Birth of limit cycles bifurcating from a nonsmooth center

Número: 
11
Ano: 
2013
Autor: 
Cláudio A. Buzzi
Tiago de Carvalho
Marco A. Teixeira
Abstract: 

This paper is concerned with a codimension analysis of a two-fold singularity of piecewise smooth planar vector fields, when it behaves itself like a center of smooth vector fields (also called nondegenerate Σ-center). We prove that any nondegenerate Σ-center is Σ-equivalent to a particular normal form Z0 . Given a positive integer number k we explicitly construct families of piecewise smooth vector fields emerging from Z0 that have k hyperbolic limit cycles bifurcating from the nondegenerate Σ-center of Z0 (the same holds for k = ∞).Moreover, we also exhibit families of piecewise smooth vector fields of codimension k emerging from Z0 . As a consequence we prove that Z0 has infinite codimension.

Keywords: 
nonsmooth vector field
bifurcation
limit cycles
centers
Mathematics Subject Classification 2010 (MSC 2010): 
Primary 34A36; 37G10; 37G05;
Observação: 
07/13
Arquivo: