| Grant number: | 17/03352-6 |
| Support type: | Research Grants - Visiting Researcher Grant - International |
| Duration: | May 15, 2017 - June 22, 2017 |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Geometry and Topology |
| Principal researcher: | Claudio Aguinaldo Buzzi |
| Grantee: | Claudio Aguinaldo Buzzi |
| Visiting researcher: | Joan Torregrosa i Arús |
| Visiting researcher institution: | Universitat Autònoma de Barcelona (UAB), Spain |
| 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 |
| Associated research grant: | 13/24541-0 - Ergodic and qualitative theory of dynamical systems, AP.TEM |
Abstract
We will study the isolated periodic orbits (limit cycles) in planar piecewise polynomial systems. The first problem is defined in four zones defined by two straight lines that crosses perpendicularly. Here we consider two or four different linear systems. We prove that in a symmetric case (two systems) there exist systems with five limit cycles bifurcating far from the origin and in the non-symmetric case (four systems) the number increase to ten. The study of the Poincaré return map, computing the well known Lyapunov quantities, is the main technique in this problem. The second problem is defined in two zones separated by a straight line. We prove that, for polynomial perturbations of degree n, no more than Nn - 1 limit cycles appear up to a study of order N. We also show that this upper bound is reached for orders one and two. The Poincaré-Pontryagin-Melnikov theory is the main technique used to prove the results in this second problem. (AU)