First and Second Integrals of Hopf–Langford-Type Systems
The work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf–Langford-type systems. First, by introducing cylindrical coordinates in its phase space, we show that the regar...
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2024-12-01
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| author | Vassil M. Vassilev Svetoslav G. Nikolov |
| author_facet | Vassil M. Vassilev Svetoslav G. Nikolov |
| author_sort | Vassil M. Vassilev |
| collection | DOAJ |
| description | The work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf–Langford-type systems. First, by introducing cylindrical coordinates in its phase space, we show that the regarded system can be reduced to a two-dimensional Liénard system, which corresponds to a second-order Liénard equation. Then, we present (in explicit form) polynomial first and second integrals of Liénard systems of the considered type identifying those values of their parameters for which these integrals exist. It is also proved that a generic Liénard equation is factorizable if and only if the corresponding Liénard system admits a second integral of a special form. It is established that each Liénard system corresponding to a Hopf–Langford system of the considered type admits such a second integral, and hence, the respective Liénard equation is factorizable. |
| format | Article |
| id | doaj-art-f2014871e66d4b8d8b79fb1614d899d5 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-f2014871e66d4b8d8b79fb1614d899d52025-01-24T13:22:08ZengMDPI AGAxioms2075-16802024-12-01141810.3390/axioms14010008First and Second Integrals of Hopf–Langford-Type SystemsVassil M. Vassilev0Svetoslav G. Nikolov1Institute of Mechanics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev St., Block 4, 1113 Sofia, BulgariaInstitute of Mechanics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev St., Block 4, 1113 Sofia, BulgariaThe work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf–Langford-type systems. First, by introducing cylindrical coordinates in its phase space, we show that the regarded system can be reduced to a two-dimensional Liénard system, which corresponds to a second-order Liénard equation. Then, we present (in explicit form) polynomial first and second integrals of Liénard systems of the considered type identifying those values of their parameters for which these integrals exist. It is also proved that a generic Liénard equation is factorizable if and only if the corresponding Liénard system admits a second integral of a special form. It is established that each Liénard system corresponding to a Hopf–Langford system of the considered type admits such a second integral, and hence, the respective Liénard equation is factorizable.https://www.mdpi.com/2075-1680/14/1/8dynamical systemsLiénard system and equationfirst integralssecond integralsDarboux polynomialsexact solutions |
| spellingShingle | Vassil M. Vassilev Svetoslav G. Nikolov First and Second Integrals of Hopf–Langford-Type Systems Axioms dynamical systems Liénard system and equation first integrals second integrals Darboux polynomials exact solutions |
| title | First and Second Integrals of Hopf–Langford-Type Systems |
| title_full | First and Second Integrals of Hopf–Langford-Type Systems |
| title_fullStr | First and Second Integrals of Hopf–Langford-Type Systems |
| title_full_unstemmed | First and Second Integrals of Hopf–Langford-Type Systems |
| title_short | First and Second Integrals of Hopf–Langford-Type Systems |
| title_sort | first and second integrals of hopf langford type systems |
| topic | dynamical systems Liénard system and equation first integrals second integrals Darboux polynomials exact solutions |
| url | https://www.mdpi.com/2075-1680/14/1/8 |
| work_keys_str_mv | AT vassilmvassilev firstandsecondintegralsofhopflangfordtypesystems AT svetoslavgnikolov firstandsecondintegralsofhopflangfordtypesystems |