Vol. 15, No. 4 (2026): Regular Issue

This paper develops a mathematical framework for modeling Bitcoin price dynamics through a system of coupled stochastic differential equations (SDEs). We capture the complex nonlinear interactions between Bitcoin price and five key factors: investor sentiment, trading volume, mining hashrate, transaction fees, and transaction counts. The model incorporates jump processes to account for sudden price movements and regime-switching to capture state-dependent dynamics. We derive the resulting partial differential equations for derivative pricing and analyze the system's behavior through simulation. Our empirical findings suggest significant feedback mechanisms between network metrics and price dynamics, with hashrate exhibiting the strongest correlation with price movements. The framework provides a foundation for understanding the complex, non-linear, and fractal-like behavior observed in cryptocurrency markets while enabling the pricing of derivatives in this emerging asset class.

In this work, we consider a nonlinear fractional Langevin equation involving the $(k,\psi)$-Hilfer proportional fractional operator, which unifies several well-known fractional derivatives as particular cases, together with nonlocal multi-point fractional boundary conditions. This setting combines the operator and the boundary conditions within a coherent mathematical framework that allows the treatment of a broad class of Langevin-type problems in a unified manner. By means of Banach’s contraction principle and Krasnoselskii’s fixed-point theorem, sufficient conditions ensuring the existence and uniqueness of solutions are derived. Moreover, Ulam-Hyers and Ulam-Hyers-Rassias stability of the solutions are investigated. Illustrative examples are included to highlight the applicability of the obtained results.

This paper presents a new framework for analyzing the stability of fractional-order digital control systems (FODCS) utilizing the N-transform method. In contrast to conventional techniques that rely on Laplace or Grunwald–Letnikov formulations, the proposed approach supports discrete-time modeling and allows for the visualization of stability regions through Riemann surface decomposition. The method derives closed-form unit step responses and introduces generalized stability criteria applicable to both linear and nonlinear systems. A comparative analysis with existing methods demonstrates improved interpret ability and computational efficiency, making this approach a valuable asset for advanced digital control applications.

In this paper, we proved that there exists a universal constant of convergence rate when a Siegel map whose rotation number $\theta_*$ has a periodic continued fraction, on the boundary of any hyperbolic component of any quadratic-like family, is approximated by functions with a parabolic or an attracting cycle in the family of quadratic-like maps. Additionally, the satellite's valuable flowers of the functions converging to the Siegel map also approximate the Siegel disk. In particular, there is a universal convergence rate in the Mandelbrot set when Siegel parameters of any period and rotation number $\theta_*$ are approximated by the centers of hyperbolic components, which are related to some of the denominators of the continued fraction that converge to $\theta_*$.

The aim of this paper is to study the linear and nonlinear stability of four equilibrium positions of a satellite with variable mass distribution, performing an elliptical orbit around a Newtonian center of attraction. To do such research, we restrict to the plane of the orbit, and focuses in the Hamiltonian perspective of the problem, expanding its Hamiltonian function around the equilibrium positions up to terms of fourth order. Regions of linear stability are found in the plane of the parameters $K\times e$, where $e$ is the eccentricity of the orbit, and $K$ is a parameter associated with the tensor of inertia of the satellite. In the regions where linear stability happens, we perform the nonlinear analysis for arbitrary values of $e$ by using numerical methods, and for sufficiently small values of $e$ by analytical methods. We conclude about the nonlinear stability for the nonresonant case and in the case of parametric resonances of orders three and four.

The mathematical modeling and analysis of radial annular fins exposed to heat radiation in a porous medium is the main focus of this work. The transformed, non-dimensional temperature equation is resolved utilizing a recently developed approximate analytical method, yielding a semi-analytical solution. An understandable format is used to display the temperature profile expression that is produced. By contrasting the outcomes with both numerical solution and current semi-analytical technique, the accuracy of the suggested method is assessed. The consistency and dependability of this comparison are very high. Graphical representations are provided to further highlight the importance of several physical parameters, including thermal conductivity, radiation effects, and heat transmission coefficients. Radiation and convection factors elevate the temperature while the internal heat generation factor elevates the fin efficiency. These visuals help people understand how the system works in different situations. Also, the fins' thermal performance is measured by calculating the fin efficiency, which is also shown in a graph to support up the study's semi-analytical results.

