On the number of limit cycles in piecewise linear systems

• Grant number: 13/00552-3
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: April 01, 2013
• End date: December 31, 2013
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Claudio Aguinaldo Buzzi
• Grantee: Yagor Romano Carvalho
• Host 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

Abstract

Introduce the student to the study of piecewise smooth dynamical systems through the study of the class of planar piecewise linear systems. In a preliminary phase conduct a study and prepare a collection of mathematical models involving special ordinary differential equations of first and second orders, solving them analytically and also developing computational experiments. Following will be an introductory study of the basic results of the Qualitative Theory of Ordinary Differential Equations, with special emphasis on planar systems. Study global aspects such as the notion of limit sets and attractors, Poincaré-Bendixon Theorem, and the Poincaré First Return Map in planar systems. In the final phase will be studied piecewise linear systems themselves. First will be the equilibrium points of the system and then study a special class of piecewise linear systems with a saddle point in each region of definition. This system has a generalized singular point in the line of separation and we prove that the maximum number of cycles that bifurcates from it is two. (AU)