A general mechanism to generate three limit cycles in planar Filippov systems with two zones

Discontinuous piecewise linear systems with two zones are considered. A general canonical form that includes all the possible configurations in planar linear systems is introduced and exploited. It is shown that the existence of a focus in one zone is sufficient to get three nested limit cycles, independently on the dynamics of the another linear zone. Perturbing a situation with only one hyperbolic limit cycle, two additional limit cycles are obtained by using an adequate parametric sector of the unfolding of a codimension-two focus-fold singularity.