Existence of periodic solutions and homoclinic orbits for third-order nonlinear differential equations
The existence of periodic solutions for the third-order differential equation x¨˙+ω2x˙=μF(x,x˙,x¨) is studied. We give some conditions for this equation in order to reduce it to a second-order nonlinear differential equation. We show that the existence of periodic solutions for the second-order equa...
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| Format: | Article |
| Language: | English |
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Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171203107089 |
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| author | O. Rabiei Motlagh Z. Afsharnezhad |
| author_facet | O. Rabiei Motlagh Z. Afsharnezhad |
| author_sort | O. Rabiei Motlagh |
| collection | DOAJ |
| description | The existence of periodic solutions for the third-order
differential equation x¨˙+ω2x˙=μF(x,x˙,x¨) is studied. We give some conditions for
this equation in order to reduce it to a second-order nonlinear
differential equation. We show that the existence of periodic
solutions for the second-order equation implies the existence of
periodic solutions for the above equation. Then we use the Hopf
bifurcation theorem for the second-order equation and obtain many
periodic solutions for it. Also we show that the above equation
has many homoclinic solutions if F(x,x˙,x¨) has a quadratic form. Finally, we compare our result to that of Mehri and Niksirat (2001). |
| format | Article |
| id | doaj-art-3bc65cfed79445d6b6027e0c384cd11e |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2003-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-3bc65cfed79445d6b6027e0c384cd11e2025-02-03T06:01:02ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252003-01-012003420922810.1155/S0161171203107089Existence of periodic solutions and homoclinic orbits for third-order nonlinear differential equationsO. Rabiei Motlagh0Z. Afsharnezhad1Department of Mathematics, Faculty of Mathematical Sciences, Ferdowsi University, Mashhad, IranDepartment of Mathematics, Faculty of Mathematical Sciences, Ferdowsi University, Mashhad, IranThe existence of periodic solutions for the third-order differential equation x¨˙+ω2x˙=μF(x,x˙,x¨) is studied. We give some conditions for this equation in order to reduce it to a second-order nonlinear differential equation. We show that the existence of periodic solutions for the second-order equation implies the existence of periodic solutions for the above equation. Then we use the Hopf bifurcation theorem for the second-order equation and obtain many periodic solutions for it. Also we show that the above equation has many homoclinic solutions if F(x,x˙,x¨) has a quadratic form. Finally, we compare our result to that of Mehri and Niksirat (2001).http://dx.doi.org/10.1155/S0161171203107089 |
| spellingShingle | O. Rabiei Motlagh Z. Afsharnezhad Existence of periodic solutions and homoclinic orbits for third-order nonlinear differential equations International Journal of Mathematics and Mathematical Sciences |
| title | Existence of periodic solutions and homoclinic orbits for
third-order nonlinear differential equations |
| title_full | Existence of periodic solutions and homoclinic orbits for
third-order nonlinear differential equations |
| title_fullStr | Existence of periodic solutions and homoclinic orbits for
third-order nonlinear differential equations |
| title_full_unstemmed | Existence of periodic solutions and homoclinic orbits for
third-order nonlinear differential equations |
| title_short | Existence of periodic solutions and homoclinic orbits for
third-order nonlinear differential equations |
| title_sort | existence of periodic solutions and homoclinic orbits for third order nonlinear differential equations |
| url | http://dx.doi.org/10.1155/S0161171203107089 |
| work_keys_str_mv | AT orabieimotlagh existenceofperiodicsolutionsandhomoclinicorbitsforthirdordernonlineardifferentialequations AT zafsharnezhad existenceofperiodicsolutionsandhomoclinicorbitsforthirdordernonlineardifferentialequations |