Algorithmic derivation of isochronicity conditions (Q880283)

A center of a planar polynomial system \((S)\) of differential equations is isochronous (all sufficiently small orbits surrounding it have the same period) if and only if it can be linearized. In 1969 I.~I.~Pleshkin found a collection of polynomials in the coefficients of the right-hand sides of \((S)\) whose simultaneous vanishing is a necessary and sufficent condition for a critical point of \((S)\) to be an isochronous center, based on the existence of a linearization. The authors use these polynomials as the basis of a computer algebra approach to find necessary conditions for a critical point in a system \((S)\) to be an isochronous center. They prove sufficiency of their conditions by other techniques, such as Urabe's condition. They test their technique by applying it to several systems for which isochronicity conditions are known, then use it to derive necessary and sufficient conditions for a critical point to be an isochronous center in two families for which such a result was not previously known.