Bifurcations of Double Critical Homoclinic Loops Connecting One Tangent Point

In this paper, we investigate the bifurcations of a double critical homoclinic loop connecting one tangent point in piecewise-smooth differential systems. Since this double homoclinic loop is composed of two critical loops connecting the same tangent point, there are crossing limit cycles and sliding loops simultaneously bifurcating from three parts under perturbations, i.e. from two critical loops and the exterior of the double homoclinic loop. We prove that the maximum sum of crossing limit cycles and sliding loops under any perturbations is 5 in the case that the two critical loops are unstable, and 4 otherwise. Further, by constructing a perturbation system with a three-dimensional unfolding parameter which independently controls bifurcations of these three parts separately, we prove the reachability of the maximum sum and obtain their coexistence numbers and locations.