Article Contents
Article Contents

# Non-formally integrable centers admitting an algebraic inverse integrating factor

• * Corresponding author: Manuel Reyes
• We study the existence of a class of inverse integrating factor for a family of non-formally integrable systems whose lowest-degree quasi-homogeneous term is a Hamiltonian vector field. Once the existence of an inverse integrating factor is established, we study the systems having a center. Among others, we characterize the centers of the perturbations of the system $-y^3\partial_x+x^3\partial_y$ having an algebraic inverse integrating factor.

Mathematics Subject Classification: 34C05, 34C14, 34C20.

 Citation:

• Table 1.  Range and co-range of operator $\ell_{j}$ for the system (8).

 Range($\ell_{2}$)=span{$-bx^2-3y^2,3ax^2+2bxy$}. If $a\ne 0,$ Cor($\ell_{2}$)=span{$xy$}. If $a=0$, Cor($\ell_{2}$)=span{$x^2$}. Range($\ell_{3}$)=span{$-2bx^3-6xy^2,6ax^3+4bx^2y-3h,6ax^2y+4bxy^2$}. Cor($\ell_{3}$)=span{$h$}. Range($\ell_{4}$)=span{$3bx^4+9x^2y^2,-9ax^4-6bx^3y+6xh,$ $-9ax^3y-6bx^2y^2+3yh$}. Cor($\ell_{4}$)=span{$xh,yh$}.

Table 2.  Range and co-range of operator $\ell_{j}$ for the system (15)

 Range($\ell_{6}$)=span{$0$}, Cor($\ell_{6}$)=span{$x^2$} Range($\ell_{7}$)=span{$0$}, Cor($\ell_{7}$)=span{$xy$} Range($\ell_{8}$)=span{$y^2$}, Cor($\ell_{8}$)=span{$0$} Range($\ell_{9}$)=span{$x^3$}, Cor($\ell_{9}$)=span{$0$} Range($\ell_{10}$)=span{$0$}, Cor($\ell_{10}$)=span{$x^2y$} Range($\ell_{11}$)=span{$xy^2$}, Cor($\ell_{11}$)=span{$0$} Range($\ell_{12}$)=span{$7x^4-12h$}, Cor($\ell_{12}$)=span{$h$} Range($\ell_{13}$)=span{$x^3y$}, Cor($\ell_{13}$)={$0$} Range($\ell_{14}$)=span{$x^2y^2$}, Cor($\ell_{14}$)={$0$} Range($\ell_{15}$)=span{$x^3-6xh$}, Cor($\ell_{15}$)=span{$xh$} Range($\ell_{16}$)=span{$11x^4y-12yh$}, Cor($\ell_{16}$)=span{$yh$} Range($\ell_{17}$)=span{$x^3y^2$}, Cor($\ell_{17}$)={$0$} Range($\ell_{18}$)=span{$13x^6-36x^2h$}, Cor($\ell_{18}$)=span{$x^2h$} Range($\ell_{19}$)=span{$7x^5-12xyh$}, Cor($\ell_{19}$)=span{$xyh$} Range($\ell_{22}$)=span{$17x^6y-9x^2yh$}, Cor($\ell_{22}$)=span{$x^2yh$}

Table 3.  Range and co-range of operator $\ell_{j}$ for the system (21).

 Range($\ell_{3}$)=span{$x^3,y^3$}. Cor($\ell_{3}$)=span{$x^2y,xy^2$}. Range($\ell_{4}$)=span{$xy^3,x^4+2h,x^3y$}. Cor($\ell_{4}$)=span{$x^2y^2,h$}. Range($\ell_{5}$)=span{$x^2y^3,3x^5+8xh,3x^4y+4yh,x^3y^2$}. Cor($\ell_{5}$)=span{$xh,yh$}. Range($\ell_{6}$)=span{$x^3y^3,x^6+3x^2h,x^5y+2xyh, x^4y^2+y^2h$}. Cor($\ell_{6}$)=span{$x^2h,xyh,y^2h$}.
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