Eight Limit Cycles in a Three-Dimensional Discontinuous Piecewise Differential Systems Separated by Two Planes

Limit cycles of planar piecewise differential systems have been intensively investigated by many authors across various fields, such as physics, biology and economics. However, limit cycles of piecewise differential systems in the space [Formula: see text] have been the subject of very few studies. This work aims to investigate three-dimensional discontinuous piecewise differential systems formed by linear vector fields separated by two planes either parallel to each other or that cross each other. We show that when these discontinuous piecewise differential systems are separated by two parallel planes, they can have at most four limit cycles, and when the separated surface is two intersecting planes, the discontinuous piecewise differential systems can have at most eight limit cycles. Furthermore, we prove that these upper bounds can be attained.