On the Periodic Orbits of Piecewise Holomorphic Systems on ℂ2

In this paper, we propose an analytic framework to characterize centers in piecewise holomorphic systems on [Formula: see text] by analyzing the structure of periodic orbits in the associated real systems. For three representative classes of such systems, we establish necessary and sufficient conditions for the existence of a simple pseudo-center in each case, and we determine the dimension of the set of periodic orbits in the corresponding real systems when the simple pseudo-center conditions are not satisfied.