Generalized pseudo-Hopf bifurcation: Limit cycle position and period

We investigate planar piecewise-smooth vector fields with a discontinuity line, focusing on the bifurcation of crossing limit cycles that arise when one of the vector fields is translated along the discontinuity set. We establish topological conditions under which such bifurcations occur and, under additional generic hypotheses, derive precise asymptotic expressions for both the position and the period of the resulting limit cycle in terms of the perturbation parameter. Our results extend the classical pseudo-Hopf bifurcation: they are not restricted to invisible folds or elementary monodromic singularities, but also apply to nilpotent centers/foci, half-monodromic singularities such as cusps, periodic orbits, and hyperbolic polycycles, thereby encompassing both local and non-local configurations. We show that the period function exhibits distinct asymptotic behaviors depending on the interacting objects. In particular, we provide a comprehensive table summarizing the leading terms of the period and position of the limit cycle for each possible configuration.