Three-Dimensional Limit Cycles Generated from Discontinuous Piecewise Differential Systems Separated by Two Intersecting Planes

Due to their widespread applications in modeling natural issues, the study of piecewise linear differential systems has gained popularity in recent years. It is well known that the qualitative theory of piecewise linear differential systems heavily relies on limit cycles. By now, most studies have only considered planar systems by examining the presence and maximum number of limit cycles for piecewise differential systems. However, there have been few studies on this issue in & Ropf;3. We recall the problem of the existence and the maximum number of limit cycles for planar discontinuous piecewise differential systems formed by linear differential centers separated by one or two parallel straight lines that have at most one limit cycle, respectively. Although, in & Ropf;3, the maximal number of limit cycles for the same problem is 0 when the separation surface is a plane and at most four if the separation surface is two parallel planes. In this paper, we mainly focus on the problem of the existence and the maximum number of limit cycles in & Ropf;3, when the separating surface is formed by two intersecting half-planes. First, we prove that when the entire space is divided into two regions, this family can have at most five limit cycles, where one limit cycle intersects the separation surface at two points and the remaining four limit cycles intersect the separation surface at four points. Second, when the entire space is divided into three regions, we prove that the maximum number of limit cycles intersecting the separation surface at three points and four points simultaneously is at most eight.