128
Views
2
CrossRef citations to date
0
Altmetric
Original Articles

Sufficient conditions for the existence of periodic solutions of the extended Duffing–Van der Pol oscillator

&
Pages 1358-1382 | Received 20 Oct 2014, Accepted 28 Mar 2015, Published online: 09 Jun 2015

References

  • W.W. Adams and P. Loustaunau, An Introduction to Gröbner Bases, Graduate Studies in Mathematics, Vol. 3, American Mathematical Society, Providence, RI, 1994.
  • K. Bod, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, R. Oliva, and W. Weckesser, The forced Van der Pol equation II: Canards in the reduced system, SIAM J. Appl. Dyn. Syst. 2 (2003), pp. 570–608. doi: 10.1137/S1111111102419130
  • A. Buica and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math. 128 (2004), pp. 7–22. doi: 10.1016/j.bulsci.2003.09.002
  • A. Buica, J.P. Françoise, and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small parameter, Comm. Pure Appl. Anal. 6 (2007), pp. 103–111.
  • A. Chen and G. Jiang, Periodic solution of the Duffing-Van der Pol oscillator by homotopy perturbation method, Int. J. Comput. Math. 87 (2010), pp. 2688–2696. doi: 10.1080/00207160802706591
  • Y.M. Chen and J.K. Liu, Uniformly valid solution of limit cycle of the Duffing-van der Pol equation, Mech. Res. Comm. 36 (2009), pp. 845–850. doi: 10.1016/j.mechrescom.2009.06.001
  • W.O. Criminale, T.L. Jackson, and P.W. Nelson, Limit cycle-strange attractor competition, Stud. Appl. Math. 112 (2004), pp. 133–160. doi: 10.1111/j.0022-2526.2004.01507.x
  • C. Egami and N. Hirano, Periodic solutions for forced van der Pol type equations, Math. Econ. 1264 (2001), pp. 159–172.
  • R.D. Euzébio and J. Llibre, Periodic solutions of El Ni no model through the Vallis differential system, Discrete Contin. Dyn. Syst. Ser. A 34 (2014), pp. 3455–3469. doi: 10.3934/dcds.2014.34.3455
  • T.H. Fay, The forced Van der Pol equation, Int. J. Math. Educ. Sci. Techol. 40 (2009), pp. 669–677. doi: 10.1080/00207390902759568
  • J. Guchkenheimer, K. Hoffman, and W. Weckesser, The forced Van der Pol equation I: The slow flow and its bifurcations, SIAM J. Appl. Dyn. Syst. 2 (2003), pp. 1–35. doi: 10.1137/S1111111102404738
  • F.M.M. Kakmeni, S. Bowong, C. Tochawoua, and E. Kaptouom, Strange attractors and chaos control in a Duffing–Van der Pol oscillator with two external periodic forces, J. Sound Vib. 227 (2004), pp. 783–799. doi: 10.1016/j.jsv.2003.09.051
  • F.M.M. Kakmeni, S. Bowong, C. Tchawoua, and E. Kaptouom, Chaos control and synchronization of a φ6-Van der Pol oscillator, Phys. Lett. A 322 (2004), pp. 305–323. doi: 10.1016/j.physleta.2004.01.016
  • H.K. Leung, Synchronization dynamics of coupled van der Pol systems, Phys. A 321 (2003), pp. 248–255. doi: 10.1016/S0378-4371(02)01797-1
  • H. Li, Gröbner Bases in Ring Theory, World Scientific Publishing, Hackensack, NJ, 2011.
  • Z.-G. Li, W. Xu, and X.-Y. Zhang, Analysis of chaotic behavior in the extended Duffing-Van der Pol system subject to additive non-symmetry biharmonical excitation, Appl. Math. Comput. 183 (2006), pp. 858–871. doi: 10.1016/j.amc.2006.06.033
  • D. Liu and H. Yamaura, Chaos control of a φ6-Van der Pol oscillator driven by external excitation, Nonlinear Dyn. 68 (2012), pp. 95–105. doi: 10.1007/s11071-011-0206-0
  • J. Llibre and C. Vidal, Periodic solutions of a periodic FitzHugh-Nagumo differential system, preprint (2014).
  • M. Ma, J. Zhou, and J. Cai, Practical synchronization of second-order nonautonomous systems with parameter mismatch and its applications, Nonlinear Dyn. 69 (2012), pp. 1285–1292. doi: 10.1007/s11071-012-0346-x
  • I.G. Malkin, Some Problems of the Theory of Nonlinear Oscillations (Russian), Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956.
  • M. Roseau, Vibrations non linéaires et théorie de la stabilité (French), Springer Tracts in Natural Philosophy, Vol. 8, Springer-Verlag, Berlin-New York, 1966.
  • H. Salarieh and A. Alasty, Control of stochastic chaos using sliding mode method, J. Comput. Appl. Math. 225 (2009), pp. 135–145. doi: 10.1016/j.cam.2008.07.032
  • J.A. Sanders, F. VerhlstT, and J. Murdock, Averaging Method in Nonlinear Dynamical Systems, Applied Mathematical Science, Vol. 59, Springer, New York, 2007.
  • M.S. Siewe, F.M.M. Kakmeni, and C. Tchawoua, Resonant oscillation and homoclinic bifurcation in a φ6-Van der Pol oscillator, Chaos Solitons and Fractals 21 (2004), pp. 841–853. doi: 10.1016/j.chaos.2003.12.014
  • W. Szemplinska-Stupnicka and J. Rudowski, Neimark bifurcation, almost-periodicity and chaos in the forced van der Pol-Duffing system in the neighbourhood of the principal resonance, Phys. Lett. A 192 (1994), pp. 201–206. doi: 10.1016/0375-9601(94)90244-5
  • R. Tchoukuegno, B.R.N. Nbendjo, and P. Woafo, Resonant oscillations and fractal basin boundaries of a particle in a φ6-potential, Phys. A 304 (2002), pp. 362–378. doi: 10.1016/S0378-4371(01)00500-3
  • Y. Ueda and N. Akamatsu, Chaotically transitional phenomena in the forced negative-resistance oscillator, IEEE Trans. Circuits Syst. 28 (1981), pp. 217–224. doi: 10.1109/TCS.1981.1084975
  • A. Venkatesan and M. Lakshmanan, Bifurcation and chaos in the double-well Duffing–Van der Pol oscillator: Numerical and analytical studies, Phys. Rev. E 56 (1997), pp. 6321–6330. doi: 10.1103/PhysRevE.56.6321
  • L. Yang, Recent advances on determining the number of real roots of parametric polynomials, J. Symbolic Comput. 28 (1999), pp. 225–242. doi: 10.1006/jsco.1998.0274
  • X. Yang, W. Xu, and Z. Sun, Effect of Gaussian white noise on the dynamical behaviours of an extended Duffing-Van der Pol oscillator, Int. J. Bifur. Chaos Appl. Sci. Eng. 16 (2006), pp. 2587–2600. doi: 10.1142/S0218127406016276
  • X. Yang, W. Xu, and Z. Sun, Effect of bounded noise on the chaotic motion of a Duffing Van der pol oscillator in a φ6 potential, Chaos Solitons Fractals 27 (2006), pp. 778–788. doi: 10.1016/j.chaos.2005.04.048

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.