A van der Pol-Duffing Oscillator with Indefinite Degree

• Hebai Chen ^[1] ; Jie Jin ^[1] ; Zhaoxia Wang ^[2] ; Baodong Zhang ^[1]
  1. [1] Central South University

     Central South University

     China

     
  2. [2] University of Electronic Science and Technology of China

     University of Electronic Science and Technology of China

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • This paper is to study the global dynamics of a van der Pol-Duffing oscillator with indefinite degree x˙ = y, y˙ = ax+x2n+1−δ(b+x2m)y, where a, b, δ ∈ R, m, n ∈ N+ and δ = 0. By qualitative and bifurcation analysis, the oscillator contains abundant nonlinear phenomena, including the heteroclinic bifurcation, degenerate Hopf bifurcation, bifurcation of equilibria at infinity and pitchfork bifurcation.

• Referencias bibliográficas
  • 1. Alsholm, P.: Existence of limit cycles for generalized Liénard equations. J. Math. Anal. Appl. 171, 242–255 (1992)
  • 2. Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equation, 2nd edn. Springer, Berlin (1987)
  • 3. Broer, H.W., Krauskopf, B., Vegter, G.: Global Analysis of Dynamical Systems. Institue of Physics Publishing, London (2001)
  • 4. Cao, Y., Liu, C.: The estimate of the amplitude of limit cycles of symmetric Liénard systems. J. Differ. Equ. 262, 2025–2038 (2017)
  • 5. Cândido, M.R., Llibre, J., Valls, C.: Non-existence, existence, and uniqueness of limit cycles for a generalization of the Van der Pol-Duffing...
  • 6. Cartwright, M.L.: Balthazar van der Pol. J. Lond. Math. Soc. 35, 367–376 (1960)
  • 7. Chen, H., Chen, X., Xie, J.: Global phase portraits of a degenerate Bogdanov–Takens system with symmetry. Discrete Contin. Dyn. Syst. Ser....
  • 8. Chen, H., Tang, Y., Xiao, D.: Global dynamics of a quintic Liénard system with symmetry I: saddle case. Nonlinearity 34, 4332–4372 (2021)
  • 9. Chow, S.-N., Li, C., Wang, D.: Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, Cambridge (1994)
  • 10. Duffing, G.: Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz. F. Vieweg u. Sohn, Braunschweig (1918)
  • 11. Dumortier, F., Herssens, C.: Polynomial Liénard equations near infinity. J. Differ. Equ. 153, 1–29 (1999)
  • 12. Feng, Z.: Duffing-van der pol-type oscillator systems. Discrete Contin. Dyn. Syst. Ser. S 7, 1231–1257 (2014)
  • 13. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations. Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York (1990)
  • 14. Holmes, P., Rand, D.A.: Phase portraits and bifurcations of the nonlinear oscillator x¨ + (α + γ x2)x˙ + βx + δx3 =...
  • 15. Horozov, E.: Versal deformations of equivariant vector fields for cases of symmetry of order 2 and 3(In Russian). Trusdy Sem. Petrov....
  • 16. Levinson, N., Smith, O.K.: A general equation for relaxation oscillations. Duke Math. J. 9, 382–403 (1942)
  • 17. Perko, L.M.: Rotated vector fields. J. Differ. Equ. 103, 127–145 (1993)
  • 18. Takens, F.: Forced oscillations and bifurcations, in Applications of Global Analysis, I, pp. 1–59. Rijksuniversiteit Utrecht, Communications...
  • 19. van der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Rev. 1(701–710), 754–762 (1920)
  • 20. van der Pol, B., van der Mark, J.: Frequency demultiplication. Nature 120, 363–364 (1927)
  • 21. Yang, L., Zeng, X.: An upper bound for the amplitude of limit cycles in Liénard systems with symmetry. J. Differ. Equ. 258, 2701–2710...
  • 22. Zhang, Z., Ding, T., Huang, W., Dong, Z.: Qualitative Theory of Differential Equations, Transl. Math. Monogr. 101, American Mathematical...