The paper studies oscillograms, phase trajectories and bifurcation diagrams of the nonlinear FitzHugh-Nagumo oscillator with variable heredity (fractional FitzHugh-Nagumo oscillator). The model equation for the fractional FitzHugh-Nagumo oscillator contains derivatives of fractional variables of the Gerasimov-Caputo type, as well as a function of external influence, which depend on time. Local initial conditions are valid for the model equation of the fractional oscillator. Due to the nonlinearity of the fractional FitzHugh-Nagumo oscillator, a numerical algorithm based on a nonlocal finite-difference scheme of the first order of accuracy was used. Then, using the Runge rule and computer experiments, an estimate of the computational accuracy is given. The numerical algorithm was implemented in the Maple 2021 environment for constructing oscillograms and phase trajectories and in Python for constructing bifurcation diagrams. It is shown that with an increase in the nodes of the computational grid, the computational accuracy tends to theoretical estimates. Further in the article the dynamic modes of the fractional FitzHugh-Nagumo oscillator are studied using bifurcation diagrams. It is shown that regular modes can be characterized not only by relaxation oscillations, but also by damped ones. The mode type depends on the values of the external action parameters, as well as on the values of the initial conditions. It is also shown that limit cycles may not always be stable. Bifurcation diagrams were constructed, which confirmed the dynamics obtained using oscillograms and phase trajectories. The question of rigorous justification of the existence of a unique stable limit cycle remains open. For a more rigorous definition of the conditions for the existence and uniqueness of a limit cycle, as well as its stability, analogs of the theorems of Liénard and Bendixson are needed. However, the results obtained in the article allow the fractional FitzHugh-Nagumo oscillator to be used to describe self-oscillatory processes. Further development of research can be associated with the construction of maps of dynamic modes, which is a rather labor-intensive task in computational terms. Such problems can be solved on a computing server using parallel programming methods.
| Mualliflar | Alimova , N.B., Parovik, R.I., Алимова , Н.Б., Паровик, Р.И. |
|---|---|
| Jurnal | Ҳисоблаш ва амалий математика муаммолари |
| Nashr sanasi | 2025-03-22 |
| Son | 1 |
| Betlar | 5-16 |
| Til | Rus |
| DOI | 10.71310/pcam.1_63.2025.01 |
DOI: 10.71310/pcam.1_63.2025.01 · Maqolaning asl sahifasi
fractional FitzHugh-Nagumo oscillator, oscillograms, phase trajectories, nonlocal explicit finite-difference scheme, limit cycle, stability, bifuptation diagram, дробный осциллятор ФитцХью-Нагумо, осциллограммы, фазовые траектории, нелокальная явная конечно-разностная схема, предельный цикл, устойчивость, бифуркационная диаграмма
The article analyzes the theoretical and practical aspects of the mathematical model for forecasting groundwater movement and level based on the studied literature. Using this model allows for efficient management…
This paper presents the Alternating Direction Method of Multipliers (ADMM) optimizer, a powerful tool for solving optimization problems, especially in the context of mixed-binary constrained optimization (MBCO). This…
The article considers the results of experimental and numerical studies of atmospheric pollution processes in industrial zones. The proposed mathematical model helps to solve problems related to monitoring and…
Models and algorithms for data interaction based on a multi-criteria evolutionary algorithm are a promising direction for improving transport logistics in rural areas. Their use allows not only to increase the…
Modeling of nonlinear filtration processes in multilayer heterogeneous porous media is a complex area that combines various mathematical and physical principles to understand the dynamics of fluid in porous structures…
The subject of this article is modeling the process of propagation of gas mixtures and aerosol particles in the surface layer of the atmosphere. A model of the process of propagation of industrial emissions in the…
Environmental pollution is one of the most critical issues of our time, as it leads to the deterioration of human health, the degradation of flora and fauna, and the suppression of vegetation. To study and predict the…
The paper studies the process of wet deposition of radioactive impurities, including their interaction with rain, snow, and fog. Mathematics models describing the deposition process are presented based on data from the…
This study presents a mathematical model for turbulent pollutant dispersion in urban airflows based on the Navier-Stokes equations and the ????−???? turbulence model. The model incorporates meteorological parameters…
The mathematical model of the cleaning process of a cylindrical device operating on centrifugal force, describing the movement of particles and their resistance coefficients, is presented in the form of a system of…