Grant number: 22/03801-3 Support type: Scholarships abroad - Research Internship - Post-doctor Effective date (Start): October 01, 2022 Effective date (End): September 30, 2023 Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology Principal researcher: Paulo Ricardo da Silva Grantee: Luiz Fernando da Silva Gouveia Supervisor abroad: Oleg Makarenkov Home Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil Research place: University of Texas at Dallas (UT Dallas), United States Associated to the scholarship: 20/04717-0 - Dynamical systems with symmetries and implicit differential equations, BP.PD Abstract Our main goal in this project is to perturbation Kolmogorov Systems on planar piecewise vector fields defined in two zones separated by a straight line and study how many limit cycles can be obtained.In addition, we intend to approach the Kolmogorov Systems through Sotomayor-Teixeira regularization. It is well known that different regularizations can lead to different ways of defining sliding solutions on. In addition, from a theoretical point of view, assuming only linear regularization would neglect a vast expanse of phenomena occurring in smooth systems approaching pointwisely to non-smooth ones. Therefore, our main goal at this point is to consider two Kolmogorov Systems and study the regularization when both $\varphi$ is a monotonic function and $\varphi$ is not a monotonic function.