Center problem for generalized lambda-omega differential systems
$\Lambda$-$\Omega$ differential systems are the real planar polynomial differential equations of degree $m$ of the form $$ \dot{x}=-y(1+\Lambda)+x\Omega,\quad \dot{y}=x(1+\Lambda)+y\Omega, $$ where $\Lambda=\Lambda(x,y)$ and $\Omega=\Omega(x,y)$ are polynomials of degree at most $m-1$ such th...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Texas State University
2018-11-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2018/184/abstr.html |
| _version_ | 1818278033061052416 |
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| author | Jaume Llibre Rafael Ramirez Valentin Ramirez |
| author_facet | Jaume Llibre Rafael Ramirez Valentin Ramirez |
| author_sort | Jaume Llibre |
| collection | DOAJ |
| description | $\Lambda$-$\Omega$ differential systems are the real planar
polynomial differential equations of degree $m$ of the form
$$
\dot{x}=-y(1+\Lambda)+x\Omega,\quad \dot{y}=x(1+\Lambda)+y\Omega,
$$
where $\Lambda=\Lambda(x,y)$ and $\Omega=\Omega(x,y)$ are
polynomials of degree at most $m-1$ such that $\Lambda(0,0)=\Omega(0,0)=0$.
A planar vector field with linear type center can be written as a
$\Lambda$-$\Omega$ system if and only if the Poincar\'e-Liapunov
first integral is of the form $F=\frac{1}{2}(x^2+y^2)(1+O(x,y))$.
The main objective of this article is to study the center problem for
$\Lambda$-$\Omega$ systems of degree $m$ with
$\Lambda=\mu(a_2x-a_1y)$, and
$\Omega=a_1x+a_2y+\sum_{j=2}^{m-1}\Omega_j$, where
$\mu,\,a_1,\,a_2$ are constants and $\Omega_j= \Omega_j(x,y)$ is
a homogenous polynomial of degree $j$, for $j=2,\dots,m-1$.
We prove the following results. Assuming that $m=2,3,4,5$ and
$$
(\mu+(m-2))(a^2_1+a^2_2)\ne 0 \quad \text{and}\quad
\sum_{j=2}^{m-2}\Omega_j\ne 0
$$
the $\Lambda$-$\Omega$ system has a weak center at the origin
if and only if these systems after a linear change of variables
$(x,y)\to (X,Y)$ are invariant under the transformations
$(X,Y,t)\to (-X,Y,-t)$.
If $(\mu+(m-2))(a^2_1+a^2_2)=0$ and $\sum_{j=1}^{m-2}\Omega_j=0$
then the origin is a weak center. We observe that the main difficulty
in proving this result for $m>6$ is related to the huge computations. |
| first_indexed | 2024-12-12T23:11:00Z |
| format | Article |
| id | doaj.art-226cef8994964671b53d14d558326f95 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-12T23:11:00Z |
| publishDate | 2018-11-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-226cef8994964671b53d14d558326f952022-12-22T00:08:35ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912018-11-012018184,123Center problem for generalized lambda-omega differential systemsJaume Llibre0Rafael Ramirez1Valentin Ramirez2 Univ. Autonoma de Barcelona, Catalonia, Spain Univ. Rovira i Virgili,Catalans, Spain Univ. Autonoma de Barcelona, Catalonia, Spain $\Lambda$-$\Omega$ differential systems are the real planar polynomial differential equations of degree $m$ of the form $$ \dot{x}=-y(1+\Lambda)+x\Omega,\quad \dot{y}=x(1+\Lambda)+y\Omega, $$ where $\Lambda=\Lambda(x,y)$ and $\Omega=\Omega(x,y)$ are polynomials of degree at most $m-1$ such that $\Lambda(0,0)=\Omega(0,0)=0$. A planar vector field with linear type center can be written as a $\Lambda$-$\Omega$ system if and only if the Poincar\'e-Liapunov first integral is of the form $F=\frac{1}{2}(x^2+y^2)(1+O(x,y))$. The main objective of this article is to study the center problem for $\Lambda$-$\Omega$ systems of degree $m$ with $\Lambda=\mu(a_2x-a_1y)$, and $\Omega=a_1x+a_2y+\sum_{j=2}^{m-1}\Omega_j$, where $\mu,\,a_1,\,a_2$ are constants and $\Omega_j= \Omega_j(x,y)$ is a homogenous polynomial of degree $j$, for $j=2,\dots,m-1$. We prove the following results. Assuming that $m=2,3,4,5$ and $$ (\mu+(m-2))(a^2_1+a^2_2)\ne 0 \quad \text{and}\quad \sum_{j=2}^{m-2}\Omega_j\ne 0 $$ the $\Lambda$-$\Omega$ system has a weak center at the origin if and only if these systems after a linear change of variables $(x,y)\to (X,Y)$ are invariant under the transformations $(X,Y,t)\to (-X,Y,-t)$. If $(\mu+(m-2))(a^2_1+a^2_2)=0$ and $\sum_{j=1}^{m-2}\Omega_j=0$ then the origin is a weak center. We observe that the main difficulty in proving this result for $m>6$ is related to the huge computations.http://ejde.math.txstate.edu/Volumes/2018/184/abstr.htmlLinear type centerDarboux first integralweak centerPoincare-Liapunov theoremReeb integrating factor |
| spellingShingle | Jaume Llibre Rafael Ramirez Valentin Ramirez Center problem for generalized lambda-omega differential systems Electronic Journal of Differential Equations Linear type center Darboux first integral weak center Poincare-Liapunov theorem Reeb integrating factor |
| title | Center problem for generalized lambda-omega differential systems |
| title_full | Center problem for generalized lambda-omega differential systems |
| title_fullStr | Center problem for generalized lambda-omega differential systems |
| title_full_unstemmed | Center problem for generalized lambda-omega differential systems |
| title_short | Center problem for generalized lambda-omega differential systems |
| title_sort | center problem for generalized lambda omega differential systems |
| topic | Linear type center Darboux first integral weak center Poincare-Liapunov theorem Reeb integrating factor |
| url | http://ejde.math.txstate.edu/Volumes/2018/184/abstr.html |
| work_keys_str_mv | AT jaumellibre centerproblemforgeneralizedlambdaomegadifferentialsystems AT rafaelramirez centerproblemforgeneralizedlambdaomegadifferentialsystems AT valentinramirez centerproblemforgeneralizedlambdaomegadifferentialsystems |