Keywords
Lyapunov function
Qualitative analysis
Numerical integration
Ecuación de Van der Pol
Función de Lyapunov
Análisis cualitativo
integración numérica
How to Cite
Abstract
This expositive paper aims at the study of nonlinear equations, focused on the van der Pol equation, including deduction, qualitative analysis, and numerical examples. The van der Pol equation is deduced using an electrical circuit as a physical model. The qualitative analysis is divided into two parts: the theoretical enunciation and its application. The main theorems used in this study are Poincaré-Bendixson’s and Lyapunov’s. The construction of a Lyapunov function is also performed. Finally, a series of numerical examples are graphically presented using computational tools such as Python and Octave. The phase portraits and temporal behavior of the van der Pol equation are exhibited, along with the basin of attraction obtained experimentally, compared with the basin of attraction yielded by the Lyapunov function. Therefore, the numerical study provides a visual representation of the results stated in the qualitative analysis
References
G. Austin, W. Hayward, C. Tsai, A. Kuykendall, Parkinsonian tremor: some aspects of an experimental model and its solution. Confinia Neurologica 26(1965), no. 3-5, 389–403. doi: 10.1159/000104056
M. L. Cartwright, Balthazar van der Pol. J. London Math. Soc. 35(1960), 367–376. doi: 10.1112/jlms/s1-35.3.367
A. Fleitas, J. A. Méndez-Bermúdez, J. E. Nápoles Valdés, J. M. Sigarreta Almira, On fractional Liénard-type systems. Rev. Mexicana Fís. 65(2019), no. 6, 618–625. doi: 10.31349/revmexfis.65.618
M. W. Hirsch, S. Smale, R. L. Devaney, Differential equations, dynamical systems, and an introduction to chaos. Third. Elsevier/Academic Press, Amsterdam, 2013, xiv+418. doi: 10.1016/B978-0-12-382010-5.00001-4
H. J. Marquez, Nonlinear control systems: analysis and design. Vol. 161. John Wiley, NJ, 2003.
J. E. Nápoles Valdes, A century of qualitative theory of ordinary differential equations. Lect. Mat. 25(2004), no. 1, 59–111.
B. van der Pol, LXXXVIII. On relaxation-oscillations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 2(1926), no. 11, 978–992. doi: 10.1080/14786442608564127
B. van der Pol, J. van der Mark, LXXII. The heartbeat considered as a relaxation oscillation, and an electrical model of the heart. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 6(1928), no. 38, 763–775. doi: 10.1080/14786441108564652
J. N. Valdés, Differential equations and contemporaneity. Revista Brasileira de História da Matemática 7(2007), no. 14, 213–232.
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