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Two Limit Cycles in Liénard Piecewise Linear Differential Systems

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Abstract

Some techniques for studying the existence of limit cycles for smooth differential systems are extended to continuous piecewise linear differential systems. Rigorous new results are provided on the existence of two limit cycles surrounding the equilibrium point at the origin for systems with three zones separated by two parallel straight lines without symmetry. As a relevant application, it is shown the existence of bistable regimes in an asymmetric memristor-based electronic oscillator.

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References

  • Amador, A., Freire, E., Ponce, E., Ros, J.: On discontinuous piecewise linear models for memristor oscillators. Int. J. Bifurc. Chaos 27, 1730022 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Carletti, T., Villari, G.: A note on existence and uniqueness of limit cycles for Liénard systems. J. Math. Anal. Appl. 307, 763–773 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Carmona, V., Freire, E., Ponce, E., Torres, F.: On simplifying and classifying piecewise linear systems. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49, 609–620 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, H., Li, D., Xie, J., Yue, Y.: Limit cycles in planar continuous piecewise linear systems. Commun. Nonlinear Sci. Numer. Simul. 47, 438–454 (2017)

    Article  MathSciNet  Google Scholar 

  • Corinto, F., Ascoli, A., Gilli, M.: Nonlinear dynamics of memristor oscillators. IEEE Trans. Ciruits Syst. I: Regul. Pap. 58, 1323–1336 (2011)

    MathSciNet  Google Scholar 

  • Dumortier, F., Li, C.: On the uniqueness of limit cycles surrounding one or more singularities for Liénard equations. Nonlinearity 9, 1489–1500 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Dumortier, F., Rousseau, C.: Cubic Liénard equations with linear damping. Nonlinearity 3, 1015–1039 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems, Universitext. Springer, Berlin (2006)

    MATH  Google Scholar 

  • Freire, E., Ponce, E., Torres, F.: Hopf-like bifurcations in planar piecewise linear systems. Publ. Mat. 41, 135–148 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Freire, E., Ponce, E., Rodrigo, F., Torres, F.: Bifurcation sets of continuous piecewise linear systems with two zones. Int. J. Bifurc. Chaos 8, 2073–2097 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Freire, E., Ponce, E., Ros, J.: Limit cycle bifurcation from center in symmetric piecewise-linear systems. Int. J. Bifurc. Chaos 9, 895–907 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Freire, E., Ponce, E., Rodrigo, F., Torres, F.: Bifurcation sets of continuous piecewise linear systems with three zones. Int. J. Bifurc. Chaos 12, 1675–1702 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Freire, E., Ponce, E., Torres, F.: A general mechanism to generate three limit cycles in planar Filippov systems with two zones. Nonlinear Dyn. (2014). https://doi.org/10.1007/s11071-014-1437-7

  • Gasull, A., Giacomini, H., Llibre, J.: New criteria for the existence and non-existence of limit cycles in Liénard differential systems. Dyn. Syst. Int. J. 24, 171–185 (2009)

    Article  MATH  Google Scholar 

  • Hilbert, D.: Mathematische Probleme, Lecture, Second Internaternational Congress in Mathematics (Paris, 1900), Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pp. 253–297 (1900); English transl., Bull. Am. Math. Soc. 8, 437–479 (1902)

  • Hogan, S.J.: Relaxation oscillations in a system with a piecewise smooth drag coefficient. J. Sound Vib. 263, 467–471 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Ilyashenko, Yu.: Centennial history of Hilbert’s 16th problem. Bull. Am. Math. Soc. 39, 301–354 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Khibnik, A.I., Krauskopf, B., Rousseau, C.: Global study of a family of cubic Liénard equations. Nonlinearity 11, 1505–1519 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Li, J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13, 47–106 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Lima, M., Pessoa, C., Pereira, W.F.: Limit cycles bifurcating from a period annulus in continuous piecewise linear differential systems with three zones. Int. J. Bifurc. Chaos Appl. Sci. Eng. 27, 1750022 (14 pages) (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Llibre, J., Ponce, E.: Bifurcation of a periodic orbit from infinity in planar piecewise linear vector fields. Nonlinear Anal. 36, 623–653 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Llibre, J., Sotomayor, J.: Phase portraits of planar control systems. Nonlinear Anal. Methods Appl. 27, 1177–1197 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Llibre, J., Teruel, A.: Introduction to the Qualitative Theory of Differential Systems. Birkhäuser, Basel (2014)

