Existence, uniqueness and other properties of the limit cycle of a generalized Van der Pol equation
In this article, we study the bifurcation of limit cycles from the linear oscillator $\dot{x}=y$, $\dot{y}=-x$ in the class $$ \dot{x}=y,\quad \dot{y}=-x+\varepsilon y^{p+1}\big(1-x^{2q}\big), $$ where $\varepsilon$ is a small positive parameter tending to 0, $p \in \mathbb{N}_0$ is even and...
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| Format: | Article |
| Language: | English |
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Texas State University
2014-01-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2014/22/abstr.html |
| _version_ | 1811323281845780480 |
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| author | Xenakis Ioakim |
| author_facet | Xenakis Ioakim |
| author_sort | Xenakis Ioakim |
| collection | DOAJ |
| description | In this article, we study the bifurcation of limit cycles from
the linear oscillator $\dot{x}=y$, $\dot{y}=-x$ in the class
$$
\dot{x}=y,\quad \dot{y}=-x+\varepsilon y^{p+1}\big(1-x^{2q}\big),
$$
where $\varepsilon$ is a small positive parameter tending to 0,
$p \in \mathbb{N}_0$ is even and $q \in \mathbb {N}$.
We prove that the above differential system, in the global plane
where $p \in \mathbb{N}_0$ is even and $q \in \mathbb{N}$,
has a unique limit cycle. More specifically, the existence
of a limit cycle, which is the main result in this work,
is obtained by using the Poincare's method, and the uniqueness
can be derived from the work of Sabatini and Villari [6].
We also investigate and some other properties of this unique
limit cycle for some special cases of this differential system.
Such special cases have been studied by Minorsky [3] and
Moremedi et al [4]. |
| first_indexed | 2024-04-13T13:51:15Z |
| format | Article |
| id | doaj.art-8b5a6ae705b44059bddc428e93d82bb1 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-04-13T13:51:15Z |
| publishDate | 2014-01-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-8b5a6ae705b44059bddc428e93d82bb12022-12-22T02:44:19ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912014-01-01201422,19Existence, uniqueness and other properties of the limit cycle of a generalized Van der Pol equationXenakis Ioakim0 Univ. of Cyprus, Nicosia, Cyprus In this article, we study the bifurcation of limit cycles from the linear oscillator $\dot{x}=y$, $\dot{y}=-x$ in the class $$ \dot{x}=y,\quad \dot{y}=-x+\varepsilon y^{p+1}\big(1-x^{2q}\big), $$ where $\varepsilon$ is a small positive parameter tending to 0, $p \in \mathbb{N}_0$ is even and $q \in \mathbb {N}$. We prove that the above differential system, in the global plane where $p \in \mathbb{N}_0$ is even and $q \in \mathbb{N}$, has a unique limit cycle. More specifically, the existence of a limit cycle, which is the main result in this work, is obtained by using the Poincare's method, and the uniqueness can be derived from the work of Sabatini and Villari [6]. We also investigate and some other properties of this unique limit cycle for some special cases of this differential system. Such special cases have been studied by Minorsky [3] and Moremedi et al [4].http://ejde.math.txstate.edu/Volumes/2014/22/abstr.htmlGeneralized Van der Pol equationlimit cyclesexistenceuniqueness |
| spellingShingle | Xenakis Ioakim Existence, uniqueness and other properties of the limit cycle of a generalized Van der Pol equation Electronic Journal of Differential Equations Generalized Van der Pol equation limit cycles existence uniqueness |
| title | Existence, uniqueness and other properties of the limit cycle of a generalized Van der Pol equation |
| title_full | Existence, uniqueness and other properties of the limit cycle of a generalized Van der Pol equation |
| title_fullStr | Existence, uniqueness and other properties of the limit cycle of a generalized Van der Pol equation |
| title_full_unstemmed | Existence, uniqueness and other properties of the limit cycle of a generalized Van der Pol equation |
| title_short | Existence, uniqueness and other properties of the limit cycle of a generalized Van der Pol equation |
| title_sort | existence uniqueness and other properties of the limit cycle of a generalized van der pol equation |
| topic | Generalized Van der Pol equation limit cycles existence uniqueness |
| url | http://ejde.math.txstate.edu/Volumes/2014/22/abstr.html |
| work_keys_str_mv | AT xenakisioakim existenceuniquenessandotherpropertiesofthelimitcycleofageneralizedvanderpolequation |