On the critical crossing cycle bifurcation in planar Filippov systems

We consider planar piecewise smooth differential systems with a discontinuity line, and characterize the critical crossing cycle bifurcation (CC bifurcation), also termed as homoclinic connection to a fold, see the celebrated papers by Kuznetzov et al. (2003) [13] and by Guardia et al. (2011) [4].The new characterization of the CC bifurcation achieved generically assures the persistence of one limit cycle, which changes from sliding to crossing, preserving its stability character. Therefore, we clarify some previous misconceptions: the case CC2 described in Kuznetzov et al. [13] is not possible. Thus, Proposition 10.1 of Guardia et al. [4], predicting a CC2 bifurcation in a piecewise linear normal form unfolding the codimension-2 focus-fold bifurcation, is not completely correct. The complete unfolding contains a region where two crossing limit cycles coexist, colliding to disappear at a new bifurcation curve.