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This study presents an innovative approach to analyzing finite-time stability (FTS) and synchronization (FTSYN) in integer-order reaction-diffusion systems (RDs), particularly in the context of epidemiological modeling. By integrating Gronwall’s inequality, Lyapunov functionals (LFs), and linear control strategies, a comprehensive framework is developed to address transient dynamics within finite time frames. The proposed methodology advances the theoretical understanding of FTS and FTSYN by addressing the relatively unexplored dynamics of spatially extended systems. MATLAB simulations validate the theoretical findings, demonstrating the effectiveness of the control schemes and their practical applicability in modeling real-world disease transmission. Integrating spatial diffusion and disease dynamics provides critical insights into the influence of parameters such as diffusion rates and mortality on system behavior. This work contributes a robust framework for enhancing the analysis and management of nonlinear systems, with significant implications for epidemiology and other fields requiring rapid convergence and synchronization.

This paper delves into a comprehensive analysis of a generalized impulsive discrete reaction–diffusion system under periodic boundary conditions. It investigates the behavior of reactant concentrations through a model governed by partial differential equations (PDEs) incorporating both diffusion mechanisms and nonlinear interactions. By employing finite difference methods for discretization, this study retains the core dynamics of the continuous model, extending into a discrete framework with impulse moments and time delays. This approach facilitates the exploration of finite-time stability (FTS) and dynamic convergence of the error system, offering robust insights into the conditions necessary for achieving equilibrium states. Numerical simulations are presented, focusing on the Lengyel–Epstein (LE) and Degn–Harrison (DH) models, which, respectively, represent the chlorite–iodide–malonic acid (CIMA) reaction and bacterial respiration in Klebsiella. Stability analysis is conducted using Matlab’s LMI toolbox, confirming FTS at equilibrium under specific conditions. The simulations showcase the capacity of the discrete model to emulate continuous dynamics, providing a validated computational approach to studying reaction-diffusion systems in chemical and biological contexts. This research underscores the utility of impulsive discrete reaction-diffusion models for capturing complex diffusion–reaction interactions and advancing applications in reaction kinetics and biological systems.

The spread of computer viruses represents a major challenge to digital security, underscoring the need for a deeper understanding of their propagation mechanisms. This study examines the stability and chaotic dynamics of a fractional discrete Susceptible-Infected (SI) model for computer viruses, incorporating commensurate and incommensurate types of fractional orders. Using the basic reproduction number R0, the derivation of stability conditions is followed by an investigation of how varying fractional orders affect the system’s behavior. To explore the system’s nonlinear chaotic behavior, the research of this study employs a suite of analytical tools, including the analysis of bifurcation diagrams, phase portraits, and the evaluation of the maximum Lyapunov exponent (MLE) for the study of chaos. The model’s complexity is confirmed through advanced complexity algorithms, including spectral entropy, approximate entropy, and the 0−1 test. These measures offer a more profound insight into the complex behavior of the system and the role of fractional order. Numerical simulations provide visual evidence of the distinct dynamics associated with commensurate and incommensurate fractional orders. These results provide insights into how fractional derivatives influence behaviors in cyberspace, which can be leveraged to design enhanced cybersecurity measures.

This work develops and analyzes an incommensurate fractional FitzHugh–Nagumo (FHN) reaction–diffusion system in which each state variable evolves with a distinct fractional order. The formulation extends the classical and commensurate fractional models by incorporating heterogeneous memory effects that break temporal symmetry between the activator and inhibitor variables. After establishing the mathematical framework, the equilibrium states of the system are derived and subjected to a detailed local stability analysis in both diffusion-free and diffusion-driven regimes. Explicit stability criteria are obtained by examining the spectral properties of the linearized operator under incommensurate fractional dynamics. Numerical simulations based on a Caputo L1 discretization scheme corroborate the theoretical results and demonstrate how asymmetric memory orders influence transient behavior, convergence rates, and the qualitative structure of the solutions. The study provides the first systematic stability characterization of an incommensurate fractional FitzHugh–Nagumo reaction–diffusion model, highlighting the role of fractional-order asymmetry in shaping the system’s dynamical response.

This paper describes a new fractional predator–prey discrete system of the Leslie type. In addition, the non-linear dynamics of the suggested model are examined within the framework of commensurate and non-commensurate orders, using different numerical techniques such as Lyapunov exponent, phase portraits, and bifurcation diagrams. These behaviours imply that the fractional predator–prey discrete system of Leslie type has rich and complex dynamical properties that are influenced by commensurate and incommensurate orders. Moreover, the sample entropy test is carried out to measure the complexity and validate the presence of chaos. Finally, nonlinear controllers are illustrated to stabilize and synchronize the proposed model.

