Center-focus and isochronous center problems for discontinuous differential equations

The study of the center focus problem and the isochronicity problem for differential equations with a line of discontinuities is usually done bycomputing the whole return map as the composition of the two maps associated to the two smooth differential equations. This leads to large formulaswhich usually are treated with algebraic manipulators. In this paper we approach to this problem from a more theoretical point of view. The results thatwe obtain relate the order of degeneracy of the critical point of the discontinuous differential equations with the order of degeneracy of the two smoothcomponent differential equations. Finally we apply them to some families ofexamples.