# American Institute of Mathematical Sciences

March  2020, 10(1): 27-45. doi: 10.3934/mcrf.2019028

## A Poincaré-Bendixson theorem for hybrid systems

 1 Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, USA 2 Instituto de Ciencias Mathemáticas (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 13-15, 28049, Madrid, Spain

* Corresponding author: William Clark

Received  April 2018 Published  April 2019

Fund Project: W. Clark was supported by NSF grant DMS-1613819. A. Bloch was supported by NSF grant DMS-1613819 and AFOSR grant FA 9550-18-0028. L. Colombo was partially supported by Ministerio de Economia, Industria y Competitividad (MINEICO, Spain) under grant MTM2016-76702-P and "Severo Ochoa Programme for Centres of Excellence" in R & D (SEV-2015-0554).

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.

Citation: William Clark, Anthony Bloch, Leonardo Colombo. A Poincaré-Bendixson theorem for hybrid systems. Mathematical Control & Related Fields, 2020, 10 (1) : 27-45. doi: 10.3934/mcrf.2019028
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The orbit of the periodic orbit for the system given by Theorem 5.2
The vector $\delta x = F(y)\delta y + f(x)\delta t$, where the horizontal line is the tangent to $S$ at the point $x$
1000 cycles of the flow from §5.1.1
Displaying the locations of the jumps after performing 1000 iterations of the system in §5.1.2
The rimless wheel
Left: The lighter region indicates values of $\alpha$ and $\delta$ where there exists a limit cycle as predicted by equation (34). Right: The lower region is the domain of attraction for the limit cycle, whose existence is guaranteed by equation (34)
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