Limit cycles in Filippov systems having a circle as switching manifold

It is known that planar discontinuous piecewise linear differential systems separated by a straight line have no limit cycles when both linear differential systems are centers. Here, we study the limit cycles of the planar discontinuous piecewise linear differential systems separated by a circle when both linear differential systems are centers. Our main results show that such discontinuous piecewise differential systems can have zero, one, two, or three limit cycles, but no more limit cycles than three