On the number of limit cycles for piecewise polynomial holomorphic systems

Abstract: In this paper we are concerned with determining lower bounds of the number of limit cycles for piecewise polynomial holomorphic systems with a straight line of discontinuity. We approach this problem with different points of view: study of the number of zeros of the first and second order averaging functions, or with the control of the limit cycles appearing from a monodromic equilibrium point via a degenerated Andronov-Hoph type bifurcation, adding at the very end the sliding effects. We also use the Poincar\'e-Miranda theorem for obtaining an explicit piecewise linear holomorphic systems with 3 limit cycles, result that improves the known examples in the literature that had a single limit cycle.