Shilnikov problem in Filippov dynamical systems

Author(s)

Novaes, Douglas D. ^[1] ; Teixeira, Marco A. ^[1]

Affiliation

^[1] Univ Estadual Campinas, Dept Matemat, Rua Sergio Buarque de Holanda 651, BR-13083859 Campinas, SP - Brazil

• Document type: Journal article
• Source: Chaos; v. 29, n. 6 JUN 2019.
• Web of Science Citations: 0

Abstract

In this paper, we introduce the concept of sliding Shilnikov orbits for 3D Filippov systems. In short, such an orbit is a piecewise smooth closed curve, composed by Filippov trajectories, which slides on the switching surface and connects a Filippov equilibrium to itself, namely, a pseudo-saddle-focus. A version of Shilnikov's theorem is provided for such systems. Particularly, we show that sliding Shilnikov orbits occur in generic one-parameter families of Filippov systems and that arbitrarily close to a sliding Shilnikov orbit there exist countably infinitely many sliding periodic orbits. Here, no additional Shilnikov-like assumption is needed in order to get this last result. In addition, we show the existence of sliding Shilnikov orbits in discontinuous piecewise linear differential systems. As far as we know, the examples of Fillippov systems provided in this paper are the first to exhibit such a sliding phenomenon. Published under license by AIP Publishing. (AU)

• FAPESP's process: 18/16430-8 - Global dynamics of nonsmooth differential equations
• Grantee: Douglas Duarte Novaes
• Support Opportunities: Regular Research Grants