On the limit cycles of a class of discontinuous piecewise linear differential systems

Jaume Llibre; Lucyjane de A. S. Menezes

doi:10.3934/dcdsb.2020005

Article Contents

2020, Volume 25, Issue 5: 1835-1858. Doi: 10.3934/dcdsb.2020005

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On the limit cycles of a class of discontinuous piecewise linear differential systems

Jaume Llibre^1, , and
Lucyjane de A. S. Menezes^2,

1.
Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2.
Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74690-900, Goiânia, GO, Brazil

^* Corresponding author: Jaume Llibre
^* Corresponding author: Jaume Llibre

Received: November 30, 2018

Revised: June 30, 2019

Early access: December 2019

Published: May 2020

The first author is partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER) and grant MDM-2014-0445, the Agència de Gestió d'Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author is partially supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brazil(CAPES) grant PDSE-88881.133794/2016-01 and a CAPES finance code 001 grant

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Abstract

In this paper we consider discontinuous piecewise linear differential systems whose discontinuity set is a straight line $ L $ which does not pass through the origin. These systems are formed by two linear differential systems of the form $ \dot{x} = Ax\pm b $. We study the limit cycles of this class of discontinuous piecewise linear differential systems. We do this study by analyzing the fixed points of the return map of the system defined on the straight line $ L $. This kind of differential systems appear in control theory.

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Mathematics Subject Classification: Primary: 34C05, 34C07.

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Figure 1. Qualitative behavior of the Poincaré map $ \pi_{++} $; $ (a) \quad t>0 $ and $ (b) \quad t<0 $

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Figure 2. Qualitative behavior of the Poincaré map $ \pi_{++} $; $ (a) \quad t>0 $ and $ (b) \quad t<0 $

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Figure 3. Qualitative behavior of the Poincaré map $ \pi_{++} $; $ (a) \quad t>0 $ and $ (b) \quad t<0 $

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Figure 4. Qualitative behavior of the Poincaré map $ \widetilde{\pi}_{++} $; $ (a) \quad t>0 $ and $ (b) \quad t<0 $

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Figure 5. Relation between the half–lines $ L^O_+ $, $ L^I_+ $, $ L^{*O}_+ $, and $ L^{*I}_+ $ depending on (a) $ e_+\in S_- $, (b) $ e_+\in S_+ $

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