Symbolic dynamics of planar piecewise smooth vector fields

Abstract

It is well known that many results obtained for piecewise smooth vector fields do not have an analogous for smooth vector fields and vice-versa. These differences are generated by the non-uniqueness of trajectory passing through a point. Inspired by the classical fact that one-dimensional discrete dynamic systems can produce chaotic behavior, we construct a conjugation between shift maps and piecewise smooth vector fields presenting homoclinic loops which are associated to symbols in such a way that the flow restricted to a homoclinic loop is codified with a symbol. The construction of the topological conjugation between the quoted piecewise smooth vector fields and the respective shift spaces needs several technicality which were solved considering a specific family of piecewise smooth vector fields (Theorem A) and then generalizing the result for an entire class of piecewise smooth vector fields (Theorem B). By means of the results obtained and the techniques employed, a new perspective on the study of piecewise smooth vector fields is brought to light and, through already established results for discrete dynamic systems, we will be able to obtain results regarding piecewise smooth vector fields.