UNA REVISIÓN DE LA ECUACIÓN DE VAN DER POL Y SUS MODIFICACIONES

Autores/as

Winter Daniel Mendoza Mendoza Instituto de Posgrado Universidad Técnica de Manabí, Portoviejo, Ecuador. https://orcid.org/0000-0001-5929-630X
Antonio Ramón Acosta Orellana Yachay Tech, 100119 Urcuquí, Ecuador. https://orcid.org/0000-0003-3616-8377
Luis Bladismir Ruiz Leal Departamento de Matemáticas y Estadística, Universidad Técnica de Manabí, Portoviejo, Ecuador. https://orcid.org/0000-0002-7737-3847

DOI:

https://doi.org/10.33936/revbasdelaciencia.v7iESPECIAL.4743

Palabras clave:

Ciclo Límite, Ecuación de Van der Pol, Ecuación Modificada de Van der Pol y Oscilador de Relajación.

Resumen

En los años veinte Van der Pol y Van der Mark descubren que la ecuación de segundo orden de oscilación de relajación modelaba el fenómeno generado cuando un marcapasos impulsa a un corazón real. Esto generó gran interés en el estudio de la ecuación de Van der Pol y Van der Mark, desde entonces se han realizado una gran variedad de modificaciones a la ecuación con la finalidad de obtener mejores resultados del estudio de la dinámica de ritmos cardiacos, modelar fenó- menos físicos y biológicos entre otros. Este trabajo exhibe una revisión y evolución de la ecuación de Van der Pol desde sus inicios hasta la actualidad enfocado en aquellas ecuaciones de segundo orden relacionadas con la dinámica del ritmo cardiaco. Se destaca la aparición de la llamada ecuación modificada de Van der Pol a partir de los años noventa hasta la actualidad.

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Citas

Acosta, A., Gallo, R., Garcı́a, P., y Peluffo-Ordóñez, D. (2022). Positive invariant regions for a modi- fied Van Der Pol equation modeling heart action. Applied Mathematics and Computation, 442, 127732.

A.P. Kuznetsov and N.V. Stankevich and L.V. Turukina. (2009). Coupled van der Pol–Duffing osci- llators: Phase dynamics and structure of synchronization tongues. Physica D: Nonlinear Phe- nomena, 238(14), 1203-1215. https://doi.org/10.1016/j.physd.2009.04.001

Cardarilli, G. C., Nunzio, L. D., Fazzolari, R., Re, M., y Silvestri, F. (2019). Improvement of the Cardiac Oscillator Based Model for the Simulation of Bundle Branch Blocks. Applied Sciences, 9(18), 3653. https://doi.org/10.3390/app9183653

Cartwright, M. L., y Littlewood, J. E. (1945). On Non-Linear Differential Equations of the Second Order: I. the Equation y ̈ - k (1-y 2 )y∙ y = b k cos(l ), k Large. Journal of the London Mathematical Society, s1-20(3), 180-189. https://doi.org/10.1112/jlms/s1-20.3.180

Dixit, S., Sharma, A., y Shrimali, M. D. (2019). The dynamics of two coupled Van der Pol oscillators with attractive and repulsive coupling. Physics Letters A, 383(32), 125930. https://doi.org/10. 1016/j.physleta.2019.125930

FitzHugh, R. (1961). Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical Journal, 1(6), 445-466. https://doi.org/10.1016/s0006-3495(61)86902-6

Giacomini, H., y Neukirch, S. (1997). Number of limit cycles of the Liénard equation. Physical Review E, 56(4), 3809-3813. https://doi.org/10.1103/physreve.56.3809

Ginoux, J.-M. (2017). History of Nonlinear Oscillations Theory in France (1880-1940). Springer In- ternational Publishing. https://doi.org/10.1007/978-3-319-55239-2

Ginoux, J.-M., y Letellier, C. (2012). Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept. Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(2), 023120. https://doi.org/10.1063/1.3670008

Grant, R. P. (1956). The mechanism of A-V arrhythmias. The American Journal of Medicine, 20(3), 334-344. https://doi.org/10.1016/0002-9343(56)90118-8

