<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>1409-2433</journal-id>
<journal-title><![CDATA[Revista de Matemática Teoría y Aplicaciones]]></journal-title>
<abbrev-journal-title><![CDATA[Rev. Mat]]></abbrev-journal-title>
<issn>1409-2433</issn>
<publisher>
<publisher-name><![CDATA[Centro de Investigaciones en Matemática Pura y Aplicada (CIMPA) y Escuela de Matemática, San José, Costa Rica.]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S1409-24332023000200229</article-id>
<article-id pub-id-type="doi">10.15517/rmta.v30i2.50545</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[The van der Pol equation: qualitative and numerical study]]></article-title>
<article-title xml:lang="es"><![CDATA[Ecuación de van der Pol: estudio cualitativo y numérico]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Pinto]]></surname>
<given-names><![CDATA[Vinícius Justen]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Salgado]]></surname>
<given-names><![CDATA[Luciana]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Universidade Federal do Rio de Janeiro  Department of Electrical Engineering]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Brazil</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,Universidade Federal do Rio de Janeiro  Department of Mathematics]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Brazil</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>12</month>
<year>2023</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>12</month>
<year>2023</year>
</pub-date>
<volume>30</volume>
<numero>2</numero>
<fpage>229</fpage>
<lpage>251</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_arttext&amp;pid=S1409-24332023000200229&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_abstract&amp;pid=S1409-24332023000200229&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_pdf&amp;pid=S1409-24332023000200229&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract This expositive paper aims at the study of nonlinear equations, focused on the van der Pol equation, including deduction, qualitative analysis, and nu- merical 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 Poincar'e-Bendixson's and Lyapunov's. The construction of a Lyapunov function is also performed. Finally, a series of numerical ex- amples 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 exper- imentally, 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.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen Este artículo expositivo tiene como objetivo el estudio de ecuaciones no lineales, enfocado en la ecuación de van der Pol, incluyendo deducción, análisis cualitativo y ejemplos numéricos. La ecuación de van der Pol es deducida utilizando un circuito eléctrico como modelo físico. El análisis cualitativo está dividido en dos partes: enunciación teórica y sus aplicaciones. Los teoremas principales usados en este estudio son los de Poincaré-Bendixson y de Lyapunov. Se hace también la construcción de una función de Lyapunov. Finalmente, una serie de ejemplos numéricos son ilustrados gráficamente utilizando herramientas computacionales como Python y Octave. Son exhibidos retratos de fase y comportamientos temporales, así como la cuenca de atracción obtenida experimentalmente, en comparación con la obtenida por la función de Lyapunov. Por consiguiente, el estudio numérico proporciona una representación visual de los resultados determinados en el análisis cualitativo.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[van der Pol equation]]></kwd>
<kwd lng="en"><![CDATA[Lyapunov function]]></kwd>
<kwd lng="en"><![CDATA[qualitative analysis]]></kwd>
<kwd lng="en"><![CDATA[numerical integration.]]></kwd>
<kwd lng="es"><![CDATA[ecuación de van der Pol]]></kwd>
<kwd lng="es"><![CDATA[función de Lyapunov]]></kwd>
<kwd lng="es"><![CDATA[análisis cualitativo]]></kwd>
<kwd lng="es"><![CDATA[integración numérica.]]></kwd>
</kwd-group>
</article-meta>
</front><back>
<ref-list>
<ref id="B1">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[G]]></surname>
<given-names><![CDATA[Austin]]></given-names>
</name>
<name>
<surname><![CDATA[W]]></surname>
<given-names><![CDATA[Hayward]]></given-names>
</name>
<name>
<surname><![CDATA[C]]></surname>
<given-names><![CDATA[Tsai]]></given-names>
</name>
<name>
<surname><![CDATA[A]]></surname>
