Fractional-order models are capable of capturing memory, nonlocality, and anomalous dispersion, making them powerful tools for describing nonlinear wave propagation in complex media. This work investigates the influence of fractional and integer-order derivatives on the multidimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili (FDSKP) equation, a higher-dimensional model widely used to characterize internal waves, optical pulses, and fluid–structure interactions. Here, the term fractional parameters refers explicitly to the orders of the fractional time derivatives appearing in the governing equations no spatial fractional derivatives are considered. Physically, fractional time derivatives introduce temporal nonlocality through memory effects, implying that the system evolution depends on its entire past history and can capture long-memory dynamics beyond the capability of classical integer-order models. Fractional beta, M-truncated, and classical integer-order derivatives are incorporated, and the resulting modifications in wave structures under different operator definitions are systematically analyzed. The FDSKP equation is reduced to an ordinary differential equation via appropriate traveling-wave transformations corresponding to each derivative type, and exact analytical solutions are obtained using the Modified Auxiliary Equation (MAE) and Jacobi Elliptic Function (JEF) methods. The proposed framework generates a rich family of wave structures, including bell-shaped, W-shaped, composite dark–bright, and periodic waves. The results demonstrate that the fractional parameters effectively regulate amplitude, steepness, and overall wave dynamics with greater flexibility than classical integer-order derivatives. Furthermore, two- and three-dimensional visualizations reveal the dependence of dispersion characteristics and symmetry on the fractional order. Overall, the findings confirm that the choice of fractional operator significantly influences the physical interpretation of the model, highlighting the importance of fractional calculus for accurately representing complex wave dynamics in higher-dimensional nonlinear systems.

This paper investigates the fractal dynamics of a discrete Caputo fractional-order hepatitis C virus (HCV) system. Due to the memory effect and the hereditary property, the discrete fractional model helps in the study of the fractal behavior of the HCV model. First, the three-dimensional integer-order HCV model is extended to the fractional-order one, and the corresponding Julia set is defined. Next, a fractional-order controller based on a coordinate transformation is designed to control the system’s Julia set. The stability interval of the controlled system is determined by calculating the spectral radius of the Jacobian matrix at the system’s fixed point, and numerical simulations are then presented to illustrate how the Julia set changes as the control parameters are increased within the stability interval. In addition, a nonlinear coupling controller is constructed and added to the three-dimensional model to achieve synchronization between two discrete fractional-order systems with different fractional orders and different parameters. Rigorous mathematical proofs are provided to establish the correctness and effectiveness of the proposed synchronization method. Moreover, numerical simulation figures are presented to illustrate the synchronization of the response system’s Julia sets toward the target system’s Julia sets as the synchronization parameters vary. These results contribute to a systematic characterization of fractal dynamical behavior of the fractional-order HCV system and provide useful insights fto inform HCV control strategies.

The stochastic P-bifurcation characteristics of a bistable Van der Pol-Rayleigh system incorporating fractional-order inertia and damping are examined under the joint effects of additive and multiplicative Gaussian colored noises. By invoking the minimum mean square error criterion, the fractional-order inertial and damping components are reformulated as the combinations of integer-order stiffness, damping, and inertial terms, allowing the original fractional system to be transformed into an isovalent integer-order dynamical system.

On this basis, the stationary probability density function of the system amplitude is obtained through the stochastic averaging technique. The parametric conditions governing the emergence of stochastic P-bifurcation are then identified using singularity theory. Furthermore, the qualitative topology evolution of the steady-state probability density is investigated by analyzing representative parameters located in distinct regions delineated by the transition curves.

To verify the analytical process, the numerical results obtained from Monte Carlo simulations and a Radial Basis Function Neural Network (RBFNN) are compared with the theoretical solutions. The high agreement among these results confirms the validity of the analytical framework and the accuracy of the transition set acquired. The findings provide theoretical insights into the control of bifurcation and vibration suppression in nonlinear fractional systems under random excitation, with potential applications in engineering design, aeroelastic stability, and vibration control of structures with memory-dependent properties.

Air pollution is one of the most significant threats to environmental sustainability and public health, with particulate matter PM2.5PM_{2.5}PM2.5​ and PM10PM_{10}PM10​ recognized as among the most hazardous pollutants. Elevated concentrations of these fine particles are strongly associated with cardiovascular and respiratory diseases, leading to increased mortality worldwide. Understanding the dynamics of particulate matter and its health impacts is therefore essential for effective mitigation strategies and policy development. In this study, we propose a novel mathematical model to investigate air pollution dynamics by incorporating population, PM2.5, PM10, and pollution-induced mortality as key variables. The model is analyzed under deterministic, fractional-order, and stochastic frameworks. Fractional-order modeling is formulated using the Caputo derivative to capture memory and hereditary effects, while stochastic differential equations are employed to account for environmental randomness and uncertainty. The well-posedness of both the fractional and stochastic models is rigorously established. Stability and boundedness properties of the fractional-order system are examined, and the feasibility of equilibrium points is analyzed using isoclines and asymptotic behavior. For numerical simulations, the Adams–Bashforth–Moulton method is applied to the fractional-order model, whereas Milstein’s scheme is utilized for the stochastic system. Sensitivity analysis is conducted to evaluate the influence of key parameters on system dynamics. To enhance predictive performance, machine learning techniques are integrated with the mathematical framework. Data-driven forecasting methods, including the ARIMA model and random forest regression, are employed to capture both short-term fluctuations and long-term trends in pollutant levels. By combining analytical modeling with data-driven approaches, the proposed framework improves forecasting accuracy. It provides deeper insights into the complex interactions among air quality, particulate matter, and associated health risks.

In this study, we propose a nutrient–plankton–fish interaction model in a marine environment incorporating the Caputo fractional operator to investigate the role of various parameters, such as the mortality rate of zooplankton, the maximal ingestion rate of fish, and the nutrient consumption rate by phytoplankton, in stabilizing the system. To investigate the qualitative properties of the model, the fundamental properties of its solutions and the stability of its equilibria are examined. Further, to investigate the effect of the order of the fractional derivative on system stability, we have arbitrarily chosen four values: 0.91, 0.94, 0.97, and 1. To verify the theoretical results fo the system's stability, we present numerical examples using MATLAB. The findings identify specific biological rates that contribute to system instability, providing critical information for control. Reducing zooplankton mortality, potentially through reduced pesticide use, may unintentionally disrupt population dynamics, leading to erratic fluctuations. Also, the negative relationship between fish ingestion rate and stability serves as a cautionary tale for fishery managers. Policies that actively promote high ingestion rates risk causing the entire fish population to collapse into chaotic patterns. Moreover, the fractional-order derivative approach enhances realism by accounting for memory effects, offering deeper insights into stability and resilience compared to classical integer-order models.