability regime for materials with \(\tilde{K} < 1\) (as illustrated by bovine brain white matter). For materials with \(\tilde{K} > 1\), the analysis revealed Hopf bifurcation results illustrating critical delay thresholds and frequencies for the onset of oscillations. Further, it was established that all calibrated delay values were significantly less than these threshold values. This indicates that all identified models functioned well below the oscillation thresholds at realistic delay times. Ultimately, the proposed DRSDDE model represents a physically intuitive, robust, and flexible method for modeling complex viscoelastic systems. Future research will involve investigating temperature-dependent effects, nonlinear bifurcations, and experimental validations of predicted oscillatory dynamics Full article

This research explores the existence of solutions for a class of random fractional differential equations characterized by bounded delay, specifically within the context of Fréchet spaces. Random fractional differential equations serve as powerful mathematical tools for modeling complex phenomena subjected to stochastic perturbations and hereditary effects. Despite their significance, establishing solution existence in infinite-dimensional spaces remains a challenging task. By integrating the properties of the noncompactness measures with a generalized Darbo fixed point approach, we establish new existence results for the associated Darboux-type problem under milder compactness conditions. To illustrate the practical utility of these analytical results and demonstrate the validity of our theoretical framework, a representative example is provided. Full article

This paper investigates the optimal control problem of time-delay backward doubly stochastic systems under partial information. Partial information widely exists in practical control systems due to monitoring constraints, communication delays, and data acquisition costs. Combined with inherent system time delays, it greatly complicates state estimation and decision-making, which requires research. A new type of anticipated backward doubly stochastic differential equations is introduced to describe the system dynamics. Using stochastic analysis and the variational methods, the corresponding maximum principle for optimal control is derived. Furthermore, a verification theorem is established that provides rigorous sufficient optimality conditions: any admissible control satisfying the necessary conditions, along with reasonable convexity assumptions, indeed optimizes the cost functional, thereby bridging the gap between necessary and sufficient optimality criteria. As an application, we solve a time-delay linear-quadratic optimal control problem and obtain explicit analytical expressions; the results demonstrate the validity of the established theoretical framework. Full article

The present paper addresses the problem of modeling the resilient operation of a cryptocurrency exchange (CEX) under delayed quantum attacks of the harvest-now-decrypt-later type. The proposed model diverges from extant approaches in its conceptualization of the quantum threat. Whereas extant approaches treat the quantum threat as an external shock, the proposed model conceptualizes the accumulation of cryptographically vulnerable data as an internal state variable of the system. The framework under consideration is formulated as a system of nonlinear differential equations linking the exchange’s liquidity, the intensity of post-quantum cryptography (PQC) adoption, and the volume of accumulated threat. The analytical conditions for asymptotic stability are derived. The resolution of the system enables the identification of a region of admissible defense strategies, which is interpreted in the paper as a “survival dome.” Numerical simulations demonstrate that both delayed and excessively aggressive migration strategies toward post-quantum cryptography may lead to the degradation of the exchange. The findings indicate that a balanced and adaptive transition strategy, aimed at mitigating quantum risks, can preserve liquidity while minimizing long-term losses. The findings establish a theoretical framework for the development of migration strategies for financial platforms undergoing a transition to post-quantum security standards. Full article

To investigate the structural dynamic response of pile-column bridge piers in mountainous regions under rockfall impact, this study takes the No. 4 double-column pier of Changba Bridge in Nanchuan, Chongqing, as a prototype. Based on similarity theory and differential equation analysis, a scaled model test was designed and conducted. By considering different rockfall impact angles (30°, 45°, 60°) and different impact positions on the pier (top, middle, bottom), the strain response characteristics of the pier concrete and reinforcing steel were systematically analyzed. The results indicate that the peak strain at the impacted location increases significantly with the increase in the rockfall impact angle; when the impact angle increases from 30° to 60°, the peak strain increases by approximately 22.8%. The peak strain decreases as the impact position approaches the bottom of the pier, with the most pronounced strain response observed at the middle position. The strain response of the reinforcing steel follows the same pattern as that of the concrete, albeit with a brief delay. Furthermore, the stirrups exhibit predominantly transverse orthogonal strain, while the longitudinal reinforcing bars exhibit predominantly longitudinal orthogonal strain. It is concluded that the impact angle is a key parameter affecting local damage to the pier, and the middle section of the pier should be regarded as a priority protection zone. This study provides a theoretical basis for the design of bridge piers in mountainous regions against rockfall impact. Full article

