The center problem for a linear center perturbed by homogeneous polynomials (Q882716)

The author studies the center problem for a special system of differential equations of the form \[ \dot x=-y+X_{s}(x,y),\qquad \dot y=x+Y_{s}(x,y), \] where \(X_{s}(x,y)\) and \(Y_{s}(x,y)\) are homogeneous polynomials of degree \(s,\) with \(s\geq2.\) It is known that the centers of the polynomial differential systems with a linear center perturbed by homogeneous polynomials have been studied for the degrees \(s=2,3,4,5\); they are completely classified for \(s=2,3\) and partially classified for \(s=4,5.\) The author recalls these results and gives new centers for \(s=6,7.\)