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

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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 R3. The approach involves the regularization process of non-smooth vector fields and a method based in the Malkin bifurcation function for C0 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 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 R3 and explicit a manner to obtain them by performing non-smooth perturbations in codimension one Euclidean manifolds.



Limit cycles, Malkin's bifurcation function, Non-smooth vector fields

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Bulletin des Sciences Mathematiques, v. 140, n. 5, p. 519-540, 2016.