Limit Cycles of Piecewise Smooth Differential Equations on Two Dimensional Torus
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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.
KeywordsPiecewise smooth differential equations Limit cycles Global dynamics in torus
Mathematics Subject ClassificationPrimary 34A36 34C07 34C23 34C60
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.
- 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
- 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
- 9.Novaes, D.D., Torregrosa, J.A.: On extended chebyshev systems with positive accuracy. J. Math. Anal. Appl. 448, 171–186 (2017)Google Scholar
- 14.Poincaré, H.: Sur les courbes définies par les équation differentielles iv. J. Math. Pures Appl. 2, 155–217 (1886)Google Scholar
- 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.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.Tonon, D.J., Martins, R.M.: Chaos in piecewise smooth vector fields on two dimensional torus and sphere. arXiv:1601.05670