Abstract
We provide lower bounds for the maximum number of limit cycles for the m-piecewise discontinuous polynomial differential equations \({\dot{x} = y+{\rm sgn}(g_m(x, y))F(x)}\), \({\dot{y} = -x}\), where the zero set of the function sgn(g m (x, y)) with m = 2, 4, 6, . . . is the product of m/2 straight lines passing through the origin of coordinates dividing the plane into sectors of angle 2π/m, and sgn(z) denotes the sign function.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. Washington (1964)
Andronov A.A., Vitt A.A., Khaikin S.E.: Theory of Ocillators. Dover, New York (1966)
Anosov D.V.: On stability of equilibrium states of relay systems (Russian). Avtomatika i Telemehanika 20, 135–149 (1959)
Barbashin, E.A.: Introduction to the theory of stability. Translated from the Russian by Transcripta Service, London. Edited by T. Lukes Wolters–Noordhoff Publishing, Groningen (1970)
Berezin I.S., Zhidkov N.P.: Computing Methods, vol. II. Pergamon Press, Oxford (1964)
Brogliato, B.: Nonsmooth impact mechanics. Models, dynamics and control. Lecture Notes in Control and Information Sciences, vol. 220. Springer, London Ltd., London (1996)
Buica A., Llibre J.: Averaging methods for finding periodic orbits via Brouwer degree. Bulletin des Sciences Mathemàtiques 128, 7–22 (2004)
De Maesschalck P., Dumortier F.: Classical Liénard equation of degree n ≥ 6 can have \({\left[\frac{n-1}{2} \right]+2}\) limit cycles. J. Differ. Equ. 250, 2162–2176 (2011)
di Bernardo M., Budd C.J., Champneys A.R., Kowalczyk P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, Berlin (2008)
Dumortier F., Panazzolo D., Roussarie R.: More limit cycles than expected in Liénard equations. Proc. Am. Math. Soc. 135, 1895–1904 (2007)
Écalle J.: Introduction Aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac. Hermann, Paris (1992)
Guckenheimer J., Holmes P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vectors Fields. Springer, Berlin (1983)
Henry P.: Differential equations with discontinuous right-hand side for planning procedures. J. Econ. Theory 4, 545–551 (1972)
Hilbert, D.: Mathematische Probleme. Lecture, Second Internat. Congr. Math. (Paris, 1900) (Nachr. Ges. Wiss. Göttingen) Math. Phys. KL. pp. 253–297 (1900); English transl., Bull. American Mathematical Society, vol. 8, pp. 437–479 (1902)
Ilyashenko, Y.: Finiteness Theorems for Limit Cycles. Translations of Math. Monographs, vol. 94. American Mathematical Society (1991)
Ilyashenko Y., Panov A.: Some upper estimates of the number of limit cycles of planar vector fields with applications to Liénard equations. Moscow Math. J. 1, 583–599 (2001)
Kunze M., Kupper T.: Qualitative bifurcation analysis of a non-smooth friction-oscillator model. Z. Angew. Math. Phys. 48, 87–101 (1997)
Li C., Llibre J.: Uniqueness of limit cycle for Liénard equations of degree four. J. Differ. Equ. 252, 3142–3162 (2012)
Lins, A., de Melo, W., Pugh, C.C.: On Liénard’s Equation. Lecture Notes in Mathematics, vol. 597, pp. 335–357. Springer, Berlin (1977)
Llibre, J., Novaes, D.D., Teixeira, M.A.: Averaging methods for studying the periodic orbits of discontinuous differential systems. IMECC Technical Report, 8 (2012)
Llibre J., Rodríguez G.: Configurations of limit cycles and planar polynomial vector fields. J. Differ. Equ. 198, 374–380 (2004)
Llibre J., Swirszcz G.: On the limit cycles of polynomial vector fields. Dyn. Contin. Discrete Impuls. Syst. 18, 203–214 (2011)
Makarenkov O., Lamb J.S.W.: Dynamics and bifurcations of nonsmooth systems: a survey. Phys. D 241, 1826–1844 (2012)
Sanders J., Vehrulst F.: Averaging Method in Nonlinear Dynamical Systems, Applied Mathematical Sciences, vol. 59. Springer, Berlin (1985)
Smale S.: Mathematical problems for the next century. Math. Intell. 20, 7–15 (1998)
Teixeira M.A.: Perturbation theory for non-smooth systems. In: Meyers, R.A., Gaeta, G. (eds.) Encyclopedia of Complexity and Systems Science, pp. 6697–6709. Springer, New York (2009)
Vehrulst F.: Nonlinear Differential Equations and Dynamical Systems, Universitext. Springer, Berlin (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is partially supported by a MICINN/FEDER Grant MTM2008-03437, by a AGAUR Grant number 2009SGR-0410 and by ICREA Academia. The second author is partially supported by a FAPESP-BRAZIL Grant 2007/06896-5. All the authors are also supported by the joint project CAPES-MECD Grant PHB-2009-0025-PC.
Rights and permissions
About this article
Cite this article
Llibre, J., Teixeira, M.A. Limit cycles for m-piecewise discontinuous polynomial Liénard differential equations. Z. Angew. Math. Phys. 66, 51–66 (2015). https://doi.org/10.1007/s00033-013-0393-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-013-0393-2