Bifurcations of limit circles and center conditions for a class of non-analytic cubic \(Z_2\) polynomial differential systems (Q1928159)

The system (1), \[ \begin{multlined} {dx\over dt}= -y+ {d+1\over 2} x^4 y+ (3B_{03}+ B_{12}- 3B_{03} d- B_{12} d) x^3 y^2+ (-\textstyle{{1\over 2}}- 14A_{03}- 4A_{12}+\\ +\textstyle{{1\over 2}} d+ 2A_{03} d) x^2 y^3+ (B_{12}- B_{03} d- B_{12}d) xy^4+ (-1-4A_{03}- 4A_{12}) y^5,\end{multlined} \] \[ \begin{multlined} {dy\over dt}= x- x^5+ (12A_{03}- \textstyle{{1\over 2}}+ 4A_{12}+ \textstyle{{1\over 2}} d) x^3 y^2+ (11B_{03}+ B_{12}- 3B_{03} d- B_{12} d) x^2 y^3+\\ +(2A_{03}+ \textstyle{{1\over 2}}+ 4A_{12}+ \textstyle{{1\over 2}} d+ 2A_{03} d) xy^4+ (B_{03}+ B_{12}- B_{03} d- B_{12}d) y^5,\end{multlined} \] is considered. Let the coefficients of system (1) satisfy the following conditions \[ \begin{aligned} A_{03} &= {1\over 384} (-128A_{12}- 129+ 2d- d^2),\;A_{12}=- {-705- 185d+ 117d^2+ 5d^3\over 640(-1+ d)},\\ B^2_{03} &= {27215+ 25520d- 9706d^2- 120d^3+ 99d^4\over 420(335- 450d+ 61d^2+ 9d^3)},\;d\approx 9.305103625335557.\end{aligned} \] Theorem. 5 limit cycles can bifurcate from each of the two fine focus points \((\pm 1,0)\) of system (1), and two limit cycles can be created from the origin at the same time. Thus, 12 limit cycles can bifurcate in the system (1), five of which are located in the neighborhood of each of the two fine focus points \((\pm 1, 0)\) and two of which are located in the neighborhood of the origin.