Strong isochronism of a center and a focus for systems with homogeneous nonlinearities
Given a real polynomial planar differential equation \[ x'=-y+p_n(x,y),~y'=x+q_n(x,y), \] where \(p_n\) and \(q_n\) are homogeneous polynomials of degree \(n\geq2,\) the author proposes a new method to detect isochronism. It is based on the existence of a transformation, \(X=x+f(x,y)\), \(Y=y,\) that converts it into a system of the form \[ X'=-Y+X\,S(X,Y),~Y'=X+Y\,S(X,Y). \] Notice that in polar coordinates this last system writes as \(\theta'=1\) and so it has at the origin either an isochronous center, or an isochronous focus. The author applies this method when \(2\leq n\leq 5\) and gives a list of isochronous centers and foci. As fas as the reviewer knows, some of the given systems exhibiting isochronism are new. It is also noticed in the paper that this method always provides so called strong isochronous systems because the transformation leaves the \(x\)-axis invariant and the systems present isochronism with respect to it.
scientific article; zbMATH DE number 5611131