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We give a new and short proof of the characterization of monodromic nilpotent critical points. We also calculate the first generalized Lyapunov constants in order to solve the stability problem. We apply the results to several families of planar systems obtaining necessary and sufficient conditions for having a center. Our method also allows us to generate limit cycles from the origin.

We study the stability of a singular point for planar discontinuous differential equations with a line of discontinuities. This is done, for the most generic cases, by computing some kind of Lyapunov constants. Our computations are based on the Ž . so called R, , p, q -generalized polar coordinates, introduced by Lyapunov, and they are essentially different from the ones used in the smooth case. These Lyapunov constants are also used to generate limit cycles for some concrete examples. ᮊ

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It is well known that the normal form theory can be applied to solve the center-focus problem for monodromic planar nilpotent singularities. In this paper we see how this theory can also be applied to generate limit cycles from this type of singularities.