Limit Cycles of Piecewise Smooth Vector Fields Formed by Rigid Systems and Linear Centers

We establish sharp upper bounds for the number of limit cycles in a class of piecewise smooth vector fields composed of two subsystems separated by a straight line. One subsystem is a rigid vector field of degree n , while the other is either a linear center or a rigid vector field of degree m . By analyzing the first return map associated with a Bernoulli differential equation, we prove that at most one limit cycle exists when the linear center is located at the origin. When the center is displaced, this upper bound increases to two for any n≥ 1 n ≥ 1 , and it is sharp. For systems formed by two rigid vector fields, we show that uniqueness holds when $$n=m$$ n = m . In the resonant case $$m=nk$$ m = n k , we obtain an upper bound of k limit cycles. We also provide a sharper bound related to an open problem on rigid smooth differential systems.