The center and cyclicity problems for some analytic maps

The center variety and bifurcations of limit cycles from the center for maps f(x)=−∑k=0∞akxk+1 arising from x+y+∑j=0nαn−j,jxn−jyj=0are considered. Motivated by a general result for n=2ℓ+1 we investigate the center and cyclicity problem for n being even. We review results for n=2 and n=4 and perform the analysis for n=6,8,10. Finally, we state some conjectures for general n=2ℓ.