Electron. J. Differential Equations, Vol. 2018 (2018), No. 184, pp. 1-23.

Center problem for generalized lambda-omega differential systems

Jaume Llibre, Rafael Ramirez, Valentin Ramirez

Abstract:
$\Lambda$-$\Omega$ 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 Poincare-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.

Submitted July 9, 2018. Published November 14, 2018.
Math Subject Classifications: 34C05, 34C07.
Key Words: Linear type center; Darboux first integral; weak center; Poincare-Liapunov theorem; Reeb integrating factor.

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Jaume Llibre
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra,
Barcelona, Catalonia, Spain
email: jllibre@mat.uab.cat
  Rafael Ramírez
Departament d'Enginyeria Informática i Matemátiques
Universitat Rovira i Virgili
Avinguda dels Pa&imul;sos Catalans 26
43007 Tarragona, Catalonia, Spain
email: rafaelorlando.ramirez@urv.cat
  Valentín Ramírez
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra
Barcelona, Catalonia, Spain
email: valentin.ramirez@e-campus.uab.cat

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