Nonlinear Dynamics

, Volume 98, Issue 3, pp 1761–1780 | Cite as

When does a periodic response exist in a periodically forced multi-degree-of-freedom mechanical system?

  • Thomas BreunungEmail author
  • George Haller
Original paper


While periodic responses of periodically forced dissipative nonlinear mechanical systems are commonly observed in experiments and numerics, their existence can rarely be concluded in rigorous mathematical terms. This lack of a priori existence criteria for mechanical systems hinders definitive conclusions about periodic orbits from approximate numerical methods, such as harmonic balance. In this work, we establish results guaranteeing the existence of a periodic response without restricting the amplitude of the forcing or the response. Our results provide a priori justification for the use of numerical methods for the detection of periodic responses. We illustrate on examples that each condition of the existence criterion we discuss is essential.


Nonlinear oscillations Periodic response Global analysis Harmonic balance Existence criterion 



We are thankful to Florian Kogelbauer and Walter Lacarbonara for fruitful discussion on this work.


We received no funding for this study.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest

Supplementary material


  1. 1.
    Antman, S., Lacarbonara, W.: Forced radial motions of nonlinearly viscoelastic shells. J. Elast. 96(2), 155–190 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ascher, U., Russell, R., Mattheij, R.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, volume 13 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1995)Google Scholar
  3. 3.
    Bobylev, N., Burman, Y., Korovin, S.: Approximation Procedures in Nonlinear Oscillation Theory, volume 2 of De Gruyter Series in Nonlinear Analysis and Applications. de Gruyter, Berlin (1994)Google Scholar
  4. 4.
    Bolotin, V.: Nonconservative Problems of the Theory of Elastic Stability. Macmillan, New York (1963)zbMATHGoogle Scholar
  5. 5.
    Breunung, T., Haller, G.: Explicit backbone curves from spectral submanifolds of forced-damped nonlinear mechanical systems. Proc. R. Soc. A 474(2213), 20180083 (2018)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cameron, T., Griffin, J.: An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems. J. Appl. Mech. 56(1), 149–154 (1989)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chu, J., Torres, P., Zhang, M.: Periodic solutions of second order non-autonomous singular dynamical systems. J. Differ. Equ. 239(1), 196–212 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chua, L., Ushida, A.: Algorithms for computing almost periodic steady-state response of nonlinear systems to multiple input frequencies. IEEE Trans. Circuits Syst. 28(10), 953–971 (1981)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cochelin, B., Vergez, C.: A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions. J. Sound Vib. 324(1–2), 243–262 (2009)Google Scholar
  10. 10.
    Coddington, E., Levinson, N.: Theory of Ordinary Differential Equations. Tata McGraw-Hill, New Delhi (1982)zbMATHGoogle Scholar
  11. 11.
    Dankowicz, H., Schilder, F.: Recipes for Continuation, volume 11 of Computational Science and Engineering. Society for Industrial and Applied Mathematics, Philadelphia (2013)zbMATHGoogle Scholar
  12. 12.
    Edwards, R.: Fourier Series: A Modern Introduction, vol. 2. Springer, New York (1979)zbMATHGoogle Scholar
  13. 13.
    Farkas, M.: Periodic Motions, volume 104 of Applied Mathematical Sciences. Springer, New York (1994)Google Scholar
  14. 14.
    Gaines, R., Mawhin, J.: Coincidence Degree, and Nonlinear Differential Equations, volume 568 of Lecture Notes in Mathematics. Springer, Berlin (1977)Google Scholar
  15. 15.
    Géradin, M., Rixen, D.: Mechanical Vibrations: Theory and Application to Structural Dynamics, 3rd edn. Wiley, Chichester (2015)Google Scholar
  16. 16.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, volume 42, Ed. 2002 of Applied Mathematical Sciences, corr. 7 printing edn. Springer, New York (2002)Google Scholar
  17. 17.
    Haro, A., de la Llave, R.: A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results. J. Differ. Equ. 228(2), 530–579 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  19. 19.
