Bifurcation of limit cycles from a non-smooth perturbation of a two-dimensional isochronous cylinder

Abstract

Detect the birth of limit cycles in non-smooth vector fields is a very important matter into the recent theory of dynamical systems and applied sciences. The goal of this paper is to study the bifurcation of limit cycles from a continuum of periodic orbits filling up a two-dimensional isochronous cylinder of a vector field in $\mathbb{R}^{3}$. The approach involves the regularization process of non-smooth vector fields and a method based in the Malkin's bifurcation function for $C^{0}$ perturbations. The results provide sufficient conditions in order to obtain limit cycles emerging from the cylinder through smooth and non-smooth perturbations of it. To the best of our knowledge they also illustrate the implementation by the first time of a new method based in the Malkin's bifurcation function. In addition, some points concerning the number of limit cycles bifurcating from non-smooth perturbations compared with smooth ones are studied. In summary the results yield a better knowledge about limit cycles in non-smooth vector fields in $\mathbb{R}^{3}$ and explicit a manner to obtain them by performing non-smooth perturbations in codimension one Euclidean manifolds.