May  2020, 25(5): 1835-1858. doi: 10.3934/dcdsb.2020005

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

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

Received  December 2018 Revised  July 2019 Published  May 2020 Early access  December 2019

Fund Project: 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.

Citation: Jaume Llibre, Lucyjane de A. S. Menezes. On the limit cycles of a class of discontinuous piecewise linear differential systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1835-1858. doi: 10.3934/dcdsb.2020005
References:
[1]

J. C. ArtésJ. LlibreJ. C. Medrado and M. A. Teixeira, Piecewise linear differential systems with two real saddles, Math. and Comp. in Simul., 95 (2014), 13-22.  doi: 10.1016/j.matcom.2013.02.007.

[2]

O. A. R. Cespedes, Ciclos Limite e Singularidades Típicas de Sistemas de Equações Diferenciais Suaves por Partes, Ph.D thesis, Universidade Federal de Goiás, 2015.

[3]

E. FreireE. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems, J. App. Dyn. Systems, 11 (2012), 182-211.  doi: 10.1137/11083928X.

[4]

M. A. Han and W. N. Zhang, On Hopf bifurcation in non-smooth planar systems, Differential Equations, 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002.

[5]

S.-M. Huan and X.-S. Yang, On the number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147.

[6]

J. H. Liang and S. Y. Tang, Global qualitative analysis of a non-smooth Gause predator-prey model with a refuge, Nonlinear Anal., 76 (2013), 165-180.  doi: 10.1016/j.na.2012.08.013.

[7]

J. LlibreD. D. Novaes and M. A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dyn., 82 (2015), 1159-1175.  doi: 10.1007/s11071-015-2223-x.

[8]

J. Llibre and A. E. Teruel, Existence of Poincaré maps in piecewise linear differential systems in $ {\mathbb{R}}^n$, Int. J. Bifurcation and Chaos, 14 (2004), 2843-2851.  doi: 10.1142/S0218127404010874.

[9]

J. Llibre and A. E. Teruel, Introduction to the Qualitative Theory of Differential Systems, Birkhäuser/Springer, Basel, 2014. doi: 10.1007/978-3-0348-0657-2.

[10]

J. Llibre and E. Ponce, Three limit cycles in discontinuous piecewise linear differential systems with two zones, Dyn. of Cont., Discr. Impul. Syst., Series B, 19 (2012), 325-335. 

[11]

J. C. Medrado and J. Torregrosa, Uniqueness of limit cycles for sewing piecewise linear systems, J. Math. Anal. Appl., 431 (2015), 529-544.  doi: 10.1016/j.jmaa.2015.05.064.

show all references

References:
[1]

J. C. ArtésJ. LlibreJ. C. Medrado and M. A. Teixeira, Piecewise linear differential systems with two real saddles, Math. and Comp. in Simul., 95 (2014), 13-22.  doi: 10.1016/j.matcom.2013.02.007.

[2]

O. A. R. Cespedes, Ciclos Limite e Singularidades Típicas de Sistemas de Equações Diferenciais Suaves por Partes, Ph.D thesis, Universidade Federal de Goiás, 2015.

[3]

E. FreireE. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems, J. App. Dyn. Systems, 11 (2012), 182-211.  doi: 10.1137/11083928X.

[4]

M. A. Han and W. N. Zhang, On Hopf bifurcation in non-smooth planar systems, Differential Equations, 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002.

[5]

S.-M. Huan and X.-S. Yang, On the number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147.

[6]

J. H. Liang and S. Y. Tang, Global qualitative analysis of a non-smooth Gause predator-prey model with a refuge, Nonlinear Anal., 76 (2013), 165-180.  doi: 10.1016/j.na.2012.08.013.

[7]

J. LlibreD. D. Novaes and M. A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dyn., 82 (2015), 1159-1175.  doi: 10.1007/s11071-015-2223-x.

[8]

J. Llibre and A. E. Teruel, Existence of Poincaré maps in piecewise linear differential systems in $ {\mathbb{R}}^n$, Int. J. Bifurcation and Chaos, 14 (2004), 2843-2851.  doi: 10.1142/S0218127404010874.

[9]

J. Llibre and A. E. Teruel, Introduction to the Qualitative Theory of Differential Systems, Birkhäuser/Springer, Basel, 2014. doi: 10.1007/978-3-0348-0657-2.

[10]

J. Llibre and E. Ponce, Three limit cycles in discontinuous piecewise linear differential systems with two zones, Dyn. of Cont., Discr. Impul. Syst., Series B, 19 (2012), 325-335. 

[11]

J. C. Medrado and J. Torregrosa, Uniqueness of limit cycles for sewing piecewise linear systems, J. Math. Anal. Appl., 431 (2015), 529-544.  doi: 10.1016/j.jmaa.2015.05.064.

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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