<?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>0121-7488</journal-id>
<journal-title><![CDATA[Ciencia en Desarrollo]]></journal-title>
<abbrev-journal-title><![CDATA[Ciencia en Desarrollo]]></abbrev-journal-title>
<issn>0121-7488</issn>
<publisher>
<publisher-name><![CDATA[Universidad Pedagógica y Tecnológica de Colombia]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S0121-74882022000100001</article-id>
<article-id pub-id-type="doi">10.19053/01217488.v13.n1.2022.12650</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[Hopf Bifurcation in the Study of Synchronous Motor Stability]]></article-title>
<article-title xml:lang="es"><![CDATA[Bifurcación de Hopf en el estudio de la estabilidad del motor síncrono]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Mesa]]></surname>
<given-names><![CDATA[Fernando]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Correa]]></surname>
<given-names><![CDATA[Germán]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Barba-Ortega]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
<xref ref-type="aff" rid="Aaf"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Universidad Tecnológica de Pereira  ]]></institution>
<addr-line><![CDATA[Pereira ]]></addr-line>
<country>Colombia</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,Universidad Nacional de Colombia Departamento de Física ]]></institution>
<addr-line><![CDATA[Bogotá ]]></addr-line>
<country>Colombia</country>
</aff>
<aff id="Af3">
<institution><![CDATA[,Foundation of Researchers in Science and Technology of Materials  ]]></institution>
<addr-line><![CDATA[Bucaramanga ]]></addr-line>
<country>Colombia</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>06</month>
<year>2022</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>06</month>
<year>2022</year>
</pub-date>
<volume>13</volume>
<numero>1</numero>
<fpage>1</fpage>
<lpage>7</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_arttext&amp;pid=S0121-74882022000100001&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_abstract&amp;pid=S0121-74882022000100001&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_pdf&amp;pid=S0121-74882022000100001&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract In this work, the dynamic model of the synchronous motor was analyzed, which has a typical structure of Lienard-type systems. For this, the theory of dynamic systems was used, especially the Hopf bifurcation. The objective is to apply this type of bifurcation to the model described in order to show the variations in the equilibrium points of the system by taking as a variable parameter the voltage of the bus to which it is connected. The conditions that the voltage of the infinite bus to which the network is connected must meet in order for it to have asymptotic or spiral stability. It can then be shown that when the bus voltage presents variations, the equilibrium points change their dynamics from asymptotic stability to spiral stability.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen En este trabajo se analizó el modelo dinámico del motor síncrono, el cual tiene una estructura típica de los sistemas tipo Lienard. Para ello se utilizó la teoría de los sistemas dinámicos, en especial la bifurcación de Hopf. El objetivo es aplicar este tipo de bifurcación al modelo descrito para mostrar las variaciones en los puntos de equilibrio del sistema tomando como parámetro variable la tensión de la barra a la que está conectado. Las condiciones que debe cumplir la tensión de la barra infinita a la que está conectada la red para que tenga estabilidad asintótica o espiral. Entonces se puede demostrar que cuando la tensión de la barra presenta variaciones, los puntos de equilibrio cambian su dinámica de estabilidad asintótica a estabilidad espiral.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[Dynamic systems]]></kwd>
<kwd lng="en"><![CDATA[Equilibrium points]]></kwd>
<kwd lng="en"><![CDATA[Periodic orbits]]></kwd>
<kwd lng="en"><![CDATA[Stable system]]></kwd>
<kwd lng="en"><![CDATA[Unstable system]]></kwd>
<kwd lng="es"><![CDATA[Sistemas dinámicos]]></kwd>
<kwd lng="es"><![CDATA[Puntos de equilibrio]]></kwd>
<kwd lng="es"><![CDATA[Orbitas periódicas]]></kwd>
<kwd lng="es"><![CDATA[Sistema estable]]></kwd>
</kwd-group>
</article-meta>
</front><back>
<ref-list>
<ref id="B1">
<label>[1]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Zhu]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<name>
<surname><![CDATA[Chen]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
<name>
<surname><![CDATA[Zhao]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
<name>
<surname><![CDATA[Ma]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Nonlinear dynamic analysis and modeling of fractional permanent magnet synchronous motors]]></article-title>
<source><![CDATA[Journal of Vibration and Control]]></source>
<year>2016</year>
<volume>22</volume>
<numero>7</numero>
<issue>7</issue>
<page-range>1855-75</page-range></nlm-citation>
</ref>
<ref id="B2">
<label>[2]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Arumugam]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
<name>
<surname><![CDATA[Hamiti]]></surname>
<given-names><![CDATA[T]]></given-names>
</name>
<name>
<surname><![CDATA[Brunson]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
<name>
