On the Number of Limit Cycles for Piecewise Polynomial Holomorphic Systems\ast

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. Initially, we study the number of zeros of the first- and second order averaging functions. We also use the Lyapunov quantities to produce limit cycles appearing from a monodromic equilibrium point via a degenerated Andronov--Hopf type bifurcation, adding at the very end the sliding effects. Finally, we use the Poincare'\--Miranda theorem for obtaining an explicit piecewise linear holomorphic system with 3 limit cycles, a result that improves the known examples in the literature that had a single limit cycle.