A Poincaré-Bendixson theorem for hybrid systems

Clark William; Bloch Anthony; Colombo Leonardo

doi:10.3934/mcrf.2019028

Article Contents

2020, Volume 10, Issue 1: 27-45. Doi: 10.3934/mcrf.2019028

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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
^* Corresponding author: William Clark

Received: April 2018

Early access: April 2019

Published: March 2020

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 34A38, 34C25, 70K05; Secondary: 34D20, 70K42.

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Figure 1. The orbit of the periodic orbit for the system given by Theorem 5.2

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Figure 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 $x$

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Figure 3. 1000 cycles of the flow from §5.1.1

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Figure 4. Displaying the locations of the jumps after performing 1000 iterations of the system in §5.1.2

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Figure 5. The rimless wheel

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Figure 6. 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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