CENTERS AND LIMIT CYCLE BIFURCATIONS IN A FAMILY OF PIECEWISE SMOOTH SEPTIC <i>Z</i><sub>2</sub>-EQUIVARIANT SYSTEMS

In this paper, we investigate the center-focus problem and the number of limit cycles bifurcating from three foci for a family of piecewise smooth planar septic <i>Z</i>₂-equivariant systems, which include (±1, 0) and infinity as their singularities. We achieve a comprehensive classification of the conditions under which (±1, 0) act as centers. Moreover, we rigorously prove that, under small <i>Z</i>₂-equivariant perturbations, the perturbed system possesses at least 15 limit cycles, comprising 14 with small amplitude and 1 large amplitude with the scheme 1 ⊃ (7 ∪7).