This paper presents a generalized epidemic modeling framework based on g-tempered Caputo fractional derivatives with discrete time delays. The proposed approach incorporates nonlocal memory effects, nonlinear temporal scaling, and delayed epidemiological responses within a unified mathematical structure. The introduction of the nonlinear time transformation $g\left(t\right)$ and the tempering parameter $\lambda$ eliminates the unrealistic infinite-memory behavior associated with classical power-law kernels while simultaneously introducing new challenges related to parameter identifiability and inverse problems. We investigate the structural properties of the resulting dynamical systems and show that the associated inverse problem is inherently ill-posed. To illustrate the practical implications of these results, the framework is applied to a delayed SIQR epidemiological model. Numerical simulations are performed using a generalized L1-type scheme adapted to delayed fractional histories, and a multi-phase parameter estimation procedure is proposed to address the ill-posedness of the reconstruction problem. The results demonstrate the ability of the model to capture both short- and long-term memory effects in epidemic evolution while highlighting the challenges of statistical identifiability in generalized fractional systems. Full article

To investigate the underlying mechanisms of talent mobility in higher-education institutions influenced by factors such as the development environment, macroeconomic policies, and evaluation mechanisms, this paper proposes a nonlinear stochastic differential equation (SDE) dynamical model that incorporates time delays and multiplicative noise. We analyze the dynamic processes of talent mobility under varying conditions regarding the number of nodes, policy implementation cycles, and noise intensity. First, we employ central manifold theory and stochastic averaging methods to reduce the system to a one-dimensional averaged $It\widehat{o}$ equation. Subsequently, with $\tau$ as a parameter, we conduct an in-depth study of the system’s stochastic bifurcation behavior using the corresponding Fok–Planck–Kolmogorov equations. Finally, we validate the theoretical conclusions through numerical simulations. The results indicate that the number of nodes, policy delay, and noise intensity all have significant effects on system stability; an increasing delay induces random P-bifurcation in the system, and when $N\le 3$ and $N>3$, the system exhibits distinctly different steady-state behaviors. We also found that excessively high noise intensity disrupts system stability, whereas moderate noise intensity has a positive effect on stability. This study not only provides theoretical insights into the dynamic evolution mechanisms of talent mobility in regional universities but also offers valuable guidance for universities in formulating talent recruitment and evaluation policies. The methodology employed in this study opens up a promising avenue for analyzing complex dynamic problems in the field of sociology. Full article

The microstructure of semicrystalline bioresorbable polymers is central to their biomedical performance because the crystalline content influences both the mechanical stability and the degradation behaviour. Experimental studies have shown that crystallinity evolves concurrently with mass loss during enzymatic degradation. However, most existing models represent the material as a single homogeneous structure, preventing them from capturing this microstructural evolution or the state-selective mechanisms that drive it. We present a one-dimensional partial differential equation model for the enzymatic degradation of thin films, which treats the crystalline and amorphous states as distinct reactive components. Calibrated to poly($\epsilon$-caprolactone) (PCL) degraded by Candida antarctica lipase in vitro, the model accurately reproduces both the observed weight-loss profile and the concurrent decline in crystallinity. Parameter uncertainty analysis indicates that while there are varying degrees of confidence in individual parameter values, the overall model predictive uncertainty is well constrained. Parameter sensitivity analysis shows that the amorphous catalytic rate (the rate at which the enzyme degrades the amorphous region) is the dominant driver of degradation dynamics. The identified model parameters are used to explore the role of film thickness on the rates of mass and crystallinity loss. It was found that thin films remain largely reaction-limited, whereas thicker specimens become increasingly transport-influenced, with slower degradation and delayed structural evolution in the material interior. The model provides a useful tool to explore the effect of changing PCL film thickness on degradation rate and crystallinity-related properties without extensive experimentation. Full article

Sepsis is a prominent cause of death in intensive care units, and delayed diagnosis greatly worsens fatal outcomes due to the complex, irregular, and uneven character of clinical time-series data. Hence we proposed an interpretable and uncertainty-aware deep learning architecture that solves data quality, temporal irregularity, and clinical explainability restrictions, which are frequently addressed separately by existing models. The suggested method combines Bidirectional Recurrent Imputation for Time Series (BRITS)-based imputation, hybrid Conditional Tabular Generative Adversarial Network-Synthetic Minority Over-sampling Technique (CTGAN-SMOTE) data augmentation, a Temporal Convolutional Networks (TCN)-Attention architecture, and continuous-time neural Ordinary Differential Equations (ODEs), along with SHapley Additive exPlanations (SHAP)-based feature attribution and uncertainty quantification. The experimental evaluation on a large ICU dataset reveals greater predictive accuracy, with an AUROC of 0.926 and accurate early warnings up to six hours before clinical onset, all while maintaining strong interpretability and calibration. The proposed framework demonstrates strong predictive performance and provides early warnings up to six hours before clinical onset, while maintaining interpretability and calibration. While the results are promising, further validation across multiple clinical settings is required to confirm its generalisability and real-world applicability. Full article

