Poincaré-Bendixson Theorem for Hybrid Systems
This work provides a foundational theoretical result for analyzing limit sets in hybrid dynamical systems, which is important for control theory and robotics.
The authors prove a Poincaré-Bendixson theorem for two-dimensional hybrid dynamical systems and describe a method for computing the derivative of the Poincaré return map, extending classical results to hybrid systems. They also prove a version for one-dimensional hybrid systems.
The Poincaré-Bendixson theorem plays an important role in the study of the qualitative behavior of dynamical systems on the plane; it describes the structure of limit sets in such systems. We prove a version of the Poincaré-Bendixson Theorem for two dimensional hybrid dynamical systems and describe a method for computing the derivative of the Poincaré return map, a useful object for the stability analysis of hybrid systems. We also prove a Poincaré-Bendixson Theorem for a class of one dimensional hybrid dynamical systems.