On the cyclicity of monodromic tangential singularities: a look beyond the pseudo-Hopf bifurcation

The cyclicity problem consists in estimating the number of limit cycles bifurcating from a monodromic singularity of planar vector fields and is usually addressed by means of Lyapunov coefficients. For nonsmooth systems, besides the limit cycles bifurcating by varying the Lyapunov coefficients, monodromic singularities lying on the switching curve can always be split apart generating, under suitable conditions, a sliding region and an extra limit cycle surrounding it. This bifurcation phenomenon is called pseudo-Hopf bifurcation and has been used to increase the lower bounds for the cyclicity of monodromic singularities in Filippov vector fields. In this paper, we aim to go beyond the pseudo-Hopf bifurcation by showing that the destruction of (2k,2k)-monodromic tangential singularities give birth to at least k limit cycles surrounding sliding segments.