On topological entropy of piecewise smooth vector fields

Antunes, André Amaral [UNESP]; Carvalho, Tiago; Varão, Régis

On topological entropy of piecewise smooth vector fields

Data

2023-07-25

Autores

Antunes, André Amaral

Carvalho, Tiago

Varão, Régis

Tipo

Artigo

Resumo

Non-smooth vector fields do not have necessarily the property of uniqueness of solution passing through a point and this is responsible to enrich the behavior of the system. Even on the plane, non-smooth vector fields can be chaotic, a feature impossible for the smooth or continuous case. We propose a new approach to better understand chaos for non-smooth vector fields by using the notion of entropy of a system. We construct a metric space of all possible trajectories of a non-smooth vector field, where we define a flow inherited by the vector field and then define the topological entropy in this scenario. As a consequence, we are able to obtain some general results and give some examples of planar non-smooth vector fields with positive (finite and infinite) entropy.