The aim of this work is to describe the dynamics of a discrete fractional-order reaction–diffusion FitzHugh–Nagumo model. We established acceptable requirements for the local asymptotic stability of the system’s unique equilibrium. Moreover, we employed a Lyapunov functional to show that the constant equilibrium solution is globally asymptotically stable. Furthermore, numerical simulations are shown to clarify and exemplify the theoretical results.

Computer viruses remain a persistent technological challenge in information security. They require mathematical frameworks that realistically capture their propagation in digital networks. Classical continuous-time, integer-order models often overlook two key aspects of cyber environments: their inherently discrete nature and the memory-dependent effects of networked interactions. In this work, we introduce a fractional-order discrete computer virus (FDCV) model, derived from a three-dimensional continuous integer-order formulation and reformulated into a two-dimensional fractional discrete framework. We analyze its rich dynamical behaviors under both commensurate and incommensurate fractional orders. Leveraging a comprehensive toolbox including bifurcation diagrams, Lyapunov spectra, phase portraits, the 0–1 test for chaos, spectral entropy, and C0 complexity measures, we demonstrate that the FDCV system exhibits persistent chaos and high dynamical complexity across broad parameter regimes. Our findings reveal that fractional-order discrete models not only enhance the dynamical richness compared to integer-order counterparts but also provide a more realistic representation of malware propagation. These insights advance the theoretical study of fractional discrete systems, supporting the development of potential technologies for cybersecurity modeling, detection, and prevention strategies.

The study of economic maps has consistently attracted researchers due to their rich dynamics and practical relevance. A deeper understanding of these systems enables the development of more effective control strategies. In this work, we examine the influence of the fractional order υ with the Caputo fractional difference on an economic market map. The primary contribution is the comprehensive analysis of how both commensurate and incommensurate fractional orders affect the stability and complexity of the map. Numerical investigations, including phase portraits, largest Lyapunov exponents, and bifurcation analysis, reveal that the system undergoes a cascade of period-doubling bifurcations before transitioning into chaos. To further characterize the dynamics, complexity is evaluated using the 0–1 test and C0 complexity, both confirming chaotic behavior. Furthermore, two-dimensional control schemes are introduced and theoretically validated to both stabilize the chaotic economic market map and achieve synchronization with a combined response map. The theoretical and numerical results are validated through MATLAB 2016 simulations.

Social media occupies an important place in people’s daily lives where users share various contents and topics such as thoughts, experiences, events and feelings. The massive use of social media has led to the generation of huge volumes of data. These data constitute a treasure trove, allowing the extraction of high volumes of relevant information particularly by involving deep learning techniques. Based on this context, various research studies have been carried out with the aim of studying the detection of mental disorders, notably depression and anxiety, through the analysis of data extracted from the Twitter platform. However, although these studies were able to achieve very satisfactory results, they nevertheless relied mainly on binary classification models by treating each mental disorder separately. Indeed, it would be better if we managed to develop systems capable of dealing with several mental disorders at the same time. To address this point, we propose a well-defined methodology involving the use of deep learning to develop effective multi-class models for detecting both depression and anxiety disorders through the analysis of tweets. The idea consists in testing a large number of deep learning models ranging from simple to hybrid variants to examine their strengths and weaknesses. Moreover, we involve the grid search technique to help find suitable values for the learning rate hyper-parameter due to its importance in training models. Our work is validated through several experiments and comparisons by considering various datasets and other binary classification models. The aim is to show the effectiveness of both the assumptions used to collect the data and the use of multi-class models rather than binary class models. Overall, the results obtained are satisfactory and very competitive compared to related works.

In this work, we present a fractional-order reaction–diffusion model for the spread of infectious diseases, incorporating incommensurate Caputo derivatives to capture memory effects and heterogeneous temporal behavior across compartments. Focusing on a generalized SI model with nonlinear incidence, we explore the local asymptotic stability of both disease-free and endemic equilibria. The model accommodates spatial diffusion, saturation effects, and varying fractional orders, yielding a more realistic depiction of epidemic propagation. Analytical techniques—ranging from linearization to spectral analysis—are employed to rigorously establish stability conditions. Numerical simulations support the theoretical findings, highlighting the impact of memory and spatial structure on long-term dynamics. This study offers a refined mathematical lens to understand the persistence or eradication of infectious diseases under memory-dependent and spatially heterogeneous environments.