Abstract
Minimal sets and chaos in planar piecewise smooth vector fields
Highlights
Dynamical systems have become one of the most promising areas of mathematics since its strong development started by Poincaré
Is mandatory to search for news methods and tools that are more realistic and feasible in theory. In this direction have emerged within the theory of dynamical system a set of methods which is widely known by piecewise smooth vector fields (PSVFs, for short)
We take into account aspects of chaotic PSVFs and how this concept relates to minimal sets. The definitions of both chaos e minimal sets are refined to consider the role of orientation and we provide a definitive characterization of chaotic PSVFs involving such objects
Summary
Dynamical systems have become one of the most promising areas of mathematics since its strong development started by Poincaré (see [22]). That leads to the behavior known as sliding motion, characterized by the collapse of distinct trajectories which combine to slide on the common frontier of each dynamic Under this scenario some behavior strange to the classical theory of dynamical systems may occur, so the study of new objects and the validation of known results is mandatory when one investigate PSVFs. For instance, we mention the Peixoto‘s Theorem (see [21]), the Closing Lemma (see [8]) and the Poincaré–Bendixson Theorem (see [5]), which posses analogues version in the context of PSVFs (see [10, 13, 17]). The definitions of both chaos e minimal sets are refined to consider the role of orientation and we provide a definitive characterization of chaotic PSVFs involving such objects.
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