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Math."],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider <jats:inline-formula><jats:alternatives><jats:tex-math>$$3\\times 3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>3<\/mml:mn>\n                    <mml:mo>\u00d7<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> partially hyperbolic linear differential systems over an ergodic flow <jats:inline-formula><jats:alternatives><jats:tex-math>$$X^t$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mi>t<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and derived from the linear homogeneous differential equation <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\dddot{x}(t)+\\beta (X^t(t))\\dot{x}(X^t(t))+\\gamma (t) x(t)=0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mover>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mo>\u20db<\/mml:mo>\n                    <\/mml:mover>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mi>\u03b2<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msup>\n                        <mml:mi>X<\/mml:mi>\n                        <mml:mi>t<\/mml:mi>\n                      <\/mml:msup>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi>t<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mover>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mo>\u02d9<\/mml:mo>\n                    <\/mml:mover>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msup>\n                        <mml:mi>X<\/mml:mi>\n                        <mml:mi>t<\/mml:mi>\n                      <\/mml:msup>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi>t<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mi>\u03b3<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Assuming that the partial hyperbolic decomposition <jats:inline-formula><jats:alternatives><jats:tex-math>$$E^s\\oplus E^c\\oplus E^u$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mi>E<\/mml:mi>\n                      <mml:mi>s<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>\u2295<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>E<\/mml:mi>\n                      <mml:mi>c<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>\u2295<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>E<\/mml:mi>\n                      <mml:mi>u<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is proper and displays a zero Lyapunov exponent along the central direction <jats:inline-formula><jats:alternatives><jats:tex-math>$$E^c$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>E<\/mml:mi>\n                    <mml:mi>c<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> we prove that some <jats:inline-formula><jats:alternatives><jats:tex-math>$$C^0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> perturbation of the parameters <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta (t)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b2<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\gamma (t)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b3<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> can be done in order to obtain non-zero Lyapunov exponents and so a chaotic behaviour of the solution.<\/jats:p>","DOI":"10.1007\/s00010-023-00948-z","type":"journal-article","created":{"date-parts":[[2023,3,29]],"date-time":"2023-03-29T11:03:31Z","timestamp":1680087811000},"page":"467-487","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Plenty of hyperbolicity on a class of linear homogeneous jerk differential equations"],"prefix":"10.1007","volume":"97","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1758-2225","authenticated-orcid":false,"given":"M\u00e1rio","family":"Bessa","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,3,29]]},"reference":[{"key":"948_CR1","doi-asserted-by":"crossref","unstructured":"Amaro, D., Bessa, M., Vilarinho, H.: The simplicity of the Lyapunov spectrum for linear homogeneous differential equations with $$L^p$$-variation on the parameters, Submitted (2023)","DOI":"10.1007\/s00010-023-00948-z"},{"key":"948_CR2","doi-asserted-by":"publisher","first-page":"1655","DOI":"10.1017\/S0143385702001773","volume":"23","author":"A Baraviera","year":"2003","unstructured":"Baraviera, A., Bonatti, C.: Removing zero Lyapunov exponents. 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