On the number of limit cycles for piecewise polynomial holomorphic systems

TL;DR

This paper investigates lower bounds on the number of limit cycles in piecewise polynomial holomorphic systems with a discontinuity line, using averaging functions, bifurcation analysis, and the Poincaré-Miranda theorem.

Contribution

It introduces new methods to estimate the number of limit cycles and constructs explicit examples with three limit cycles, surpassing previous known results.

Findings

Established lower bounds for limit cycles in the systems studied.

Developed techniques combining averaging functions and bifurcation analysis.

Constructed explicit systems with three limit cycles.

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.