Piecewise holomorphic systems

Abstract

Holomorphic systems are well known in the literature due to their important properties: reversibility, integrability, non-existence of limit cycles and complete knowledge of the phase portraits around their non-essential singularities. In this paper we are concerned about studying the piecewise holomorphic systems (PWHS). Specifically, we study the sewing, sliding, and tangential regions of the PWHS and we classify the typical singularities of these systems. Also, we are interested in understanding how the trajectories of the regularized system associated with the PWHS transit through the region of regularization. In addition, we know that holomorphic systems have no limit cycles, but piecewise holomorphic systems do, so we provide conditions to ensure the existence of limit cycles of these systems. Additional conditions are provided to guarantee the stability and uniqueness of such limit cycles.