Grudziński, K., y Żebrowski, J. J. (2004). Modeling cardiac pacemakers with relaxation oscillators. Physica A: Statistical Mechanics and its Applications, 336(1-2), 153-162. https://doi.org/10. 1016/j.physa.2004.01.020

Jawarneh, I., y Staffeldt, R. (2019). CONLEY INDEX METHODS DETECTING BIFURCATIONS IN AMODIFIED VAN DER POL OSCILLATOR APPEARING IN HEARTACTION MO- DELS. https://www.researchgate.net/publication/330775277

J.J.O’Connor y E.F.Robertson. (2008). Balthasar van der Pol. School of Mathematics and Statistics- University of St Andrews, Scotland. https://mathshistory.st-andrews.ac.uk/Biographies/Van_ der_Pol/

Katholi, C., Urthaler, F., Macy, J., y James, T. (1977). A mathematical model of automaticity in the sinus node and AV junction based on weakly coupled relaxation oscillators. Computers and Biomedical Research, 10(6), 529-543. https://doi.org/10.1016/0010-4809(77)90011-8

Levinson, N. (1949). A Second Order Differential Equation with Singular Solutions. The Annals of Mathematics, 50(1), 127. https://doi.org/10.2307/1969357

Levinson, N., y Smith, O. K. (1942). A general equation for relaxation oscillations. Duke Mathematical Journal, 9(2). https://doi.org/10.1215/s0012-7094-42-00928-1

Low, L. A., Reinhall, P. G., Storti, D. W., y Goldman, E. B. (2006). Coupled van der Pol oscillators as a simplified model for generation of neural patterns for jellyfish locomotion. Structural Control and Health Monitoring, 13(1), 417-429. https://doi.org/10.1002/stc.133

Martı́n, S. G., y Lafuente, V. (2017). Referencias bibliográficas: indicadores para su evaluación en trabajos cientı́ ficos (Vol. 31). Universidad Nacional Autonoma de Mexico. https://doi.org/10. 22201/iibi.0187358xp.2017.71.57814

Postnov, D., Han, S. K., y Kook, H. (1999). Synchronization of diffusively coupled oscillators near the homoclinic bifurcation. Physical Review E, 60(3), 2799-2807. https://doi.org/10.1103/ physreve.60.2799

Ryzhii, E., y Ryzhii, M. (2014). Modeling of Heartbeat Dynamics with a System of Coupled Nonlinear Oscillators. En Communications in Computer and Information Science (pp. 67-75). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-54121-6_6

Suarez, C., Mesahuanca, C., Hernandez, F., y Felipe Tolentino, W. (2020). El Oscilador de Van Der Pol.

van der Pol, B. [B.]. (1934). The Nonlinear Theory of Electric Oscillations. Proceedings of the IRE, 22(9), 1051-1086. https://doi.org/10.1109/jrproc.1934.226781

van der Pol, B. [Balth.]. (1926). LXXXVIII.On “relaxation-oscillations”. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11), 978-992. https://doi.org/10. 1080/14786442608564127

van der Pol, B. [Balth], y Bremmer, H. (1937). LXXVI. The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to ra- diotelegraphy and the theory of the rainbow.—Part II. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 24(164), 825-864. https://doi.org/10.1080/ 14786443708565149

van der Pol, B. [Balth.], y van der Mark, J. (1928). LXXII.The heartbeat considered as a relaxa- tion oscillation, and an electrical model of the heart. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 6(38), 763-775. https://doi.org/10.1080/ 14786441108564652

West, B. J., Goldberger, A. L., Rovner, G., y Bhargava, V. (1985). Nonlinear dynamics of the heartbeat. Physica D: Nonlinear Phenomena, 17(2), 198-206. https://doi.org/10.1016/0167-2789(85) 90004-1

Zduniak, B., Bodnar, M., y Foryś, U. (2014). A modified van der Pol equation with delay in a descrip- tion of the heart action. International Journal of Applied Mathematics and Computer Science, 24(4), 853-863. https://doi.org/10.2478/amcs-2014-0063