The discontinuous matching of two globally asymptotically stable crossing piecewise smooth systems in the plane do not produce in general a piecewise differential system globally asymptotically stable

Author(s)

Braga, Denis de Carvalho ; Dias, Fabio Scalco ; Llibre, Jaume ; Mello, Luis Fernando

• Document type: Journal article
• Source: SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES; v. N/A, p. 8-pg., 2023-05-30.

Abstract

A differential system in the plane R2 is globally asymptotically stable if it has an equilibrium point p and all the other orbits of the system tend top in forward time. In other words if the basin of attraction of p is R2. The problem of determining the basin of attraction of an equilibrium point is one of the main problems in the qualitative theory of differential equations. We prove that planar crossing piecewise smooth systems with two zones formed by two globally asymptotically stable differential systems sharing the same equilibrium point localized in the separation line are not necessarily globally asymptotically stable. (AU)

• FAPESP's process: 19/07316-0 - Singularity theory and its applications to differential geometry, differential equations and computer vision
• Grantee: Farid Tari
• Support Opportunities: Research Projects - Thematic Grants