Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities

Authors: Hongbin Chen and Yi Li
Journal: Proc. Amer. Math. Soc. 135 (2007), 3925-3932
MSC (2000): Primary 34C10, 34C25
Published electronically: September 7, 2007
MathSciNet review: 2341942
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the stability and exact multiplicity of periodic solutions of the Duffing equation with cubic nonlinearities,

$\displaystyle x''+cx'+ax-x^{3}=h(t),\tag{$*$} $

where $ a$ and $ c>0$ are positive constants and $ h(t)$ is a positive $ T$-periodic function. We obtain sharp bounds for $ h$ such that $ (*)$ has exactly three ordered $ T$-periodic solutions. Moreover, when $ h$ is within these bounds, one of the three solutions is negative, while the other two are positive. The middle solution is asymptotically stable, and the remaining two are unstable.

References [Enhancements On Off] (What's this?)

  • 1. J.M. Alonso, Optimal intervals of stability of a forced oscillator, Proc. Amer. Math. Soc. 123 (1995), 2031-2040. MR 1301005 (95k:34057)
  • 2. J.M. Alonso and R. Ortega, Boundedness and global asymptotic stability of a forced oscillator, Nonlinear Anal. 25 (1995), 297-309. MR 1336527 (96e:34076)
  • 3. H.B. Chen and Yi Li, Exact multiplicity for periodic solutions of a first-order differential equation, J. Math. Anal. Appl. 292 (2004), 415-422. MR 2047621 (2004m:34090)
  • 4. -, Existence, uniqueness, and stability of periodic solutions of an equation of Duffing type, Discrete Contin. Dyn. Syst., accepted.
  • 5. H.B. Chen, Yi Li, and X.J. Hou, Exact multiplicity for periodic solutions of Duffing type, Nonlinear Anal. 55 (2003), 115-124. MR 2001635 (2004g:34072)
  • 6. P.T. Church and J.G. Timourian, Global cusp maps in differential and integral equations, Nonlinear Anal. 20 (1993), 1319-1343. MR 1220838 (94h:58043)
  • 7. C. Fabry, J. Mawhin, and M.N. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations, Bull. London Math. Soc. 18 (1986), 173-180. MR 818822 (87e:34072)
  • 8. G. Fournier and J. Mawhin, On periodic solutions of forced pendulum-like equations, J. Differential Equations 60 (1985), no. 3, 381-395. MR 811773 (87a:34039)
  • 9. S. Gaete and R.F. Manásevich, Existence of a pair of periodic solutions of an O.D.E. generalizing a problem in nonlinear elasticity, via variational methods, J. Math. Anal. Appl. 134 (1988), 257-271. MR 961337 (89k:34059)
  • 10. G. Katriel, Periodic solutions of the forced pendulum: Exchange of stability and bifurcations, J. Differential Equations 182 (2002), no. 1, 1-50. MR 1912068 (2003e:34072)
  • 11. A.C. Lazer and P.J. McKenna, On the existence of stable periodic solutions of differential equations of Duffing type, Proc. Amer. Math. Soc. 110 (1990), 125-133. MR 1013974 (90m:34093)
  • 12. -, Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities, Trans. Amer. Math. Soc. 315 (1989), 721-739. MR 979963 (90a:34011)
  • 13. J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991) (M. Furi and P. Zecca, eds.), Lecture Notes in Math., vol. 1537, Springer, Berlin, 1993, pp. 74-142. MR 1226930 (94h:47121)
  • 14. F.I. Njoku and P. Omari, Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions, Appl. Math. Comput. 135 (2003), 471-490. MR 1937268 (2003h:34110)
  • 15. P. Omari and M. Trombetta, Remarks on the lower and upper solutions method for second- and third-order periodic boundary value problems, Appl. Math. Comput. 50 (1992) 1-21. MR 1164490 (93d:34040)
  • 16. R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Ital. B (7) 3 (1989), 533-546. MR 1010522 (90g:34045)
  • 17. G. Tarantello, On the number of solutions for the forced pendulum equation, J. Differential Equations 80 (1989), 79-93. MR 1003251 (90h:34058)
  • 18. A. Tineo, Existence of two periodic solutions for the periodic equation $ \ddot x=g(t,x)$, J. Math. Anal. Appl. 156 (1991), 588-596. MR 1103031 (92c:34050)
  • 19. -, A result of Ambrosetti-Prodi type for first order ODEs with cubic non-linearities, Part I, Ann. Mat. Pura Appl. (4) 182 (2003), 113-128. MR 1985559 (2004c:34129)
  • 20. P.J. Torres and M. Zhang, A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. Nachr. 251 (2003), 101-107. MR 1960807 (2004a:34032)
  • 21. A. Zitan and R. Ortega, Existence of asymptotically stable periodic solutions of a forced equation of Liénard type, Nonlinear Anal. 22 (1994), no. 8, 993-1003. MR 1277595 (95f:34047)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34C10, 34C25

Retrieve articles in all journals with MSC (2000): 34C10, 34C25

Additional Information

Hongbin Chen
Affiliation: Department of Mathematics, Xi’an Jiaotong University, Xi’an, People’s Republic of China

Yi Li
Affiliation: Department of Mathematics, Hunan Normal University, Changsha, Hunan, People’s Republic of China
Address at time of publication: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242

Keywords: Duffing equation, periodic solution, stability
Received by editor(s): September 27, 2006
Published electronically: September 7, 2007
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society