{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:18Z","timestamp":1740134538859,"version":"3.37.3"},"reference-count":22,"publisher":"Wiley","license":[{"start":{"date-parts":[[2022,9,2]],"date-time":"2022-09-02T00:00:00Z","timestamp":1662076800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100009543","name":"Pontificia Universidad Javeriana","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100009543","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2022,9,2]]},"abstract":"<jats:p>We provide sufficient conditions for the existence of periodic solutions for an idealized electrostatic actuator modeled by the Li\u00e9nard-type equation <jats:inline-formula>\n                     <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                        <a:mover accent=\"true\">\n                           <a:mi>x<\/a:mi>\n                           <a:mo>\u00a8<\/a:mo>\n                        <\/a:mover>\n                        <a:mo>+<\/a:mo>\n                        <a:msub>\n                           <a:mrow>\n                              <a:mi>F<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mi>D<\/a:mi>\n                           <\/a:mrow>\n                        <\/a:msub>\n                        <a:mfenced open=\"(\" close=\")\">\n                           <a:mrow>\n                              <a:mi>x<\/a:mi>\n                              <a:mo>,<\/a:mo>\n                              <a:mover accent=\"true\">\n                                 <a:mi>x<\/a:mi>\n                                 <a:mo>\u0307<\/a:mo>\n                              <\/a:mover>\n                           <\/a:mrow>\n                        <\/a:mfenced>\n                        <a:mo>+<\/a:mo>\n                        <a:mi>x<\/a:mi>\n                        <a:mo>=<\/a:mo>\n                        <a:mi>\u03b2<\/a:mi>\n                        <a:msup>\n                           <a:mrow>\n                              <a:mi mathvariant=\"script\">V<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mn>2<\/a:mn>\n                           <\/a:mrow>\n                        <\/a:msup>\n                        <a:mfenced open=\"(\" close=\")\">\n                           <a:mrow>\n                              <a:mi>t<\/a:mi>\n                           <\/a:mrow>\n                        <\/a:mfenced>\n                        <a:mo>\/<\/a:mo>\n                        <a:msup>\n                           <a:mrow>\n                              <a:mfenced open=\"(\" close=\")\">\n                                 <a:mrow>\n                                    <a:mn>1<\/a:mn>\n                                    <a:mo>\u2212<\/a:mo>\n                                    <a:mi>x<\/a:mi>\n                                 <\/a:mrow>\n                              <\/a:mfenced>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mn>2<\/a:mn>\n                           <\/a:mrow>\n                        <\/a:msup>\n                        <a:mo>,<\/a:mo>\n                        <a:mi>x<\/a:mi>\n                        <a:mo>\u2208<\/a:mo>\n                        <a:mfenced open=\"]\" close=\"[\">\n                           <a:mrow>\n                              <a:mo>\u2212<\/a:mo>\n                              <a:mrow>\n                                 <a:mo>\u221e<\/a:mo>\n                              <\/a:mrow>\n                              <a:mrow>\n                                 <a:mo>,<\/a:mo>\n                              <\/a:mrow>\n                              <a:mn>1<\/a:mn>\n                           <\/a:mrow>\n                        <\/a:mfenced>\n                     <\/a:math>\n                  <\/jats:inline-formula> with <jats:inline-formula>\n                     <n:math xmlns:n=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\">\n                        <n:mi>\u03b2<\/n:mi>\n                        <n:mo>\u2208<\/n:mo>\n                        <n:msup>\n                           <n:mrow>\n                              <n:mi>\u211d<\/n:mi>\n                           <\/n:mrow>\n                           <n:mrow>\n                              <n:mo>+<\/n:mo>\n                           <\/n:mrow>\n                        <\/n:msup>\n                     <\/n:math>\n                  <\/jats:inline-formula>, <jats:inline-formula>\n                     <p:math xmlns:p=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\">\n                        <p:mi mathvariant=\"script\">V<\/p:mi>\n                        <p:mo>\u2208<\/p:mo>\n                        <p:mi>C<\/p:mi>\n                        <p:mfenced open=\"(\" close=\")\">\n                           <p:mrow>\n                              <p:mi>\u211d<\/p:mi>\n                              <p:mo>\/<\/p:mo>\n                              <p:mi>T<\/p:mi>\n                              <p:mi>\u2124<\/p:mi>\n                           <\/p:mrow>\n                        <\/p:mfenced>\n                     <\/p:math>\n                  <\/jats:inline-formula>, and <jats:inline-formula>\n                     <u:math xmlns:u=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\">\n                        <u:msub>\n                           <u:mrow>\n                              <u:mi>F<\/u:mi>\n                           <\/u:mrow>\n                           <u:mrow>\n                              <u:mi>D<\/u:mi>\n                           <\/u:mrow>\n                        <\/u:msub>\n                        <u:mfenced open=\"(\" close=\")\">\n                           <u:mrow>\n                              <u:mi>x<\/u:mi>\n                              <u:mo>,<\/u:mo>\n                              <u:mover accent=\"true\">\n                                 <u:mi>x<\/u:mi>\n                                 <u:mo>\u0307<\/u:mo>\n                              <\/u:mover>\n                           <\/u:mrow>\n                        <\/u:mfenced>\n                        <u:mo>=<\/u:mo>\n                        <u:mi>\u03ba<\/u:mi>\n                        <u:mover accent=\"true\">\n                           <u:mi>x<\/u:mi>\n                           <u:mo>\u0307<\/u:mo>\n                        <\/u:mover>\n                        <u:mo>\/<\/u:mo>\n                        <u:msup>\n                           <u:mrow>\n                              <u:mfenced open=\"(\" close=\")\">\n                                 <u:mrow>\n                                    <u:mn>1<\/u:mn>\n                                    <u:mo>\u2212<\/u:mo>\n                                    <u:mi>x<\/u:mi>\n                                 <\/u:mrow>\n                              <\/u:mfenced>\n                           <\/u:mrow>\n                           <u:mrow>\n                              <u:mn>3<\/u:mn>\n                           <\/u:mrow>\n                        <\/u:msup>\n                     <\/u:math>\n                  <\/jats:inline-formula>, <jats:inline-formula>\n                     <cb:math xmlns:cb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\">\n                        <cb:mi>\u03ba<\/cb:mi>\n                        <cb:mo>\u2208<\/cb:mo>\n                        <cb:msup>\n                           <cb:mrow>\n                              <cb:mi>\u211d<\/cb:mi>\n                           <\/cb:mrow>\n                           <cb:mrow>\n                              <cb:mo>+<\/cb:mo>\n                           <\/cb:mrow>\n                        <\/cb:msup>\n                     <\/cb:math>\n                  <\/jats:inline-formula> (called squeeze film damping force), or <jats:inline-formula>\n                     <eb:math xmlns:eb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\">\n                        <eb:msub>\n                           <eb:mrow>\n                              <eb:mi>F<\/eb:mi>\n                           <\/eb:mrow>\n                           <eb:mrow>\n                              <eb:mi>D<\/eb:mi>\n                           <\/eb:mrow>\n                        <\/eb:msub>\n                        <eb:mfenced open=\"(\" close=\")\">\n                           <eb:mrow>\n                              <eb:mi>x<\/eb:mi>\n                              <eb:mo>,<\/eb:mo>\n                              <eb:mover accent=\"true\">\n                                 <eb:mi>x<\/eb:mi>\n                                 <eb:mo>\u0307<\/eb:mo>\n                              <\/eb:mover>\n                           <\/eb:mrow>\n                        <\/eb:mfenced>\n                        <eb:mo>=<\/eb:mo>\n                        <eb:mi>c<\/eb:mi>\n                        <eb:mover accent=\"true\">\n                           <eb:mi>x<\/eb:mi>\n                           <eb:mo>\u0307<\/eb:mo>\n                        <\/eb:mover>\n                     <\/eb:math>\n                  <\/jats:inline-formula>, <jats:inline-formula>\n                     <kb:math xmlns:kb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\">\n                        <kb:mi>c<\/kb:mi>\n                        <kb:mo>\u2208<\/kb:mo>\n                        <kb:msup>\n                           <kb:mrow>\n                              <kb:mi>\u211d<\/kb:mi>\n                           <\/kb:mrow>\n                           <kb:mrow>\n                              <kb:mo>+<\/kb:mo>\n                           <\/kb:mrow>\n                        <\/kb:msup>\n                     <\/kb:math>\n                  <\/jats:inline-formula> (called linear damping force). If <jats:inline-formula>\n                     <mb:math xmlns:mb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\">\n                        <mb:msub>\n                           <mb:mrow>\n                              <mb:mi>F<\/mb:mi>\n                           <\/mb:mrow>\n                           <mb:mrow>\n                              <mb:mi>D<\/mb:mi>\n                           <\/mb:mrow>\n                        <\/mb:msub>\n                     <\/mb:math>\n                  <\/jats:inline-formula> is of squeeze film type, we have proven that there exists at least two positive periodic solutions, one of them locally asymptotically stable. Meanwhile, if <jats:inline-formula>\n                     <ob:math xmlns:ob=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\">\n                        <ob:msub>\n                           <ob:mrow>\n                              <ob:mi>F<\/ob:mi>\n                           <\/ob:mrow>\n                           <ob:mrow>\n                              <ob:mi>D<\/ob:mi>\n                           <\/ob:mrow>\n                        <\/ob:msub>\n                     <\/ob:math>\n                  <\/jats:inline-formula> is a linear damping force, we have proven that there are only two positive periodic solutions. One is unstable, and the other is locally exponentially asymptotically stable with rate of decay of <jats:inline-formula>\n                     <qb:math xmlns:qb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\">\n                        <qb:mi>c<\/qb:mi>\n                        <qb:mo>\/<\/qb:mo>\n                        <qb:mn>2<\/qb:mn>\n                     <\/qb:math>\n                  <\/jats:inline-formula>. Our technique can be applied to a class of Li\u00e9nard equations that model several microelectromechanical system devices, including the comb-drive finger model and torsional actuators.<\/jats:p>","DOI":"10.1155\/2022\/1498981","type":"journal-article","created":{"date-parts":[[2022,9,2]],"date-time":"2022-09-02T15:36:19Z","timestamp":1662132979000},"page":"1-15","source":"Crossref","is-referenced-by-count":5,"title":["Periodic Oscillations in MEMS under Squeeze Film Damping Force"],"prefix":"10.1155","volume":"2022","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4000-9718","authenticated-orcid":true,"given":"Juan","family":"Beron","sequence":"first","affiliation":[{"name":"Departamento de Ciencias Naturales y Matem\u00e1ticas Pontificia Universidad Javeriana Cali, Facultad de Ingenier\u00eda y Ciencias, Calle 18 No. 118\u2013250 Cali, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6225-2519","authenticated-orcid":true,"given":"Andr\u00e9s","family":"Rivera","sequence":"additional","affiliation":[{"name":"Departamento de Ciencias Naturales y Matem\u00e1ticas Pontificia Universidad Javeriana Cali, Facultad de Ingenier\u00eda y Ciencias, Calle 18 No. 118\u2013250 Cali, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/j.sna.2014.04.025"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-6020-7"},{"key":"3","article-title":"Bosch Sensortec Gmbh, Acceleration sensors"},{"key":"4","article-title":"Bosch Sensortec Gmbh, MEMS scanner"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1109\/T-ED.1967.15912"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1007\/s10884-007-9094-x"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127413500880"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijnonlinmec.2018.05.009"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2018.09.010"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1088\/1361-665x\/ab2c40"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2018.07.025"},{"volume-title":"Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering","year":"2006","author":"C. D. Coster","key":"12"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0085073"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1007\/s00030-016-0358-1"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1007\/s00009-004-0025-3"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.11650\/tjm.19.2015.3992"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1007\/BF02418013"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1002\/mana.200310033"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-42.3.505"},{"volume-title":"Hill's Equation, Dover Books on Mathematics Series","year":"2004","author":"W. Magnus","key":"20"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-12-804117-8.00003-5"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-02-06462-6"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2022\/1498981.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2022\/1498981.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2022\/1498981.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,2]],"date-time":"2022-09-02T15:36:23Z","timestamp":1662132983000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/jam\/2022\/1498981\/"}},"subtitle":[],"editor":[{"given":"Fernando","family":"Simoes","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2022,9,2]]},"references-count":22,"alternative-id":["1498981","1498981"],"URL":"https:\/\/doi.org\/10.1155\/2022\/1498981","relation":{},"ISSN":["1687-0042","1110-757X"],"issn-type":[{"type":"electronic","value":"1687-0042"},{"type":"print","value":"1110-757X"}],"subject":[],"published":{"date-parts":[[2022,9,2]]}}}