FracDynZe is a Computer Program for Studying the Dynamics of Cardiac Function using the Fractional Zeeman Oscillator
DOI:
https://doi.org/10.71310/pcam.2_72.2026.01Keywords:
fractional derivatives, modeling, phase trajectory, stability, memory, python, numerical methodsAbstract
This paper examines the dynamic modes of the fractional Zeeman oscillator. The fractional Zeeman oscillator is a system of two ordinary differential equations with fractional derivatives, understood in the Gerasimov-Caputo sense, for which local initial conditions are valid. The fractional Zeeman oscillator describes the dynamics of heart contractions using an electrochemical potential. Due to the nonlinearity of the fractional Zeeman oscillator, a numerical algorithm based on a nonlocal explicit finite-difference scheme was used to obtain a solution. The numerical algorithm was implemented in the FracDynZe computer program, written in Python in the PyCharm environment. The software package allows for visualization and saving of simulation results. This article describes the FracDynZe computer program, which implements a numerical algorithm for a nonlocal explicit finite-difference scheme. Using FracDynZe, test examples demonstrate that the numerical scheme has first-order accuracy. Examples and their visualizations are provided for various values of the Zeeman fractional oscillator parameters. The FracDynZe computer program can be supplemented with a module for the qualitative analysis of the Zeeman fractional oscillator. For example, this module can implement the ability to construct bifurcation diagrams for studying regular and chaotic regimes.
References
Кошелев В.Н., Воробьев В.А., Шляхтенко А.А. Математическое моделирование кровообращения. – М.: Наука, 2001. – 256 с.
Киселев К.А. Модульное моделирование физиологических процессов. – М.: Физматлит, 2012. – 312 с.
Fenner J.W., Brook B., Clapworthy G., et al. The EuroPhysiome: towards a European virtual physiological human // Philosophical Transactions of the Royal Society A. – 2008. – Vol. 366. – P. 2979-2999.
Niederer S.A., Hunter P.J., Smith N.P. A quantitative analysis of cardiac myocyte relaxation: a simulation study // Biophysical journal. – 2006. – Vol. 90. – No. 5. – P. 1697-1722.
Georgieva-Tsaneva G., Gospodinova E. Mathematical Technologies for Modeling Cardiological Data: Heart Rate Variability // International Journal Bioautomation. – 2021. – Vol. 25. – No. 2. – P. 133-142.
Zeeman E.C. Differential equations for the heartbeat and nerve impulse // Salvador symposium on Dynamical Systems. – Academic Press, 1973. – P. 683-741.
Novozhenova O.G. Life And Science of Alexey Gerasimov, One of the Pioneers of Fractional Calculus in Soviet Union // FCAA. – 2017. – Vol. 20. – P. 790-809. – doi: http://dx.doi.org/10.1515/fca-2017-0040.
Caputo M., Fabrizio M. On the notion of fractional derivative and applications to the hysteresis phenomena // Meccanica. – 2017. – Vol. 52. – P. 3043-3052. – doi: http://dx.doi.org/10.1007/s11012-017-0652-y.
Нахушев А.М. Дробное исчисление и его применение. – М.: Физматлит, 2003. – 272 с.
Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. – Amsterdam: Elsevier, 2006. – 523 p.
Traver J.E., Nuevo-Gallardo C., Tejado I., Fern´andez-Portales J., Ortega-Mor´an J.F., Pagador J.B., Vinagre B.M. Cardiovascular Circulatory System and Left Carotid Model: A Fractional Approach to Disease Modeling // Fractal and Fractional. – 2022. – Vol. 6. – No. 2. – 64. – doi: http://dx.doi.org/10.3390/fractalfract6020064.
Alimov Kh.T., Parovik R.I. Simulation of artificial ECGs of a healthy person using the fractional McSherry model // AIP conference proceedings. – 2024. – Vol. 3244. – No. 1. – 020005. – doi: http://dx.doi.org/10.1063/5.0241420.
Исрайилжанова Г.С., Каримов Ш.Т., Паровик Р.И. Математическая дробная модель Зимана для описания сердечных сокращений // Вестн. КРАУНЦ. Физ.-мат. науки. – 2024. – Т. 48. – № 3. – С. 83-84.
Israyiljanova G., Karimov Sh., Parovik R.I. Zeeman’s mathematical fractional model for describing cardiac contractions // AIP Conference Proceedings. – 2025. – Vol. 3356. – No. 1. – 020010. – doi: http://dx.doi.org/10.1063/5.0296360.
Parovik R.I. On a Finite-Difference Scheme for an Hereditary Oscillatory Equation // Journal of Mathematical Sciences. – 2021. – Vol. 253. – No. 4. – P. 547-557.
Diethelm K. The Analysis of Fractional Differential Equations. – Berlin: Springer, 2010. – 260 p.
Shaw Z.A. Learn Python the hard way. – Addison-Wesley Professional, 2024. – 352 p.
Van Horn B.M., Nguyen Q. Hands-On Application Development with PyCharm: Build Applications like a Pro with the Ultimate Python Development Tool. – Birmingham, UK: Packt Publishing Ltd., 2023. – 652 p.
Li´enard A. Etude des oscillations entretenues // Revue g´en´erale de l’´electricit´e. – 1928. – No. 23. – P. 901-912.
Паровик Р.И. Исследование бифуркационных диаграмм дробной динамической системы Селькова для описания автоколебательных режимов микросейсм // Вестник КРАУНЦ. Физико-математические науки. – 2024. – Т. 49. – № 4. – С. 24-35. – doi: http: //dx.doi.org/10.26117/2079-6641-2024-49-4-24-35.
Parovik R.I., Yakovleva T.P. Construction of maps for dynamic modes and bifurcation diagrams in nonlinear dynamics using the Maple computer mathematics software package // Journal of Physics: Conference Series. – 2022. – Vol. 2373. – 052022. – doi: http://dx.doi.org/10.1088/1742-6596/2373/5/052022.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 R.I. Parovik
This work is licensed under a Creative Commons Attribution 4.0 International License.