<surname><![CDATA[Gerada]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Analysis of vertical strip wound fault-tolerant permanent magnet synchronous machines]]></article-title>
<source><![CDATA[IEEE Transactions on Industrial Electronics]]></source>
<year>2014</year>
<volume>61</volume>
<page-range>1158-68</page-range></nlm-citation>
</ref>
<ref id="B3">
<label>[3]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Choi]]></surname>
<given-names><![CDATA[H. H.]]></given-names>
</name>
<name>
<surname><![CDATA[Jung]]></surname>
<given-names><![CDATA[J. W]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Fuzzy speed control with an acceleration observer for a permanent magnet synchronous motor]]></article-title>
<source><![CDATA[Nonlinear Dynamics]]></source>
<year>2012</year>
<volume>77</volume>
<page-range>1717-27</page-range></nlm-citation>
</ref>
<ref id="B4">
<label>[4]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Kundur]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
<name>
<surname><![CDATA[Pascrba]]></surname>
<given-names><![CDATA[J]]></given-names>
</name>
<name>
<surname><![CDATA[Ajjarapu]]></surname>
<given-names><![CDATA[V]]></given-names>
</name>
<name>
<surname><![CDATA[Andersson]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
<name>
<surname><![CDATA[Bose]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
<name>
<surname><![CDATA[Cazinares]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
<name>
<surname><![CDATA[Cutsem]]></surname>
<given-names><![CDATA[T. Van]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Definition and classification of power system stability]]></article-title>
<source><![CDATA[IEEE transactions on Power Systems]]></source>
<year>2004</year>
<volume>19</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>1387-401</page-range></nlm-citation>
</ref>
<ref id="B5">
<label>[5]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[DeCarlo]]></surname>
<given-names><![CDATA[R. A.]]></given-names>
</name>
<name>
<surname><![CDATA[Branicky]]></surname>
<given-names><![CDATA[M. S.]]></given-names>
</name>
<name>
<surname><![CDATA[Pettersson]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
<name>
<surname><![CDATA[Lennartson]]></surname>
<given-names><![CDATA[B]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Perspectives and results on the stability and stabilizability of hybrid systems]]></article-title>
<source><![CDATA[Proc. /EEE]]></source>
<year>2000</year>
<volume>88</volume>
<numero>7</numero>
<issue>7</issue>
<page-range>10691082</page-range></nlm-citation>
</ref>
<ref id="B6">
<label>[6]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Lee]]></surname>
<given-names><![CDATA[T. S.]]></given-names>
</name>
<name>
<surname><![CDATA[Ghosh]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[The concept of stability in asynchronous distributed decision-making systems]]></article-title>
<source><![CDATA[IEEE Trans. Systems, Man, and Cybernetics-B: Cybernetics]]></source>
<year>2000</year>
<volume>30</volume>
<page-range>549-61</page-range></nlm-citation>
</ref>
<ref id="B7">
<label>[7]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Fernandopulle]]></surname>
<given-names><![CDATA[N]]></given-names>
</name>
<name>
<surname><![CDATA[Ramshaw]]></surname>
<given-names><![CDATA[R. S.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Analysis of synchronous motor stability using Hopf bifurcation]]></article-title>
<source><![CDATA[Electric Machines and Power Systems]]></source>
<year>1991</year>
<volume>19</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>239-50</page-range></nlm-citation>
</ref>
<ref id="B8">
<label>[8]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Li]]></surname>
<given-names><![CDATA[Z]]></given-names>
</name>
<name>
<surname><![CDATA[Park]]></surname>
<given-names><![CDATA[J. B.]]></given-names>
</name>
<name>
<surname><![CDATA[Joo]]></surname>
<given-names><![CDATA[Y. H.]]></given-names>
</name>
<name>
<surname><![CDATA[Zhang]]></surname>
<given-names><![CDATA[B]]></given-names>
</name>
<name>
<surname><![CDATA[Chen]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Bifurcations and chaos in a permanent-magnet synchronous motor]]></article-title>
<source><![CDATA[IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications]]></source>
<year>2002</year>
<volume>49</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>383-7</page-range></nlm-citation>
</ref>
<ref id="B9">
<label>[9]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Guckenheimer]]></surname>
<given-names><![CDATA[J]]></given-names>
</name>
<name>
<surname><![CDATA[Holmes]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
</person-group>
<source><![CDATA[Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields]]></source>
<year></year>
<page-range>198</page-range><publisher-loc><![CDATA[New York ]]></publisher-loc>
<publisher-name><![CDATA[Springer-Verlag]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B10">
<label>[10]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Strogatz]]></surname>
<given-names><![CDATA[S. H.]]></given-names>
</name>
<name>
<surname><![CDATA[Fox]]></surname>
<given-names><![CDATA[R. F.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Nonlinear dynamics and chaos: With applications to physics, biology, chemistry and engineering]]></article-title>
<source><![CDATA[Physics Today]]></source>
<year>1995</year>
