The center-focus problem of a class of polynomial differential systems with degenerate critical points

The paper discusses the center-focus problem with a degenerate critical point at the origin, and proposes a method to Study the properties of the degenerate critical point by transforming the degenerate critical point to an elementary critical point of its dual system, so that the classical approaches to the center-focus problem can be applied. The center-focus problem with a degenerate quintic system is considered as an example. Finally, a non-polynomial system with an isochronous center at the original is constructed. As far as we know, this is the first example of a non-polynomial system with an isochronous center at a degenerate critical point.