    Book  MATH  Google Scholar 

  • Llibre, J., Valls, C.: Limit cycles for a generalization of Liénard polynomial differential systems. Chaos Solitons Fractals 46, 65–74 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Llibre, J., Mereu, A.C., Teixeira, M.A.: Limit cycles of the generalized polynomial Liénard differential equations. Math. Proc. Cambr. Philos. Soc. 148, 363–383 (2009)

    Article  MATH  Google Scholar 

  • Llibre, J., Ordóñez, M., Ponce, E.: On the existence and uniqueness of limit cycles in a planar piecewise linear systems without symmetry. Nonlinear Anal. Ser. B Real World Appl. 14, 2002–2012 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Llibre, J., Ponce, E., Valls, C.: Uniqueness and non-uniqueness of limit cycles of piecewise linear differential systems with three zones and no symmetry. J. Nonlinear Sci. 25, 861–887 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Ponce, E., Ros, J., Vela, E.: The focus-center-limit cycle bifurcation in discontinuous planar piecewise linear systems without sliding. In: Ibáñez, S., Pérez del Río, J. S., Pumariño, A., Rodríguez, J. A. (eds.) Progress and Challenges in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol. 54, pp. 335–349. Springer, Berlin, Heidelberg (2013)

  • Ponce, E., Ros, J., Vela, E.: Limit cycle and boundary equilibrium bifurcations in continuous planar piecewise linear systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 25(3), 1530008 (2015)

  • Ponce, E., Ros, J., Freire, E., Amador, A.: Unravelling the dynamical richness of 3D canonical memristor oscillators. Microelectron. Eng. 182, 1524 (2017)

    Article  Google Scholar 

  • Smale, S.: Mathematical problems for the next century. Math. Intell. 20, 7–15 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • van Horssen, W.T.: On oscillations in a system with a piecewise smooth coefficient. J. Sound Vib. 283, 1229–1234 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Xiao, D., Zhang, Z.: On the uniqueness and nonexistence of limit cycles for predator–prey systems. Nonlinearity 16, 1185–1201 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Ye, Y.-Q., et al.: Theory of Limit Cycles, Translations of Mathematical Monographs, vol. 66. American Mathematical Society, Providence (1986)

    Google Scholar 

  • Zhang, Z., Ding, T., Huang, W., Dong, Z.: Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, vol. 101. American Mathematical Society, Providence (1992)

    Google Scholar 

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Acknowledgements

J.L. is partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación Grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca Grant 2017 SGR 1617, and the European Project Dynamics-H2020-MSCA-RISE-2017-777911. E.P. was supported by MINECO/FEDER Grant MTM2015-65608-P and by the Consejería de Economía y Conocimiento de la Junta de Andalucía under Grant P12-FQM-1658. C.V. was partially supported by FCT - Fundação para a Ciência e a Tecnologia within the Project UID/MAT/04459/2013.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Catalonia, Spain

    Jaume Llibre

  2. Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería, Avda. de los Descubrimientos, 41092, Seville, Spain

    Enrique Ponce

  3. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001, Lisboa, Portugal

    Clàudia Valls

Authors
  1. Jaume Llibre
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  2. Enrique Ponce
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  3. Clàudia Valls
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Correspondence to Jaume Llibre.

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Communicated by Paul Newton.

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Llibre, J., Ponce, E. & Valls, C. Two Limit Cycles in Liénard Piecewise Linear Differential Systems. J Nonlinear Sci 29, 1499–1522 (2019). https://doi.org/10.1007/s00332-018-9523-5

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  • DOI: https://doi.org/10.1007/s00332-018-9523-5

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