Existence of Periodic Solutions for a Class of the Generalized Liénard Equations
We study analytically the existence of periodic solutions of the generalized Liénard differential equations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true">...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-05-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/5/944 |
| _version_ | 1797495130294321152 |
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| author | María Teresa de Bustos Zouhair Diab Juan Luis G. Guirao Miguel A. López Raquel Martínez |
| author_facet | María Teresa de Bustos Zouhair Diab Juan Luis G. Guirao Miguel A. López Raquel Martínez |
| author_sort | María Teresa de Bustos |
| collection | DOAJ |
| description | We study analytically the existence of periodic solutions of the generalized Liénard differential equations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mo>+</mo><mi>f</mi><mfenced separators="" open="(" close=")"><mi>x</mi><mo>,</mo><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mfenced><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mi>x</mi><mo>+</mo><mi>g</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>=</mo><msup><mi>ε</mi><mn>2</mn></msup><msub><mi>p</mi><mn>1</mn></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>+</mo><msup><mi>ε</mi><mn>3</mn></msup><msub><mi>p</mi><mn>2</mn></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> where <i>n</i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∈</mo><msup><mi mathvariant="double-struck">N</mi><mo>*</mo></msup></mrow></semantics></math></inline-formula>, the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow></semantics></math></inline-formula> are of class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">C</mi><mn>3</mn></msup><mo>,</mo><mspace width="4pt"></mspace><msup><mi mathvariant="script">C</mi><mn>4</mn></msup></mrow></semantics></math></inline-formula> in a neighborhood of the origin, respectively, the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>i</mi></msub></semantics></math></inline-formula> are of class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">C</mi><mn>0</mn></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>π</mi><mo>−</mo></mrow></semantics></math></inline-formula>periodic in the variable <i>t</i>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>=</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> is a small parameter as usual. The mathematical tool that we have used is the averaging theory of dynamical systems of second order. |
| first_indexed | 2024-03-10T01:44:02Z |
| format | Article |
| id | doaj.art-2f6db772e2dc45d6ad2a11927956032e |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T01:44:02Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-2f6db772e2dc45d6ad2a11927956032e2023-11-23T13:18:40ZengMDPI AGSymmetry2073-89942022-05-0114594410.3390/sym14050944Existence of Periodic Solutions for a Class of the Generalized Liénard EquationsMaría Teresa de Bustos0Zouhair Diab1Juan Luis G. Guirao2Miguel A. López3Raquel Martínez4Department of Applied Mathematics, University of Salamanca, Casas del Parque, 2, 37008 Salamanca, SpainDepartment of Mathematics and Computer Science, Larbi Tebessi University, 12002 Tebessa, AlgeriaDepartment of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina, 30203 Cartagena, SpainSIDIS Research Group, Department of Mathematics, Institute of Applied Mathematics in Science and Engineering (IMACI), Polytechnic School of Cuenca, University of Castilla-La Mancha, 16071 Cuenca, SpainSIDIS Research Group, Department of Mathematics, Institute of Applied Mathematics in Science and Engineering (IMACI), Polytechnic School of Cuenca, University of Castilla-La Mancha, 16071 Cuenca, SpainWe study analytically the existence of periodic solutions of the generalized Liénard differential equations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mo>+</mo><mi>f</mi><mfenced separators="" open="(" close=")"><mi>x</mi><mo>,</mo><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mfenced><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mi>x</mi><mo>+</mo><mi>g</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>=</mo><msup><mi>ε</mi><mn>2</mn></msup><msub><mi>p</mi><mn>1</mn></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>+</mo><msup><mi>ε</mi><mn>3</mn></msup><msub><mi>p</mi><mn>2</mn></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> where <i>n</i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∈</mo><msup><mi mathvariant="double-struck">N</mi><mo>*</mo></msup></mrow></semantics></math></inline-formula>, the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow></semantics></math></inline-formula> are of class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">C</mi><mn>3</mn></msup><mo>,</mo><mspace width="4pt"></mspace><msup><mi mathvariant="script">C</mi><mn>4</mn></msup></mrow></semantics></math></inline-formula> in a neighborhood of the origin, respectively, the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>i</mi></msub></semantics></math></inline-formula> are of class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">C</mi><mn>0</mn></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>π</mi><mo>−</mo></mrow></semantics></math></inline-formula>periodic in the variable <i>t</i>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>=</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> is a small parameter as usual. The mathematical tool that we have used is the averaging theory of dynamical systems of second order.https://www.mdpi.com/2073-8994/14/5/944averaging theorygeneralized Liénard differential equationsperiodic solution |
| spellingShingle | María Teresa de Bustos Zouhair Diab Juan Luis G. Guirao Miguel A. López Raquel Martínez Existence of Periodic Solutions for a Class of the Generalized Liénard Equations Symmetry averaging theory generalized Liénard differential equations periodic solution |
| title | Existence of Periodic Solutions for a Class of the Generalized Liénard Equations |
| title_full | Existence of Periodic Solutions for a Class of the Generalized Liénard Equations |
| title_fullStr | Existence of Periodic Solutions for a Class of the Generalized Liénard Equations |
| title_full_unstemmed | Existence of Periodic Solutions for a Class of the Generalized Liénard Equations |
| title_short | Existence of Periodic Solutions for a Class of the Generalized Liénard Equations |
| title_sort | existence of periodic solutions for a class of the generalized lienard equations |
| topic | averaging theory generalized Liénard differential equations periodic solution |
| url | https://www.mdpi.com/2073-8994/14/5/944 |
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