<given-names><![CDATA[Kuykendall]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Parkinsonian tremor: some aspects of an experimental model and its solution]]></article-title>
<source><![CDATA[Confinia Neurologica]]></source>
<year>1965</year>
<volume>26</volume>
<numero>3-5</numero>
<issue>3-5</issue>
<page-range>389-403</page-range></nlm-citation>
</ref>
<ref id="B2">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[M, L]]></surname>
<given-names><![CDATA[Cartwright]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Balthazar van der Pol]]></article-title>
<source><![CDATA[J. London Math. Soc.]]></source>
<year>1960</year>
<volume>35</volume>
<page-range>367-76</page-range></nlm-citation>
</ref>
<ref id="B3">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[A]]></surname>
<given-names><![CDATA[Fleitas]]></given-names>
</name>
<name>
<surname><![CDATA[J, A]]></surname>
<given-names><![CDATA[Méndez-Bermúdez]]></given-names>
</name>
<name>
<surname><![CDATA[J, E]]></surname>
<given-names><![CDATA[Nápoles Valdés]]></given-names>
</name>
<name>
<surname><![CDATA[J, M]]></surname>
<given-names><![CDATA[Sigarreta Almira]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[On fractional Liénard-type systems]]></article-title>
<source><![CDATA[Rev. Mexicana Fís.]]></source>
<year>2019</year>
<volume>65</volume>
<page-range>618-25</page-range></nlm-citation>
</ref>
<ref id="B4">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[M, W]]></surname>
<given-names><![CDATA[Hirsch]]></given-names>
</name>
<name>
<surname><![CDATA[S]]></surname>
<given-names><![CDATA[Smale]]></given-names>
</name>
<name>
<surname><![CDATA[R, L]]></surname>
<given-names><![CDATA[Devaney]]></given-names>
</name>
</person-group>
<source><![CDATA[Differential equations, dynamical systems, and an introduction to chaos]]></source>
<year>2013</year>
<edition>Third</edition>
<publisher-loc><![CDATA[Amsterdam ]]></publisher-loc>
<publisher-name><![CDATA[Elsevier/Academic Press]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B5">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[H, J]]></surname>
<given-names><![CDATA[Marquez.]]></given-names>
</name>
</person-group>
<source><![CDATA[Nonlinear control systems: analysis and design]]></source>
<year>2003</year>
<volume>161</volume>
<publisher-loc><![CDATA[NJ ]]></publisher-loc>
<publisher-name><![CDATA[John Wiley]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B6">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[J, E]]></surname>
<given-names><![CDATA[Nápoles Valdes]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A century of qualitative theory of ordinary differential equations]]></article-title>
<source><![CDATA[Lect. Mat.]]></source>
<year>2004</year>
<volume>25</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>59-111</page-range></nlm-citation>
</ref>
<ref id="B7">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[B]]></surname>
<given-names><![CDATA[van der Pol]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[LXXXVIII. On relaxation-oscillations]]></article-title>
<source><![CDATA[The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science]]></source>
<year>1926</year>
<volume>2</volume>
<numero>11</numero>
<issue>11</issue>
<page-range>978-92</page-range></nlm-citation>
</ref>
<ref id="B8">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[B]]></surname>
<given-names><![CDATA[van der Pol]]></given-names>
</name>
<name>
<surname><![CDATA[J]]></surname>
<given-names><![CDATA[van der Mark]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[LXXII. The heartbeat considered as a relaxation oscillation, and an electrical model of the heart. The London]]></article-title>
<source><![CDATA[Edinburgh, and Dublin Philosophical Magazine and Journal of Science]]></source>
<year>1928</year>
<volume>6</volume>
<numero>38</numero>
<issue>38</issue>
<page-range>763-75</page-range></nlm-citation>
</ref>
<ref id="B9">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[J, N]]></surname>
<given-names><![CDATA[Valdés]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Differential equations and contemporaneity]]></article-title>
<source><![CDATA[Revista Brasileira de História da Matemática]]></source>
<year>2007</year>
<volume>7</volume>
<numero>14</numero>
<issue>14</issue>
<page-range>213-32</page-range></nlm-citation>
</ref>
</ref-list>
</back>
</article>
