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Maximum number of limit cycles for certain piecewise linear dynamical systems

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Abstract

This paper deals with the question of the determinacy of the maximum number of limit cycles of some classes of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line \(\Sigma \). We restrict ourselves to the non-sliding limit cycles case, i.e., limit cycles that do not contain any sliding segment. Among all cases treated here, it is proved that the maximum number of limit cycles is at most 2 if one of the two linear differential systems of the discontinuous piecewise linear differential system has a focus in \(\Sigma \), a center, or a weak saddle. We use the theory of Chebyshev systems for establishing sharp upper bounds for the number of limit cycles. Some normal forms are also provided for these systems.

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References

  1. Andronov, A., Vitt, A., Khaikin, S.: Theory of Oscillations. Pergamon Press, Oxford (1966)

    Google Scholar 

  2. Artés, J.C., Llibre, J., Medrado, J.C., Teixeira, M.A.: Piecewise linear differential systems with two real saddles. Math. Comput. Simul. 95, 13–22 (2013)

    Article  Google Scholar 

  3. Bazykin, A.D.: Nonlinear Dynamics of Interacting Populations. World Scientific, River-Edge (1998)

    Google Scholar 

  4. Barbashin, E.A.: Introduction to the Theory of Stability (T. Lukes, Ed.). Noordhoff, Groningen (1970)

    Google Scholar 

  5. Bothe, D.: Periodic solutions of non-smooth friction oscillators. Z. Angew. Math. Phys. 50, 779808 (1999)

    Article  MathSciNet  Google Scholar 

  6. Braga, D.C., Mello, L.F.: More than three limit cycles in discontinuous piecewise linear differential systems with two zones in the plane. Int. J. Bifurc. Chaos 24, 1450056 (2014)

    Article  MathSciNet  Google Scholar 

  7. Braga, D.C., Mello, L.F.: Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane. Nonlinear Dyn. 73, 1283–1288 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brogliato, B.: Nonsmooth Mechanics. Springer, New York (1999)

    Book  MATH  Google Scholar 

  9. Buzzi, C., Pessoa, C., Torregrosa, J.: Piecewise linear perturbations of a linear center. Discrete Contin. Dyn. Syst. 33, 3915–3936 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Coombes, S.: Neuronal networks with gap junctions: a study of piecewise linear planar neuron models. SIAM Appl. Math. 7, 1101–1129 (2008)

    MathSciNet  MATH  Google Scholar 

  11. di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Applied Mathematical Science Series, vol. 163. Springer, London (2008)

    Google Scholar 

  12. Euzébio, R.D., Llibre, J.: On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line. J. Math. Anal. Appl. 424, 475–486 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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 

  14. Freire, E., Ponce, E., Torres, F.: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11, 181–211 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Freire, E., Ponce, E., Torres, F.: The discontinuous matching of two planar linear foci can have three nested crossing limit cycles. Publ. Mat. Vol. extra, pp. 221–253 (2014)

  16. Freire, E., Ponce, E., Torres, F.: A general mechanism to generate three limit cycles in planar Filippov systems with two zones. Nonlinear Dyn. 78, 251–263 (2014)

    Article  MathSciNet  Google Scholar 

  17. Giannakopoulos, F., Pliete, K.: Planar systems of piecewise linear differential equations with a line of discontinuity. Nonlinearity 14, 1611–1632 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Han, M., Zhang, W.: On Hopf bifurcation in non-smooth planar systems. J. Differ. Equ. 248, 2399–2416 (2010)

    Article  MATH  Google Scholar 

  19. Henry, C.: Differential equations with discontinuous righthand side for planning procedure. J. Econ. Theory 4, 541–551 (1972)

    Article  MathSciNet  Google Scholar 

  20. Huan, S.M., Yang, X.S.: On the number of limit cycles in general planar piecewise linear systems. Discrete Contin. Dyn. Syst. A 32, 2147–2164 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Huan, S.M., Yang, X.S.: Existence of limit cycles in general planar piecewise linear systems of saddle–saddle dynamics. Nonlinear Anal. 92, 82–95 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Huan, S.M., Yang, X.S.: On the number of limit cycles in general planar piecewise linear systems of node-node types. J. Math. Anal. Appl. 411, 340–353 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Ito, T.: A Filippov solution of a system of differential equations with discontinuous right-hand sides. Econ. Lett. 4, 349–354 (1979)

