The center-focus problem and small amplitude limit cycles in rigid systems (Q414772)

Consider the planar system NEWLINE\[NEWLINE{dx\over dt}= -y+ xF(x,y),\quad {dy\over dt}= x+ y F(x,y),NEWLINE\]NEWLINE where \(F: \mathbb{R}^2\to\mathbb{R}\) can be represented in one of the forms NEWLINE\[NEWLINEF(x,y)= \sum^N_{i=0} a_ix^i,\quad F(x,y)= \sum^N_{i=0} b_iy^i,\quad N= 2n\text{ or }N= 2n+1,NEWLINE\]NEWLINE NEWLINE\[NEWLINEF(x,y)= \sum^N_{i=0} (a_ix^i+ b_iy^i),\quad N= 2n.NEWLINE\]NEWLINE The authors prove that in the first two cases the sets \(\{a_0,a_1,\dots, a_{2n}\}\) and \(\{b_0,b_2,\dots, b_{2n}\}\) form a focal basis for the equilibrium \((0,0)\), respectively; in the last case the set \(\{a_0+ b_0,a_2+ b_2,\dots, a_{2n}+ b_{2n}\}\) forms a focal basis. These results are used to formulate necessary and sufficient conditions for the origin to be a center and to estimate the number of small amplitude limit cycles.