Duffing Oscillators

  • Leonid I. ManevitchEmail author
  • Agnessa Kovaleva
  • Valeri Smirnov
  • Yuli Starosvetsky
Part of the Foundations of Engineering Mechanics book series (FOUNDATIONS)


In this chapter, we illustrate the role of LPTs in the analysis of nonlinear non-stationary oscillations in non-stationary systems by a simple example of a periodically forced single-degree-of-freedom (SDOF) Duffing oscillator. The main difference from the conservative system described in the preliminary section is that we deal here with resonance energy flow from the source of energy instead of internal (intermodal) resonance.


  1. Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillators. Dover, New York (1966)Google Scholar
  2. Gendelman, O.V., Gourdon, E., Lamarque, C.H.: Quasiperiodic energy pumping in coupled oscillators under periodic forcing. J. Sound Vib. 294, 651–662 (2006)Google Scholar
  3. Gendeman, O.V., Starosvetsky, Y., Feldman, M.: Attractors of harmonically forced linear oscillator with attached nonlinear energy sink I: description of response regimes. Nonlinear Dyn. 51, 31–46 (2008)CrossRefzbMATHGoogle Scholar
  4. Guckenheimer, J., Hoffman, K., Weckesser, W.: Bifurcations of relaxation oscillations near folded saddles. Int. J. Bifurcat. Chaos 15, 3411 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Guckenheimer, J., Wechselberger, M., Young, L.S.: Chaotic attractors of relaxation oscillators. Nonlinearity 19, 701–720 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Korn, G.A., Korn, T.M.: Mathematical handbook for scientists and engineers, 2nd edn. Dover Publications, New York (2000)zbMATHGoogle Scholar
  7. Kovaleva, A., Manevitch, L.I.: Resonance energy transport and exchange in oscillator arrays. Phys. Rev. E 88, 022904 (2013)Google Scholar
  8. Manevitch, L.I.: New approach to beating phenomenon in coupling nonlinear oscillatory chains. Arch. Appl. Mech. 77, 301–312 (2007)CrossRefzbMATHGoogle Scholar
  9. Manevitch, L.I., Musienko, A.I.: Limiting phase trajectories and energy exchange between anharmonic oscillator and external force, Nonlinear Dyn. 58, 633–642 (2009)Google Scholar
  10. Manevitch, L.I., Kovaleva, A., Manevitch, E.L., Shepelev, D.S.: Limiting phase trajectories and non-stationary resonance oscillations of the duffing oscillator, part 1. A non-dissipative oscillator. Commun. Nonlinear Sci. Numer. Simul. 16, 1089–1097 (2011a)Google Scholar
  11. Manevitch, L.I., Kovaleva, A., Manevitch, E.L., Shepelev, D.S.: Limiting phase trajectories and non-stationary resonance oscillations of the Duffing oscillator. part 2. A dissipative oscillator. Commun. Nonlinear Sci. Numer. Simulat. 16, 1098–1105 (2011b)Google Scholar
  12. Manevitch, L.I., Kovaleva, A., Shepelev, D.S.: Non-smooth approximations of the limiting phase trajectories for the Duffing oscillator near 1:1 resonance. Physica D 240, 1–12 (2011c)Google Scholar
  13. Mirkina (Kovaleva), A.S.: On a modification of the averaging method and estimates of higher approximations. J. Appl. Math. Mech. 41, 901–909 (1977)Google Scholar
  14. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley-VCH, Weinheim (2004)zbMATHGoogle Scholar
  15. Neishtadt, A.I.: Passage through a separatrix in a resonance problem with slowly varying parameter. J. Appl. Math. Mech. 39, 594–605 (1975)MathSciNetCrossRefGoogle Scholar
  16. Pilipchuk, V.N.: Nonlinear dynamics: between linear and impact limits. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  17. Raghavan, S., Smerzi, A., Fantoni, S., Shenoy, S.R.: Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, π-oscillations, and macroscopic quantum self-trapping. Phys. Rev. A 59, 620–633 (1999)CrossRefGoogle Scholar
  18. Sagdeev, R.Z., Usikov, D.A., Zaslavsky, G.M.: Nonlinear physics: from the pendulum to turbulence and chaos. Harwood Academic Publishers, New York (1988)Google Scholar
  19. Starosvetsky, Y., Gendelman, O.V.: Strongly modulated response in forced 2DOF oscillatory system with essential mass and potential asymmetry. Physica D 237, 1719–1733 (2008a)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Starosvetsky, Y., Gendelman, O.V.: Attractors of harmonically forced linear oscillator with attached nonlinear energy sink II: optimization of a nonlinear vibration absorber. Nonlinear Dynam 51, 47–57 (2008b)CrossRefzbMATHGoogle Scholar
  21. Starosvetsky, Y., Gendelman, O.V.: Dynamics of a strongly nonlinear vibration absorber coupled to a harmonically excited two-degree-of-freedom system. J. Sound Vibr 312, 234–256 (2008c)CrossRefGoogle Scholar
  22. Starosvetsky, Y., Gendelman, O.V.: Interaction of nonlinear energy sink with a two degrees of freedom linear system: Internal resonance. J. Sound Vibr 329, 1836–1852 (2009a)CrossRefGoogle Scholar
  23. Starosvetsky, Y., Gendelman, O.V.: Vibration absorption in systems with a nonlinear energy sink: nonlinear damping. J. Sound Vibr 324, 916–939 (2009b)CrossRefGoogle Scholar
  24. Szmolyan, P., Wechselberger, M.: Relaxation oscillations in R3J. Diff. Eqns 200, 69–104 (2004)CrossRefzbMATHGoogle Scholar
  25. Theocharis, G., Boechler, N., Kevrekidis, P.G., Job, S., Porter, M.A., Daraio, C.: Intrinsic energy localization through discrete gap breathers in one-dimensional diatomic granular crystals. Phys. Rev. E 820 56604 (1–11) (2010)Google Scholar
  26. Vakakis, A.F., Manevitch, L.I., Mikhlin, Yu.V., Pilipchuk, V.N., Zevin, A.A.: Normal Modes and Localization in Nonlinear Systems. Wiley, New York (1996)Google Scholar
  27. Vakakis, A.F., Gendelman, O., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Nonlinear targeted energy transfer in mechanical and structural systems. Springer, Berlin (2008)zbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Leonid I. Manevitch
    • 1
    Email author
  • Agnessa Kovaleva
    • 2
  • Valeri Smirnov
    • 1
  • Yuli Starosvetsky
    • 3
  1. 1.Institute of Chemical PhysicsRussian Academy of ScienceMoscowRussia
  2. 2.Space Research InstituteRussian Academy of ScienceMoscowRussia
  3. 3.Technion—Israel Institute of TechnologyFaculty of Mechanical EngineeringHaifaIsrael

Personalised recommendations