    Article  Google Scholar 

  25. Karlin, S.J., Studden, W.J.: T-Systems: With Applications in Analysis and Statistics. Pure and Applied Mathematics. Interscience Publishers, New York (1966)

    Google Scholar 

  26. Krivan, V.: On the Gause predator-prey model with a refuge: a fresh look at the history. J. Theor. Biol. 274, 67–73 (2011)

    Article  MathSciNet  Google Scholar 

  27. Kunze, M., Küpper, T.: Qualitative bifurcation analysis of a non-smooth friction-oscillator model. Z. Angew. Math. Phys. 48, 7–101 (1997)

    Article  Google Scholar 

  28. Leine, R.E., van Campen, D.H.: Discontinuous bifurcations of periodic solutions. Math. Comput. Model. 36, 259–273 (2002)

  29. Llibre, J., Novaes, D.D., Teixeira, M.A.: On the birth of limit cycles for non-smooth dynamical systems. Bull. Sci. Math. 139, 229–244 (2015)

    Article  MathSciNet  Google Scholar 

  30. Llibre, J., Novaes, D.D., Teixeira, M.A.: Limit cycles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones. Int. J. Bifurc. Chaos (2015)

  31. Llibre, J., Ponce, E.: Three nested limit cycles in discontinuous piecewise linear differential systems with two zones. Dyn. Contin. Discrete Impuls. Syst. Ser. B 19, 325–335 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Llibre, J., Teixeira, M.A., Torregrosa, J.: Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation. Int. J. Bifurc. Chaos 23, 1350066 (2013)

    Article  MathSciNet  Google Scholar 

  33. Lum, R., Chua, L.O.: Global properties of continuous piecewise-linear vector fields. Part I: simplest case in \(R^2\), memorandum UCB/ERL M90/22, University of California at Berkeley (1990)

  34. Minorski, N.: Nonlinear Oscillations. Van Nostrand, New York (1962)

    Google Scholar 

  35. Novaes, D.D., Ponce, H.: A simple solution to the Braga–Mello conjecture. Int. J. Bifurc. Chaos 25, 1550009 (2015). 7 pp

    Article  MathSciNet  Google Scholar 

  36. Novaes, D.D., Torregrosa, J.: On the extended Chebyshev systems with positive accuracy. Departament de Matemátiques, preprint no. 13 (2015)

  37. Tonnelier, A.: The McKean’s caricature of the FitzHugh–Nagumo model I. The space-clamped system. SIAM J. Appl. Math. 63, 459–484 (2003)

    Article  MathSciNet  Google Scholar 

  38. Tonnelier, A., Gerstner, W.: Piecewise linear differential equations and integrate-and-fire neurons: insights from two-dimensional membrane models. Phys. Rev. E 67, 021908 (2003)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We thank to the referees for their helpful comments and suggestions. The first author was partially supported by a MiNECO Grant MTM2013-40998-P, an AGAUR Grant Number 2013SGR-568, and the Grants FP7-PEOPLE-2012-IRSES 318999 and 316338. The second author was partially supported by a FAPESP Grant 2012/10231-7. The third authors was partially supported by a FAPESP Grant 2012/18780-0. The three authors are also supported by a CAPES CSF-PVE Grant 88881.030454/2013-01 from the program CSF-PVE.

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Authors and Affiliations

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

    Jaume Llibre

  2. Departamento de Matemática, Universidade Estadual de Campinas, Rua Sérgio Baruque de Holanda, 651, Cidade Universitária Zeferino Vaz, Campinas, SP, 13083-859, Brazil

    Douglas D. Novaes & Marco A. Teixeira

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  1. Jaume Llibre
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  2. Douglas D. Novaes
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  3. Marco A. Teixeira
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Correspondence to Douglas D. Novaes.

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

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  • DOI: https://doi.org/10.1007/s11071-015-2223-x

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