Nonexistence of Periodic Orbits for Forced–Damped Potential Systems in Bounded Domains

  • Florian KogelbauerEmail author


We prove \(L^r\)-estimates on periodic solutions of periodically forced, linearly damped mechanical systems with polynomially bounded potentials. The estimates are applied to obtain a nonexistence result of periodic solutions in bounded domains, depending on an upper bound on the gradient of the potential. The results are illustrated on examples.


Nonexistence of periodic orbits Forced-damped mechanical systems \(L^p\)-estimates 

Mathematics Subject Classification

37N05 70K40 



The author would like to thank Thomas Breunung and George Haller for several useful comments and suggestions. The author would like to thank the anonymous reviewers for their helpful comments and suggestions.


No funding has been receive.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Arnol’d, V.I.: Mathematical Methods of Classical Mechanics, vol. 60. Springer, Berlin (2013)Google Scholar
  2. Bajaj, A.: Bifurcations in a parametrically excited non-linear oscillator. Int. J. Non-Linear Mech. 22(1), 47–59 (1987)CrossRefGoogle Scholar
  3. Bendixson, I.: Sur les courbes définies par des équations différentielles. Acta Math. 24(1), 1–88 (1901)MathSciNetCrossRefGoogle Scholar
  4. Breunung, T., Haller, G.: When does a Steady-State Response Exist in a Periodically Forced Multi-Degree-of-Freedom Mechanical System? arXiv e-prints arXiv:1907.03605 (2019)
  5. Busenberg, S., Vandendriessche, P.: A method for proving the non-existence of limit cycles. J. Math. Anal. Appl. 172(2), 463–479 (1993)MathSciNetCrossRefGoogle Scholar
  6. Capietto, A., Wang, Z., et al.: Periodic solutions of liénard equations at resonance. Differ. Integral Eq. 16(5), 605–624 (2003)zbMATHGoogle Scholar
  7. Demidowitsch, W.: Eine Verallgemeinerung des Kriteriums von Bendixson. ZAMM-J. Appl. Math. Mech. 46(2), 145–146 (1966)MathSciNetCrossRefGoogle Scholar
  8. Denegri, C.M.: Limit cycle oscillation flight test results of a fighter with external stores. J. Aircr. 37(5), 761–769 (2000)CrossRefGoogle Scholar
  9. Dulac, H.: Recherche des cycles limites. C. R. Acad. Sci. Paris 204, 1703–1706 (1937)zbMATHGoogle Scholar
  10. Fečkan, M.: A generalization of Bendixsons criterion. Proc. Am. Math. Soc. 129(11), 3395–3399 (2001)MathSciNetCrossRefGoogle Scholar
  11. Gaines, R., Mawhin, J.: Coincidence degree and nonlinear differential equations. In: Lecture Notes in Mathematics. Springer, Berlin (1977)Google Scholar
  12. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. In: Applied Mathematical Sciences. Springer, New York (2002)Google Scholar
  13. Habib, G., Kerschen, G.: Suppression of limit cycle oscillations using the nonlinear tuned vibration absorber. Proc. R. Soc. A Math. Phys. Eng. Sci. 471(2176), 20140976 (2015)MathSciNetCrossRefGoogle Scholar
  14. Insperger, T., Stépán, G.: Updated semi-discretization method for periodic delay-differential equations with discrete delay. Int. J. Numer. Methods Eng. 61(1), 117–141 (2004)MathSciNetCrossRefGoogle Scholar
  15. Kelley, A.: Changes of variables near a periodic orbit. Trans. Am. Math. Soc. 126(2), 316–334 (1967)MathSciNetCrossRefGoogle Scholar
  16. Kelley, A.: Changes of variables near a periodic surface or invariant manifold. Trans. Am. Math. Soc. 131(2), 356–364 (1968)MathSciNetCrossRefGoogle Scholar
  17. Ko, J., Kurdila, A.J., Strganac, T.W.: Nonlinear control of a prototypical wing section with torsional nonlinearity. J. Guid. Control Dyn. 20(6), 1181–1189 (1997)CrossRefGoogle Scholar
  18. Lagrange, J., Poinsot, L.: Traité de la résolution des équations numériques de tous les degrés: aved des notes sur plusieurs points de la théorie des équatins algébriques. Bachelier (1826)Google Scholar
  19. Liang, S.: The rate of decay of stable periodic solutions for Duffing equation with L\(^p\)-conditions. Nonlinear Differ. Equ. Appl. NoDEA 23(2), 15 (2016)MathSciNetCrossRefGoogle Scholar
  20. Liapounoff, A.: Problème général de la stabilité du mouvement. In Annales de la Faculté des sciences de Toulouse: Mathématiques, vol. 9, pp. 203–474. Gauthier-Villars, Imperium-Editeur; ed. privat, Imperium-Libraire (1907)MathSciNetCrossRefGoogle Scholar
  21. Smith, R.A.: An index theorem and Bendixson’s negative criterion for certain differential equations of higher dimension. Proc. R. Soc. Edinb. Sect. A Math. 91(1–2), 63–77 (1981)MathSciNetCrossRefGoogle Scholar
  22. Ward Jr., J.R., et al.: Periodic solutions of ordinary differential equations with bounded nonlinearities. Topol. Methods Nonlinear Anal. 19(2), 275–282 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Institute for Mechanical SystemsETH ZürichZurichSwitzerland

Personalised recommendations