Taiwanese Journal of Mathematics

Stable and Unstable Periodic Solutions of the Forced Pendulum of Variable Length

Zaitao Liang and Zhongcheng Zhou

Full-text: Open access


In this paper, we study the existence of stable and unstable periodic solutions of the forced pendulum of variable length. The proof is based on a stability criterion which was obtained in [11] by using the third order approximation method and a generalized version of the Poincaré-Birkhoff fixed point theorem.

Article information

Taiwanese J. Math., Volume 21, Number 4 (2017), 791-806.

Received: 8 July 2016
Revised: 22 October 2016
Accepted: 23 October 2016
First available in Project Euclid: 27 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34D20: Stability
Secondary: 34C25: Periodic solutions

forced pendulum of variable length third order approximation method Poincaré-Birkhoff fixed point theorem


Liang, Zaitao; Zhou, Zhongcheng. Stable and Unstable Periodic Solutions of the Forced Pendulum of Variable Length. Taiwanese J. Math. 21 (2017), no. 4, 791--806. doi:10.11650/tjm/7829. https://projecteuclid.org/euclid.twjm/1501120835

Export citation


  • G. D. Birkhoff, Proof of Poincaré's geometric theorem, Trans. Amer. Math. Soc. 14 (1913), no. 1, 14–22.
  • ––––, An extension of Poincaré's last geometric theorem, Acta Math. 47 (1926), no. 4, 297–311.
  • J. Chu, J. Ding and Y. Jiang, Lyapunov stability of elliptic periodic solutions of nonlinear damped equations, J. Math. Anal. Appl. 396 (2012), no. 1, 294–301.
  • J. Chu, J. Lei and M. Zhang, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator, J. Differential Equations 247 (2009), no. 2, 530–542.
  • J. Chu, P. J. Torres and F. Wang, Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem, Discrete Contin. Dyn. Syst. 35 (2015), no. 5, 1921–1932.
  • ––––, Twist periodic solutions for differential equations with a combined attractive-repulsive singularity, J. Math. Anal. Appl. 437 (2016), no. 2, 1070–1083.
  • A. Fonda, M. Sabatini and F. Zanolin, Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré-Birkhoff theorem, Topol. Methods Nonlinear Anal. 40 (2012), no. 1, 29–52.
  • J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math. (2) 128 (1988), no. 1, 139–151.
  • ––––, Erratum to: “Generalizations of the Poincaré-Birkhoff theorem", Ann. of Math. (2) 164 (2006), no. 3, 1097–1098.
  • J. Lei, X. Li, P. Yan and M. Zhang, Twist character of the least amplitude periodic solution of the forced pendulum, SIAM J. Math. Anal. 35 (2003), no. 4, 844–867.
  • J. Lei and P. J. Torres, $L^{1}$ criteria for stability of periodic solutions of a Newtonian equation, Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 2, 359–368.
  • J. Lei, P. J. Torres and M. Zhang, Twist character of the fourth order resonant periodic solution, J. Dynam. Differential Equations 17 (2005), no. 1, 21–50.
  • S. Marò, Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Methods Nonlinear Anal. 42 (2013), no. 1, 51–75.
  • J. Mawhin, Global results for the forced pendulum equation, in Handbook of Differential Equations, 533–589, Elsevier/North-Holland, Amsterdam, 2004.
  • R. Ortega, Periodic solutions of a Newtonian equation: stability by the third approximation, J. Differential Equations 128 (1996), no. 2, 491–518.
  • ––––, Stable periodic solutions in the forced pendulum equation, Regul. Chaotic Dyn. 18 (2013), no. 6, 585–599.
  • ––––, A forced pendulum equation without stable periodic solutions of a fixed period, Port. Math. 71 (2014), no. 3-4, 193–216.
  • H. Poincaré, Sur un théorème de géométrie, Rend. Circ. Mat. Palermo 33 (1912), no. 1, 375–407.
  • P. J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations 190 (2003), no. 2, 643–662.
  • ––––, Existence and stability of periodic solutions for second-order semilinear differential equations with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 1, 195–201.
  • P. J. Torres and M. Zhang, Twist periodic solutions of repulsive singular equations, Nonlinear Anal. 56 (2004), no. 4, 591–599.
  • A. A. Zevin and M. A. Pinsky, Qualitative analysis of periodic oscillations of an earth satellite with magnetic attitude stabilization, Discrete Contin. Dynam. Systems 6 (2000), no. 2, 293–297.
  • M. Zhang, The best bound on the rotations in the stability of periodic solutions of a Newtonian equation, J. London Math. Soc. (2) 67 (2003), no. 1, 137–148.