Publication Cover
Dynamical Systems
An International Journal
Volume 31, 2016 - Issue 3
179
Views
7
CrossRef citations to date
0
Altmetric
Original Articles

Birth of limit cycles for a class of continuous and discontinuous differential systems in (d + 2)–dimension

, &
Pages 237-250 | Received 28 Apr 2014, Accepted 14 Sep 2015, Published online: 20 Nov 2015

References

  • Hilbert D. Mathematische probleme. Lecture presented at the Second International Congress of Mathematicians, Paris, 1900. Nachr. Ges. Wiss. Göttingen Math. Phys. KL. (1900), 253–297 (English translation: Bull Amer Math Soc. 1902;8:437–479; Bull (New Series) Amer Math Soc. 2000;37:407–436).
  • Ilyashenko Y. Centennial history of Hilbert's 16th problem. Bull (New Series) Amer Math Soc. 2002;39:301–354.
  • Li J. Hilbert's 16th problem and bifurcations of planar polynomial vector fields. Int J Bifurcat Chaos Appl Sci Eng. 2003;13:47–106.
  • Andronov A, Vitt A, Khaikin S. Theory of oscillations. Oxford: Pergamon Press; 1966.
  • Barbashin EA. Introduction to the theory of stability. In: Lukes T, editor. Groningen: Noordhoff; 1970.
  • Bizzari F, Storace M, Colombo A. Bifurcation analysis of an impact model for forest fire prediction. Int J Bifurcat. Chaos. 2008;18:2275–2288.
  • Brogliato B. Nonsmooth mechanics. New York: Springer-Verlag; 1999.
  • Chillingworth DRJ. Discontinuity geometry for an impact oscillator. Dyn Syst. 2002;17:389–420.
  • Ito T. A Filippov solution of a system of differential equations with discontinuous right-hand sides. Econ. Lett. 1979;4:349–354.
  • Minorski N. Nonlinear oscillations. New York: Van Nostrand; 1962.
  • Krivan V. On the Gause predator–prey model with a refuge: a fresh look at the history. J Theor. Biol. 2011;274:67–73.
  • Coombes S. Neuronal networks with gap junctions: a study of piecewise linear planar neuron models. SIAM Appl Math. 2008;7:1101–1129.
  • Tonnelier A. The McKean's caricature of the FitzHugh–Nagumo model I. The space-clamped system. SIAM J Appl Math. 2003;63:459–484.
  • Tonnelier A, Gerstner W. Piecewise linear differential equations and integrate-and-fire neurons: insights from two-dimensional membrane models. Phys Rev E. 2003;67:021908.
  • Teixeira MA. Perturbation theory for non-smooth systems. Vol. 22, Encyclopedia of complexity and systems science. New York: Springer; 2009. p. 6697–6719.
  • Freire E, Ponce E, Rodrigo F, et al. Bifurcation sets of continuous piecewise linear systems with two zones. Int J Bifurcat Chaos. 1998;8:2073–2097.
  • Llibre J, Ordóñez M, Ponce E. On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry. Nonlinear Anal B. 2013;14:2002–2012.
  • Han M, Zhang W. On Hopf bifurcation in non–smooth planar systems. J Differ Equ. 2010;248:2399–2416.
  • Huan SM, Yang XS. On the number of limit cycles in general planar piecewise linear systems. Discrete Contin Dyn Syst A. 2012;32:2147–2164.
  • Llibre J, Ponce E. Three nested limit cycles in discontinuous piecewise linear differential systems with two zones. Dyn Contin Discrete Impulsive Syst B. 2012;19:325–335.
  • Cima A, Llibre J, Teixeira MA. Limit cycles of some polynomial differential systems in dimension 2, 3 and 4, via averaging. Appl Anal Int J. 2007;87:149–164.
  • Llibre J, Ponce E. Global first harmonic bifurcation diagram for nonlinear control systems. Dyn Stab Syst. 1996;11:49–88.
  • Llibre J, Ponce E, Ros J. Algebraic determination of limit cycles in 3–dimensional piecewise linear differential systems. Nonlinear Anal. 2011;74:6712–6727.
  • Llibre J, Ponce E, Ros J, et al. On the fold-Hopf bifurcation for continuous piecewise differential systems with symmetry. Chaos. 2010;20:033119.
  • Llibre J, Ponce E, Teruel A. Horseshoes near homoclinic orbits for piecewise linear differential systems in . Int J Bifurcat Chaos. 2007;17:1171–1184.
  • Llibre J, Zhang X. Hopf bifurcation in higher dimensional differential systems via the averaging method. Pac J Math. 2009;240:321–341.
  • Han M, Sheng L. Bifurcation of limit cycles in piecewise smooth systems via Melnikov function. J Appl Anal Comput. 2015;5:809–815.
  • Sanders JA, Verhulst F, Murdock J. Averaging methods in nonlinear dynamical systems. Vol. 59, Applied mathematical science. New York: Springer; 2007.
  • Verhulst F. Nonlinear differential equations and dynamical systems. Universitext. Springer; 1991.
  • Llibre J, Novaes DD, Teixeira MA. On the birth of limit cycles for non-smooth dynamical systems. Bull Sci Math. 2015;139:229–244.
  • Llibre J, Lopes BD, de Moraes JR. Limit cycles for a class of continuous and discontinuous cubic polynomial differential systems. Qual Theory Dyn Syst. 2014;13:129–148.
  • Llibre J, Mereu AC. Limit cycles for discontinuous quadratic differential systems with two zones. J Math Anal Appl. 2014;413:763–775.
  • Novaes DD. On nonsmooth perturbations of nondegenerate planar centers. Publ Mate. 2014;EXTRA:395–420.
  • Lloyd NG. Degree theory. Cambridge: Cambridge University Press; 1978.
  • Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products. 7th ed. Amsterdam: Academic Press; 2007.
  • Fulton W. Algebraic curves. Mathematics Lecture (Note Series). New York (NY): Benjamin; 1974.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.