We study a perturbed coupled system of generalized hybrid pantograph equations involving the deformable derivative of Zulfeqarr–Ujlayan–Ahuja. A central point of the revision is made explicit: for classically differentiable functions this derivative is local and satisfies $D^{\tau }u=(1-\tau )u+\tau u^{\prime }$. Therefore, in the present differentiable setting the memory or aftereffect is produced by the proportional pantograph delays, while the deformable order $\tau$ supplies an order-dependent local relaxation/drift term. After rewriting the system as an equivalent integral equation on $\mathcal{X}=C(I,{\mathbb{R}}^2)$, we establish invariant-ball conditions, existence and uniqueness within invariant balls, generalized Ulam–Hyers stability, and Lipschitz continuous dependence on the perturbation amplitude $\epsilon$. The assumptions and constants are stated so that the restrictive roles of the Lipschitz bounds, the interval length, and $\left|\epsilon \right|$ are transparent. We then provide numerical parameter sensitivity diagrams for illustrative pantograph systems and include step-size refinement checks and performance indices. The numerical and plasma-inspired sections are deliberately framed as exploratory delay-feedback examples rather than as first-principles plasma models or rigorous bifurcation theory. Full article

We analyze parameter identifiability in a Marchuk-type immune-response model using longitudinal whole-blood transcriptomic signatures from the influenza challenge. Latent states are extracted from curated gene signatures derived from nine symptomatic and eight asymptomatic subjects. The governing delay differential equations are cast in a linear-in-parameters form; derivatives are estimated by smoothing splines, coefficients are fit by ridge regression, and the delay $\tau$ is selected by grid search. We find that the parameters governing viral and innate dynamics are consistently identifiable, with low relative error, and are highly determined, whereas adaptive-immunity and tissue-damage parameters are poorly constrained by transcriptomics alone. Introducing a small additive background term and tissue dependence markedly reduces residual variance and stabilizes estimates. Symptomatic patients exhibit a characteristic regulatory delay near 21 h. These results show that aggregated transcriptomic time series can reliably identify some subsystems of classical immune models, but that adaptive immunity and damage dynamics require explicit structural extensions or additional data modalities. The study provides a practical identification pipeline and concrete guidance on model extensions needed for transcriptomic-driven mechanistic inference. Full article

Taking the Singularly Perturbed System (SPS) as a model of ODE system separation into fast and slow subsystems by an arbitrarily small parameter, we state and prove a theorem on the decomposition of an Ordinary Differential Equations (ODE) system without the aforementioned arbitrarily small parameter. In accordance with the proven theorem, we implemented an algorithm to decompose an ODE system into fast and slow subsystems by coordinate transformation. A similar algorithm is called the Singular Perturbed Vector Field (SPVF) algorithm; however, it is not justified by any stated theorem. Since we have not found any theorem to propose a similar ODE decomposition in the literature, we have tried to fill the gap with our theorem and algorithm explanations through examples. Finally, we propose our concept on Marchuk’s infectious diseases model, which allows a different analysis of the original Marchuk’s ODE system with delay. Full article

This study examines the distances between consecutive generalized zeros of first-order difference equations with several variable delays. We develop new techniques to obtain the upper bounds for these distances. The results are not only new, but they also extend and improve several earlier contributions in this area. In particular, our analysis covers the case of nonmonotonic delay sequences, which, to the best of our knowledge, has not been previously investigated, even for the corresponding delay differential equations. To illustrate the effectiveness of our findings, three numerical examples will be presented. Full article

In this paper, we present a mathematical analysis of within-host CHIKV dynamics by developing and studying a novel delay differential equation model that incorporates persistent infected monocytes, discrete time delays, and an antibody-mediated humoral immune response. The model includes five compartments: susceptible monocytes, persistent infected monocytes, actively infected monocytes, CHIKV pathogens, and neutralizing antibodies. To reflect key biological latencies, we introduce four distinct discrete delays accounting for the periods between viral entry and the emergence of infected cell populations, intracellular virion production, and antibody activation. We analyze the model, establishing the positivity, boundedness, and invariance of solutions, and derive the basic reproduction number ${\mathcal{R}}_0$ via the next-generation matrix method. Using Lyapunov functions and LaSalle’s Invariance Principle, we prove a threshold dynamic: the infection-free equilibrium is globally asymptotically stable (GAS) when ${\mathcal{R}}_0\le 1$, while a unique endemic equilibrium is GAS when ${\mathcal{R}}_0>1$. Numerical simulations validate the analytical results and illustrate threshold behavior. A detailed local sensitivity analysis of ${\mathcal{R}}_0$ identifies the most influential parameters, offering theoretical insights into potential intervention strategies. We further investigate the effects of antiviral therapy as a theoretical intervention, deriving a treatment-dependent reproduction number and the critical drug efficacy required for eradication, and explore how the intracellular production delay can itself serve as a critical threshold for infection clearance. The study provides a rigorous theoretical framework that highlights the roles of latency, immune response, and biological delays in CHIKV pathogenesis and offers qualitative insights that may inform future experimental and treatment design studies. Full article