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Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1011–1027 | Cite as

Limit Cycles of Piecewise Smooth Differential Equations on Two Dimensional Torus

  • Jaume Llibre
  • Ricardo Miranda Martins
  • Durval José Tonon
Article

Abstract

In this paper we study the limit cycles of some classes of piecewise smooth vector fields defined in the two dimensional torus. The piecewise smooth vector fields that we consider are composed by linear, Ricatti with constant coefficients and perturbations of these one, which are given in (3). Considering these piecewise smooth vector fields we characterize the global dynamics, studying the upper bound of number of limit cycles, the existence of non-trivial recurrence and a continuum of periodic orbits. We also present a family of piecewise smooth vector fields that posses a finite number of fold points and, for this family we prove that for any 2k number of limit cycles there exists a piecewise smooth vector fields in this family that presents k number of limit cycles and prove that some classes of piecewise smooth vector fields presents a non-trivial recurrence or a continuum of periodic orbits.

Keywords

Piecewise smooth differential equations Limit cycles Global dynamics in torus 

Mathematics Subject Classification

Primary 34A36 34C07 34C23 34C60 

Notes

Acknowledgements

The first author is partially supported by the MINECO grants MTM2016-77278-P and MTM2013-40998-P, an AGAUR grant number 2014SGR-568, the grants FP7-PEOPLE-2012-IRSES 318999, and a CAPES grant number 88881.030454/ 2013-01 from the program CSF-PVE. D. J. Tonon is supported by grant#2012/10 26 7000 803, Goiás Research Foundation (FAPEG), PROCAD/CAPES grant 88881.0 68462/2014-01 and by CNPq-Brazil. R. M. Martins is supported by FAPESP-Brazil project 2015/06903-8. The authors would like to thank Matheus Manzatto for the help with the figures.

References

  1. 1.
    Adler, R., Kitchens, B., Tresser, C.: Dynamics of non-ergodic piecewise affine maps of the torus. Ergod. Theory Dynam. Syst. 21(4), 959–999 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ashwin, P., Fu, X., Lin, C.: On planar piecewise and two-torus parabolic maps. Int. J. Bifurc. Chaos Appl. Sci. Eng. 19(7), 2383–2390 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berezin, I.S., Shidkov, N.P.: Computing methods. Vols. I, II. Translated by O. M. Blunn; translation edited by A. D. Booth. Pergamon Press, Oxford-Edinburgh-New York-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. Vol. I: xxxiv+464 pp. $ 15.00; Vol. II, (1965)Google Scholar
  4. 4.
    Chernov, N.I.: Ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the \(2\)-torus. J. Stat. Phys. 69(1–2), 111–134 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-smooth dynamical systems: Theory and applications. Applied Mathematical Sciences, vol. 163. Springer, London (2008)zbMATHGoogle Scholar
  6. 6.
    Filippov, A.F.: Differential equations with discontinuous righthand sides, volume 18 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, (1988). Translated from the RussianGoogle Scholar
  7. 7.
    Karlin, S., Studden, W.J.: Tchebycheff Systems: With Applications in Analysis and Statistics. Pure and Applied Mathematics, vol. XV. Interscience Publishers Wiley, New York (1966)zbMATHGoogle Scholar
  8. 8.
    Makarenkov, O., Lamb, J.S.W.: Dynamics and bifurcations of nonsmooth systems: a survey. Phys. D 241(22), 1826–1844 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Novaes, D.D., Torregrosa, J.A.: On extended chebyshev systems with positive accuracy. J. Math. Anal. Appl. 448, 171–186 (2017)Google Scholar
  10. 10.
    Orlov, Y.V.: Discontinuous Systems: Lyapunov Analysis and Robust Synthesis Under Uncertainty Conditions. Communications and Control Engineering Series. Springer, London (2009)zbMATHGoogle Scholar
  11. 11.
    Poincaré, H.: Mémoire sur les courbes d éfinies par une équation differentielle i. J. Math. Pures Appl. 7, 375–422 (1881)zbMATHGoogle Scholar
  12. 12.
    Poincaré, H.: Mémoire sur les courbes d éfinies par une équation differentielle ii. J. Math. Pures Appl. 8, 251–296 (1882)zbMATHGoogle Scholar
  13. 13.
    Poincaré, H.: Sur les courbes définies par les équation differentielles iii. J. Math. Pures Appl. 1, 167–244 (1885)zbMATHGoogle Scholar
  14. 14.
    Poincaré, H.: Sur les courbes définies par les équation differentielles iv. J. Math. Pures Appl. 2, 155–217 (1886)Google Scholar
  15. 15.
    Teixeira, M.A.: Perturbation theory for non-smooth systems. In: Mathematics of Complexity and Dynamical Systems, Vols. 1–3, pp. 1325–1336. Springer, New York, (2012)Google Scholar
  16. 16.
    Tkachenko, V.I.: On the existence of a piecewise-smooth invariant torus of an impulse system. In: Methods for investigating differential and functional-differential equations (Russian), pp. 91–96. Inst. Math. Acad. Nauk Ukrain. SSR, Kiev (1990)Google Scholar
  17. 17.
    Tonon, D.J., Martins, R.M.: Chaos in piecewise smooth vector fields on two dimensional torus and sphere. arXiv:1601.05670

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat AutoÌnoma de BarcelonaBellaterra, BarcelonaSpain
  2. 2.IMECC–UNICAMPCampinasBrazil
  3. 3.Institute of Mathematics and Statistics of Federal University of GoiásGoiâniaBrazil

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