The center-focus problem for a system with homogeneous nonlinearities in the case of zero eigenvalues of the linear part (Q1768957)

The authors consider the center-focus problem for system \[ \dot x=y+X_{2n+1}(x,y),\quad \dot y=Y_{2n+1}(x,y).\tag{1} \] They solve the center-focus problem for system (1) for \(n=3\) using the formal series method and prove the existence of systems (1) for \(n=1,2,3\) with 2, 5 and 9 limit cycles, respectively, in a neighborhood of the origin.