Conditions for the existence of a center for the Kukles homogeneous systems (Q1609062)

Here, necessary and sufficient conditions for the existence of centers of two-dimensional linear Hamiltonian systems with homogeneous quartic and quintic nonlinearities, so-called Kukles homogeneous systems, \[ \dot{x}=-y,\quad \dot{y}=x+Q_{n}(x,y),\tag{1} \] are studied, where \(Q_{n}(x,y):=b_{n0}x^n+\cdots+b_{i,n-i}x^iy^{n-i}+\cdots+b_{0n}y^n\) is a homogeneous polynomial of degree \(n=4\) or \(n=5\). For \(n=4\), theorem 2 claims that the origin is a center iff \(b_{40}=b_{22}=b_{04}=0 \vee b_{31}=b_{13}=0\), for \(n=5\), theorem 3 claims that the origin is a center iff \(b_{41}=b_{23}=b_{05}=0\). Estimates on the number of small-amplitude limit cycles bifurcating from the origin are given, and an open problem due to \textit{E. P. Volokitin} and \textit{V. V. Ivanov} [Sib. Math. J. 40, No. 1, 23-38 (1999); translation from Sib. Mat. Zh. 40, No. 1, 30-48 (1999; Zbl 0921.58053)] concerning the existence of a center of (1) in the case when its direction field is symmetric on one of the coordinate axes is solved.