Piecewise Smooth Holomorphic Systems

TL;DR

This paper studies the classification, dynamics, and limit cycle behavior of piecewise smooth holomorphic systems, extending known results from holomorphic systems to the nonsmooth case and exploring their complex trajectories.

Contribution

It introduces a classification of phase portraits for PWHS, analyzes trajectory transitions through regularization, and provides conditions for the existence, stability, and uniqueness of limit cycles.

Findings

Classified phase portraits of PWHS based on normal forms and singularities.

Derived conditions for existence and stability of limit cycles in PWHS.

Identified families of PWHS with homoclinic orbits.

Abstract

The normal forms associated with holomorphic systems are well known in the literature. In this paper we are concerned about studying the piecewise smooth holomorphic systems (PWHS). Specifically, we classify the possible phase portraits of these systems from the known normal forms and the typical singularities of PWHS. Also, we are interested in understanding how the trajectories of the regularized system associated with the PWHS transits through the region of regularization. In addition, we know that holomorphic systems have no limit cycles, but piecewise smooth 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. Finally, we give some families of PWHS that have homoclinic orbits.