Regular and Chaotic Dynamics

, Volume 21, Issue 2, pp 147–159 | Cite as

Nonstationary energy localization vs conventional stationary localization in weakly coupled nonlinear oscillators

  • Leonid I. Manevitch
  • Agnessa Kovaleva
  • Grigori Sigalov


In this paper we study the effect of nonstationary energy localization in a nonlinear conservative resonant system of two weakly coupled oscillators. This effect is alternative to the well-known stationary energy localization associated with the existence of localized normal modes and resulting from a local topological transformation of the phase portraits of the system. In this work we show that nonstationary energy localization results from a global transformation of the phase portrait. A key to solving the problem is the introduction of the concept of limiting phase trajectories (LPTs) corresponding to maximum possible energy exchange between the oscillators. We present two scenarios of nonstationary energy localization under the condition of 1:1 resonance. It is demonstrated that the conditions of nonstationary localization determine the conditions of efficient targeted energy transfer in a generating dynamical system. A possible extension to multi-particle systems is briefly discussed.


nonlinear oscillations coupled oscillators nonlinear resonances systems with slow and fast motions 

MSC2010 numbers

34C15 70K30 70K70 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, P.W., Absence of Diffusion in Certain Random Lattices, Phys. Rev., 1958, vol. 109, no. 5, pp. 1492–1505.CrossRefGoogle Scholar
  2. 2.
    Lagendijk, A., van Tiggelen, B.A., and Wiersma, D. S., Fifty Years of Anderson Localization, Phys. Today, 2009, vol. 62, no. 8, pp. 24–29.CrossRefGoogle Scholar
  3. 3.
    Rosenberg, R.M., On Nonlinear Vibrations of Systems with Many Degrees of Freedom, Adv. Appl. Mech., 1966, vol. 9, pp. 155–242.CrossRefGoogle Scholar
  4. 4.
    Manevitch, L. I., Mikhlin, Yu. V., and Pilipchuk, V.V., Method of Normal Vibrations for Essentially Nonlinear Systems, Moscow: Nauka, 1989 (Russian).Google Scholar
  5. 5.
    Vakakis, A. F., Manevitch, L. I., Mikhlin, Yu. V., Pilipchuk, V.V., and Zevin, A.A., Normal Modes and Localization in Nonlinear Systems, New York: Wiley, 1996.CrossRefMATHGoogle Scholar
  6. 6.
    Manevitch, L. I. and Smirnov, V.V., Limiting Phase Trajectories and the Origin of Energy Localization in Nonlinear Oscillatory Chains, Phys. Rev. E, 2010, vol. 82, no. 3, 036602, 9 pp.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Campbell, D.K., Flach, S., and Kivshar, Y. S., Localizing Energy through Nonlinearity and Discreteness, Phys. Today, 2004, vol. 57, no. 1, pp. 43–49.CrossRefGoogle Scholar
  8. 8.
    Focus Issue: Nonlinear Localized Modes: Physics and Applications, Chaos, 2003, vol. 13, no. 2, pp. 586–799.CrossRefGoogle Scholar
  9. 9.
    Localization and Energy Transfer in Nonlinear Systems: Proc. of the 3rd Conference (San Lorenzo de El Escorial, Madrid, Spain, 17–21 June 2002), L. Vazquez, R. MacKay, and M.P. Zorzano (Eds.), Singapore: World Sci., 2003.Google Scholar
  10. 10.
    Dauxois, Th. and Peyrard, M., Physics of Solitons, Cambridge: Cambridge Univ. Press, 2006.MATHGoogle Scholar
  11. 11.
    Manevitch, L. I., New Approach to Beating Phenomenon in Coupled Nonlinear Oscillatory Chains, Arch. Appl. Mech., 2007, vol. 77, no. 5, pp. 301–312.CrossRefMATHGoogle Scholar