This paper investigates the controllability of networked systems with heterogeneous node-specific semilinear control systems. The nonlinearities are assumed to be Hölder continuous with bounded growth. A compact Kronecker-based formulation is developed to represent the system dynamics, and algebraic conditions are derived to ensure controllability under structural and dynamical heterogeneity. The analysis leverages fixed point theorems to guaranty the existence and uniqueness of solutions. The proposed framework generalizes classical PBH-based controllability results to include nonlinear effects and nonuniform actuation. Three illustrative examples - a bidirectional 2-node network, a 3-node ring, and a 3-node star with partial actuation-demonstrate the effectiveness of the proposed controllability conditions. This work contributes a rigorous and flexible extension of controllability theory for realistic semilinear networked control systems.

We consider a fractional predator-prey system under the Caputo fractional derivative describing ecological mathematical models in the presence of environmental influences like the Allee effect, fear effect, and immigration. Particularly, this system arises in the context of interacting animals. We formulate the approximate method namely, fuzzy fractional fourth order Runge-Kutta method (FFRK4) to the underlying system of fuzzy fractional nonlinear differential equations. This approach enables us to study the fractional system over a range of initial values. Our study reveals non-trivial growth rate in fuzzy fractional predator-prey system. The results of the model indicate that this method is effective and easy to use in Fuzzy Fractional Differential Equation (FFDE) systems.

This paper introduces the concept of generalized conformable shift (delay) for the functions defined on an entire time scales by formulating the concept of generalized conformable shifting problem. We establish the existence of solutions and use this framework to define the generalized conformable convolution. A generalized conformable convolution theorem is then proved.

In this paper, a fractional model for glucose-insulin with beta cells is suggested. The model considered involves the new Hattaf mixed fractional (HMF) derivative incorporating well-known types, in particular, Caputo (C), Caputo-Fabrizio (CF), Atangana-Baleanu (AB) and generalized Hattaf fractional (GHF) derivatives. We obtain two theoretical results. Firstly, based on suitable assumptions, it is shown that there is a solution for the system in question. In addition, local stability is demonstrated. {Furthermore, an experimental validation of the model is conducted using real clinical glucose and insulin data, confirming the model's ability to closely replicate the observed physiological behavior and demonstrating the relevance of the proposed HMF formulation in capturing key features of glucose–insulin dynamics}. Finally, in order to confirm our conclusions, numerical simulations are carried out to demonstrate the local stability of the metabolism dynamics. The effectiveness of the proposed method in providing the best approximation of the fractional derivative parameters relies on the local stability of the metabolic dynamics.

The locomotion efficiency of capsule robots is highly sensitive to environmental resistance forces, especially in fluid-filled conditions. This study investigates a millimeter-scale vibro-impact capsule robot designed for gastrointestinal applications. Building on a validated dry friction model, we incorporate viscous drag and buoyancy to simulate realistic fluidic environments. A comprehensive physical and mathematical framework is developed to characterize the capsule's nonlinear dynamics. Numerical simulations and analytical tools, including time history analysis, phase trajectories, Poincaré maps, and bifurcation diagrams, are used to evaluate displacement efficiency across varying friction thresholds and fluid properties. Results show that high dry friction suppresses mobility, voltage decay reduces propulsion, and moderate fluid viscosity can enhance travel distance. These findings help identify optimal operating conditions for stable bidirectional motion and offer practical guidelines for the design and control of capsule robots in complex environments.

The class of Burgers equations is examined within the context of time scales that incorporate forward jump operators. The study establishes the existence of at least one solution, as well as a minimum of two nonnegative solutions and at least three nonnegative solutions. For this aim we use some recent fixed-point theorems. We construct two operators so that any fixed-point of their sum is a solution of the initial value problem (IVP) for the considered class of Burgers equations. The results of this paper are provided with a suitable example to illustrate the main findings.