Piecewise smooth vector fields: Closing Lemmas, shifts and horseshoe dynamics.

Abstract

In this research project we will beinterested in studying unpublished topics related to the Qualitative Theory of Ordinary Differential Equations. When we consider smooth vector fields, it is a classic and very relevant research topic, to determine the possible existence of points to which the orbit returns infinite many times in its neighborhood. The Closing Lemma seeks to establish when perturbations of the initial system have a periodic orbit and thus the orbit "closes", hence the name of the lemma. In fact, for the smooth case, there are several formulations of Closing Lemmas, where the type of domain or the differentiability of the function used varies. For some of these formulations it is known that the response about the existence of the closed orbit is positive, for other formulations it is known that the answer is negative and there are still other formulations where there is no definitive answer. With regard to piecewise smooth vector fields, this theme is still little explored and will be the object of study throughout this project. We will look for results related to the possibility (or impossibility) to obtain a Closing Lemma. Furthermore, in the study of such recurrences we will establish conjugations between shifts (with finite and infinite symbols) and flows of piecewise smooth vector fields (via the use of first return maps); besides, we will show dynamics that resemble that obtained in Smale's Horseshoe. (AU)