Cyclicity of some analytic maps

In this paper, we describe an approach to estimate the cyclicity of centers in maps given by f(x)=−x−∑k=1∞akxk+1. The main motivation for this problem originates from the study of cyclicity of planar systems of ODEs. We also consider the bifurcation of limit cycles from each component of the center variety of some particular cases of maps f(x)=−x−∑k=1∞akxk+1 arising from algebraic equations of the form x+y+h.o.t.=0x+y+h.o.t.=0 where higher order terms up to degree four are present.