    Jain, S., Breunung, T., Haller, G.: Fast computation of steady-state response for high-degree-of-freedom nonlinear systems. Nonlinear Dyn. 97, 313–341 (2019)Google Scholar
  20. 20.
    Kogelbauer, F., Breunung, T., Haller, G.: When does the method of harmonic balance give a correct prediction for mechanical systems? (2018) (submitted) Google Scholar
  21. 21.
    Krasnosel’skij, M.: The Operator of Translation Along the Trajectories of Differential Equations, volume 19 of Translations of Mathematical Monographs. American Mathematical Society, Providence (1968)Google Scholar
  22. 22.
    Lazer, A.: On Schauder’s fixed point theorem and forced second-order nonlinear oscillations. J. Math. Anal. Appl. 21(2), 421–425 (1968)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lefschetz, S.: Existence of periodic solutions for certain differential equations. Proc. Natl. Acad. Sci. U. S. A. 29(1), 29 (1943)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Leipholz, H.H.: Direct Variational Methods and Eigen Value Problems in Engineering, volume Mechanics of Elastic Stability, volume 5 of Monographs and Textbooks on Mechanics of Solids and Fluid. Noordhoff, Leyden (1977)Google Scholar
  25. 25.
    Manásevich, R., Mawhin, J.: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Differ. Equ. 145(2), 367–393 (1998)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Martelli, M.: On forced nonlinear oscillations. J. Math. Anal. Appl. 69(2), 496–504 (1979)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mawhin, J.: An extension of a theorem of A. C. Lazer on forced nonlinear oscillations. J. Math. Anal. Appl. 40(1), 20–29 (1972)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mawhin, J.: Periodic Solutions of Systems with p-Laplacian-Like Operators, pp. 37–63. Birkhäuser, Boston (2001)zbMATHGoogle Scholar
  29. 29.
    Mickens, R.: An Introduction to Nonlinear Oscillations. Cambridge University Press, Cambridge (1981)zbMATHGoogle Scholar
  30. 30.
    Mickens, R.: Comments on the method of harmonic balance. J. Sound Vib. 94(3), 456–460 (1984)MathSciNetGoogle Scholar
  31. 31.
    Mickens, R.: Truly Nonlinear Oscillations: Harmonic Balance, Parameter Expansions, Iteration, and Averaging Methods. World Scientific, Singapore (2010)zbMATHGoogle Scholar
  32. 32.
    Murdock, J.: Normal Forms and Unfoldings for Local Dynamical Systems. Springer Monographs in Mathematics. Springer, New York (2003)zbMATHGoogle Scholar
  33. 33.
    Nayfeh, A.: Perturbation Methods. Physics Textbook. Wiley, Weinheim (2007)Google Scholar
  34. 34.
    Nayfeh, A., Mook, D.: Nonlinear Oscillations. Physics Textbook. Wiley, Weinheim (2007)Google Scholar
  35. 35.
    Precup, R.: Methods in Nonlinear Integral Equations. Kluwer Academic Publishers, Dordrecht (2002)zbMATHGoogle Scholar
  36. 36.
    Rouche, N., Mawhin, J.: Ordinary Differential Equations: Stability and Periodic Solutions, volume 5 of Surveys and Reference Works in Mathematics. Pitman, Boston (1980)zbMATHGoogle Scholar
  37. 37.
    Sanders, J., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems, volume 59 of Applied Mathematical Sciences, 2nd edn. Springer, New York (2007)zbMATHGoogle Scholar
  38. 38.
    Shaw, S., Pierre, C.: Normal modes for non-linear vibratory systems. J. Sound Vib. 164(1), 85–124 (1993)zbMATHGoogle Scholar
  39. 39.
    Stokes, A.: On the approximation of nonlinear oscillations. J. Differ. Equ. 12(3), 535–558 (1972)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Thompson, J.: Designing against capsize in beam seas: recent advances and new insights. Appl. Mech. Rev. 50(5), 307–325 (1997)Google Scholar
  41. 41.
    Thompson, J., Steward, B.: Nonlinear Dynamics and Chaos, 2nd edn. Wiley, Chichester (2002)Google Scholar
  42. 42.
    Torres, P.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 190(2), 643–662 (2003)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Touzé, C., Thomas, O., Chaigne, A.: Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes. J. Sound Vib. 273(1), 77–101 (2004)Google Scholar
  44. 44.
    Urabe, M.: Galerkin’s procedure for nonlinear periodic systems. Arch. Ration. Mech. Anal. 20(2), 120–152 (1965)zbMATHGoogle Scholar
  45. 45.
    van den Berg, J., Lessard, J.-P.: Rigorous numerics in dynamics. Not. AMS 62(9), 1057–1061 (2015)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute for Mechanical SystemsZurichSwitzerland

Personalised recommendations