On limit cycles in regularized Filippov systems bifurcating from homoclinic-like connections to regular-tangential singularities

TL;DR

This paper investigates the emergence and stability of limit cycles in regularized Filippov systems near homoclinic-like connections to regular-tangential singularities, providing conditions for their existence, stability, and uniqueness.

Contribution

It offers new criteria for the existence and stability of limit cycles in regularized Filippov systems near complex singularities, using a novel approach to construct the first return map.

Findings

Conditions for bifurcation of limit cycles are established.

Criteria for stability and uniqueness of limit cycles are provided.

A new method for constructing the first return map in this context is introduced.

Abstract

In this paper, we are concerned about smoothing of Filippov systems around homoclinic-like connections to regular-tangential singularities. We provide conditions to guarantee the existence of limit cycles bifurcating from such connections. Additional conditions are also provided to ensured the stability and uniqueness of such limit cycles. All the proofs are based on the construction of the first return map of the regularized Filippov system around homoclinic-like connections. Such a map is obtained by using a recent characterization of the local behaviour of the regularized Filippov system around regular-tangential singularities. Fixed point theorems and Poincar\'{e}-Bendixson arguments are also employed.