  12. 12.
    Manevitch, L. I., Kovaleva, A., and Shepelev, D. S., Non-Smooth Approximations of the Limiting Phase Trajectories for the Duffing Oscillator Near 1: 1 Resonance, Phys. D, 2011, vol. 240, no. 1, pp. 1–12.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kovaleva, A. and Manevitch, L. I., Resonance Energy Transport and Exchange in Oscillator Arrays, Phys. Rev. E, 2013, vol. 88, no. 2, 022904, 10 pp.CrossRefGoogle Scholar
  14. 14.
    Manevitch, L. I. and Kovaleva, A., Nonlinear Energy Transfer in Classical and Quantum Systems, Phys. Rev. E, 2013, vol. 87, no. 2, 022904, 12 pp.CrossRefGoogle Scholar
  15. 15.
    Smirnov, V. V., Shepelev, D. S., and Manevitch, L. I., Energy Exchange and Transition to Localization in the Asymmetric Fermi–Pasta–Ulam Oscillatory Chain, Eur. Phys. J. B, 2013, vol. 86, no. 1, Art. 10, 9 pp.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Manevitch, L. I., Kovaleva, M. A., and Pilipchuk, V.N., Non-Conventional Synchronization of Weakly Coupled Active Oscillators, Europhys. Lett., 2013, vol. 101, no. 5, 50002, 5 pp.CrossRefGoogle Scholar
  17. 17.
    Maniadis, P., Kopidakis, G., and Aubry, S., Classical and Quantum Targeted Energy Transfer between Nonlinear Oscillators, Phys. D, 2004, vol. 188, nos. 3-4, pp. 153–177.CrossRefMATHGoogle Scholar
  18. 18.
    Raghavan, S., Smerzi, A., Fantoni, S., and Shenoy, S. R., Coherent Oscillations between Two Weakly Coupled Bose–Einstein Condensates: Josephson Effects, p-Oscillations, and Macroscopic Quantum Self-Trapping, Phys. Rev. A, 1999, vol. 59, no. 1, pp. 620–633.Google Scholar
  19. 19.
    Pinnington, R. J., Energy Dissipation Prediction in a Line of Colliding Oscillators, J. Sound Vibration, 2003, vol. 268, no. 2, pp. 361–384.CrossRefGoogle Scholar
  20. 20.
    Nayfeh, A., Introduction to Perturbation Techniques, New York: Wiley, 1993.MATHGoogle Scholar
  21. 21.
    Manevitch, L. I., Vibro-Impact Models for Smooth Non-Linear Systems, in Vibro-Impact Dynamics of Ocean Systems and Related Problems, R. A. Ibrahim, V. I. Babitsky, M. Okuma (Eds.), Lecture Notes in Applied and Computational Mechanics, vol. 44, Berlin: Springer, 2009, pp. 191–201.CrossRefGoogle Scholar
  22. 22.
    Manevitch, L. I. and Smirnov, V.V., Resonant Energy Exchange in Nonlinear Oscillatory Chains and Limiting Phase Trajectories: from Small to Large Systems, in Advanced Nonlinear Strategies for Vibration Mitigation and System Identification, A. F. Vakakis (Ed.), CISM International Centre for Mechanical Sciences, vol. 518, Vienna: Springer, 2010, pp. 207–258.CrossRefGoogle Scholar
  23. 23.
    Smirnov, V. V. and Manevitch, L. I., Limiting Phase Trajectories and Dynamic Transitions in Nonlinear Periodic Systems, Acoust. Phys., 2011, vol. 57, no. 2, pp. 271–276.CrossRefGoogle Scholar
  24. 24.
    Smirnov, V. V., Shepelev, D. S., and Manevitch, L. I., Localization of Low-Frequency Oscillations in Single-Walled Carbon Nanotubes, Phys. Rev. Lett., 2014, vol. 113, no. 13, 135502, 4 pp.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • Leonid I. Manevitch
    • 1
  • Agnessa Kovaleva
    • 2
  • Grigori Sigalov
    • 3
  1. 1.Institute of Chemical PhysicsRussian Academy of SciencesMoscowRussia
  2. 2.Space Research InstituteRussian Academy of SciencesMoscowRussia
  3. 3.University of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations