Article Contents
Article Contents

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

• * Corresponding author: Jaume Llibre

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

• 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.

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

 Citation:

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

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

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

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

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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