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.

Heart rate variability (HRV) is an essential physiological marker widely used to assess cardiovascular regulation and autonomic nervous system activity, reflecting complex, nonlinear dynamics in biological control mechanisms. HRV signals demonstrate long-range correlations, scale-invariant properties, and memory effects that challenge traditional integer-order modeling approaches. Fractal geometry and fractional calculus have emerged as powerful frameworks for analyzing such physiological signals, enabling a deeper understanding of temporal and structural complexity. This work explores the application of fractal measures and fractional-order modeling techniques to HRV analysis, emphasizing their ability to capture multiscale structures, self-similarity, and non-local temporal dynamics. Various fractal descriptors and fractional formulations reported in the literature are conceptually compared in terms of their biological interpretability, analytical performance, and potential application under different physiological or pathological conditions. Special attention is given to how non-integer order operators enhance the representation of physiological variability, anomalous fluctuations, and adaptive responses in cardiac rhythms. Furthermore, the discussion includes potential clinical and research implications, highlighting how these approaches may contribute to improved monitoring, diagnosis, and understanding of cardiovascular health. The synthesis highlights the methodological trends, advantages, and current limitations of fractal and fractional approaches, providing a comprehensive perspective on the study of complex biological rhythms. Overall, this work supports the integration of advanced mathematical tools in quantitative biology and underscores their potential for improving understanding and monitoring of cardiovascular dynamics.

This study [Cong, N. D., & Tuan, H. T. (2017) "Generation of nonlocal fractional dynamical systems by fractional differential equations" examines nonlinear Caputo fractional-order differential equations D^α x = f(t,x) for initial conditions x₀ in the reals R^d of dimension d and the extent to which this differential equation generates a nonlocal dynamical system on R^d . Our main conclusion is that this is generally not possible for dimensions, d, larger than one, as different trajectories may meet in finite time.

On the other hand, linear time-periodic systems, D^α x = A(t) x with A(t) = A(t+T), and period time T are yet to be rigorously analyzed in the fractional-order case. In the case of functional differential equations in the sense of Hale, the respective generated dynamical system on the space of continuous functions C plays an important role in the development of Floquet theory for linear time-periodic systems. Both functional and fractional differential equations show nonlocal behavior.

Thus, we study a framework of Caputo-type differential equations for the lower bound - ∞, namely D_-∞^α x = f(t,x), where an initial condition φ ∈ S must be prescribed to formulate an initial value problem and S is a subset of the space of continuous functions C(- ∞,0]. We show that this initial condition manifests as a time-dependent forcing term |F φ (t)| ≤ b (t+ η)^-α on the right-hand side of the differential equation, decaying algebraically, where b, η > 0. Furthermore, we use this bound to analyze solutions of linear time-autonomous systems.

Finally, we investigate how such a system may be proposed to generate a nonlocal dynamical system on S. In particular, we provide nontrivial conditions under which a solution trajectory could belong to S.

Introduction. Neuronal synchronization is critical to various brain processes, such as memory formation and motor control, and to brain disorders such as epilepsy. Neuronal functions and synchronization are achieved through the dynamics of action potentials and chemical neurotransmitters, which are intricately coupled with the mechanical responses of neurons and depend on neuronal structure and interactions with their shared environment. Mathematically, the propagation of action potentials can be described by the Hodgkin-Huxley model, and neuronal synchronization may be modelled using Hodgkin-Huxley models coupled by prescribed controller-based synchronizers. These mathematical models depict the transport of specific ions across neuronal membranes, but ignore neuronal mechanics. In particular, various studies have shown that neuronal functions are highly sensitive to changes in neuronal volume. The homeostatic mechanisms that conserve neuronal volume are facilitated by the mechanical interactions of neurons with the cerebrospinal fluid surrounding them. This work aims to develop a mathematical model that links the functions and volumes of synchronized neurons.

Methods. A mathematical model of two adjacent neurons, functionally synchronized and volumetrically coupled, is proposed. Neuronal functions are described by a fractional Hodgkin-Huxley model where the ion channels are assumed to behave mechanically as variable-order fractional Maxwell linear viscoelastic materials. Neuronal synchronization is characterized by a synchronization time, and the coupled volumetric dynamics are modeled by a modified coupled Lotka-Volterra model in which the rates depend on action potentials, and regulatory volumetric processes are represented as Michaelis–Menten-like terms.

Results. Computer simulations in MATLAB show oscillatory volumetric dynamics similar to those of action potentials, and the return of volumes to their initial values when regulatory mechanisms are present.

Conclusion. The proposed model allows the study of neuronal function-volume dynamics of brain health and disease that may help with the design of better therapies for various disorders.

For decades, anomalous diffusion has attracted growing attention as a fundamental transport mechanism in complex and heterogeneous systems. In media such as porous materials, polymers, and biological tissues, transport often deviates from classical diffusion, so conventional diffusion models fail to describe these phenomena. Memory effects, nonlocal interactions, structural heterogeneity, and strong correlations between constituents typically cause these deviations. Fractional diffusion equations have emerged as powerful tools to overcome these limitations, extending classical approaches by incorporating memory kernels and nonlocality. In this study, we investigate and compare solutions of fractional diffusion equations formulated using the Caputo, Caputo–Fabrizi, and Atangana–Baleanu operators. Analytical solutions are derived and systematically analyzed to investigate the specific impact of different fractional operators on diffusion profiles, transport dynamics, and memory-dependent behavior, highlighting how the choice of operator influences these characteristics. Our comparative analysis reveals notable differences in transport patterns across operators, offering further insights into the interplay among memory effects, structural heterogeneity, and boundary-induced effects. This study provides a comprehensive framework for modeling subdiffusive transport and demonstrates the effectiveness of different fractional operators in capturing anomalous diffusion phenomena in complex media. The findings have potential applications in photothermal characterization and imaging of various functional materials and biological tissues.

This research presents an impulse-based control scheme capable of stabilizing the unstable period-1 orbit of the fractional-difference logistic map. In 1876, a biologist named Robert May popularized the first known simple mathematical model with chaotic solutions. This system utilizes ordinary differences and has a memory horizon of one step. A few decades later, the fractional-difference logistic map was introduced, exhibiting a memory horizon that reaches the initial condition by using Caputo fractional differences. Due to its valuable properties, the fractional difference logistic map has already been adapted in economics, steganography, epidemiology, and engineering. In recent years, we have explored some impulse-based control schemes to stabilize this map's unstable period-1 orbit. In our recent research, it was shown that a naive control scheme does not achieve finite-time stabilization of the unstable period-1 orbit. Instead, an H-rank-based approach is used to achieve finite-time stabilization. However, this version of the control scheme exhibited violent oscillatory transient processes following the control impulse. To improve the stabilization scheme, several findings were made. Firstly, at the cost of some stabilization duration, it is possible to stabilize the unstable period-1 orbit by finding a suitable initial condition within the tolerance corridor δ. Secondly, the coordinate for the unstable period-1 orbit drifts each time a control impulse is applied. And finally, this drift is caused not only by perturbations arising from numerous stabilization impulses but also by computational effects. In conclusion, this proposed control scheme is minimally invasive compared to continuous feedback control because it preserves the system model and requires only a series of small, sparse, and instantaneous control impulses to achieve continuous adaptive stabilization of the unstable period-1 orbit of the fractional difference logistic map.