<volume>48</volume>
<page-range>196</page-range></nlm-citation>
</ref>
<ref id="B11">
<label>[11]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Lenci]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
<name>
<surname><![CDATA[Regga]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
</person-group>
<source><![CDATA[Global Nonlinear Dynamics for Engineering Design and System Safety]]></source>
<year>2019</year>
<publisher-name><![CDATA[Springer]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B12">
<label>[12]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Roldán]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
<name>
<surname><![CDATA[Martinez-Moreno]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<name>
<surname><![CDATA[Roldán]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
<name>
<surname><![CDATA[Karapinar]]></surname>
<given-names><![CDATA[E]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Some remarks on multidimensional fixed-point theorems]]></article-title>
<source><![CDATA[Fixed Point Theory]]></source>
<year>2014</year>
<volume>15</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>545-58</page-range></nlm-citation>
</ref>
<ref id="B13">
<label>[13]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Shaddad]]></surname>
<given-names><![CDATA[F]]></given-names>
</name>
<name>
<surname><![CDATA[Noorani]]></surname>
<given-names><![CDATA[M. S]]></given-names>
</name>
<name>
<surname><![CDATA[Alsulami]]></surname>
<given-names><![CDATA[S. M.]]></given-names>
</name>
<name>
<surname><![CDATA[Akhadkulov]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Coupled point results in partially ordered metric spaces without compatibility]]></article-title>
<source><![CDATA[Fixed Point Theory and Applications]]></source>
<year>2014</year>
<page-range>197-204</page-range></nlm-citation>
</ref>
<ref id="B14">
<label>[14]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Guysinsky]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
<name>
<surname><![CDATA[Hasselblatt]]></surname>
<given-names><![CDATA[B]]></given-names>
</name>
<name>
<surname><![CDATA[Rayskin]]></surname>
<given-names><![CDATA[V]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Differentiability of the Hartman-Grobman linearization]]></article-title>
<source><![CDATA[Discrete and Continuous Dynamical Systems]]></source>
<year>2003</year>
<volume>9</volume>
<numero>4</numero>
<issue>4</issue>
<page-range>979-84</page-range></nlm-citation>
</ref>
<ref id="B15">
<label>[15]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Coayla-Teran]]></surname>
<given-names><![CDATA[E. A.]]></given-names>
</name>
<name>
<surname><![CDATA[Mohammed]]></surname>
<given-names><![CDATA[S. E. A.]]></given-names>
</name>
<name>
<surname><![CDATA[Ruffino]]></surname>
<given-names><![CDATA[P. R. C.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Hartman-Grobman theorems along hyperbolic stationary trajectories]]></article-title>
<source><![CDATA[Discrete and Continuous Dynamical Systems]]></source>
<year>2007</year>
<volume>17</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>281</page-range></nlm-citation>
</ref>
<ref id="B16">
<label>[16]</label><nlm-citation citation-type="confpro">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Kolmogorov]]></surname>
<given-names><![CDATA[V]]></given-names>
</name>
<name>
<surname><![CDATA[Zabih]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
</person-group>
<source><![CDATA[What energy functions can be minimized via graph cuts?]]></source>
<year>2002</year>
<conf-name><![CDATA[ European conference on computer vision]]></conf-name>
<conf-loc>Berlin, Heidelberg </conf-loc>
<page-range>65-81</page-range></nlm-citation>
</ref>
<ref id="B17">
<label>[17]</label><nlm-citation citation-type="confpro">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Kolmogorov]]></surname>
<given-names><![CDATA[V]]></given-names>
</name>
<name>
<surname><![CDATA[Zabih]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
</person-group>
<source><![CDATA[Visual correspondence with occlusions usinggraph cuts]]></source>
<year>2001</year>
<conf-name><![CDATA[ Proceedings Eight International Conference on Computer Vision]]></conf-name>
<conf-loc>Canada </conf-loc>
<page-range>508-15</page-range></nlm-citation>
</ref>
<ref id="B18">
<label>[18]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Rosehart]]></surname>
<given-names><![CDATA[W. D.]]></given-names>
</name>
<name>
<surname><![CDATA[Cañizares]]></surname>
<given-names><![CDATA[C. A.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Bifurcation analysis of various power system models]]></article-title>
<source><![CDATA[International Journal of Electrical Power &amp; Energy Systems]]></source>
<year>1999</year>
<volume>21</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>171-82</page-range></nlm-citation>
</ref>
<ref id="B19">
<label>[19]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Meesa]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
<name>
<surname><![CDATA[L. Chua]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems]]></article-title>
<source><![CDATA[IEEE transactions on circuits and systems]]></source>
<year>1979</year>
<volume>26</volume>
<numero>4</numero>
<issue>4</issue>
<page-range>235-54</page-range></nlm-citation>
</ref>
</ref-list>
</back>
</article>
