On persistent centers
A system of analytic ordinary differential equations on the plane with a monodromic critical point at the origin at which the linear part has purely imaginary eigenvalues can be written in the form \[ \dot x = -y + P(x,y),\quad \dot y = x + Q(x,y), \] or in complex coordinates \(z = x + iy\) as \[ \dot z = iz + F(z, \bar z),\tag{1} \] Fix such a system. The authors define the origin to be a \textit{persistent center} for that system if it is a center for the system \(\dot z = iz + \lambda F(z, \bar z)\) for all complex \(\lambda\). The origin is termed a \textit{weakly persistent center} for that system if it is a center for the system \(\dot z = iz + c F(z, \bar z)\) for all real \(c\). The authors identify five broad classes of general persistent centers (e.g., the \textit{holomorphic} class \(\dot z = iz + f(z)\) with \(f(0) = f'(0) = 0\)). They also completely characterize collections of persistent centers within some general classes of systems, e.g., systems (1) in which \(F\) is a polynomial function of degree at most three.
scientific article; zbMATH DE number 5608813