Nonlinear Normal Modes and Chaotic Motions in Oscillatory Chains

  • Leonid I. Manevitch
  • Oleg V. Gendelman
  • Alexander V. Savin
Conference paper
Part of the Solid Mechanics and its Applications book series (SMIA, volume 122)


We present the results of analytical and numerical study of the random vibrations in several nonlinear one-dimensional oscillatory chains which model significant mechanical and physical systems. It is shown that existence and propagation of the random excitations in the considered models is strongly dependent on the properties of nonlinear normal modes. In particular, nonlinear vibrations in attenuation zone of linearized system initiated by global random excitations, correspond to localized nonlinear normal modes — breathers. In the localization region periodic contraction-extension of interparticle bonds occurs which is accompanied by decrease-increase of angles between the bonds. It is shown that the breathers present in thermalized chain and their contribution dependent on temperature has been revealed. Process of heat conduction in the chain with periodic potential of nearest-neighbor interaction is investigated by means of computer simulation. It is demonstrated that the periodic potential of nearest-neighbor interaction allows to obtain normal heat conductivity in isolated one-dimensional chain with conserved momentum. The system exhibits transition from infinite to normal heat conductivity with growth of its temperature. The physical reason for normal heat conductivity is excitation of high-frequency stationary localized rotational modes. These modes absorb the momentum and facilitate locking of the heat flux. Concentration and lifetime of the localized modes grow with the growth of the temperature and the heat conductivity monotonically decreases. The process of heat conduction in one-dimensional lattice with on-site potential is also studied by means of numerical simulation. Using the discrete Frenkel-Kontorova, φ4 and sinh-Gordon models we demonstrate that contrary to previously expressed opinions the sole anharmonicity of the on-site potential is insufficient to ensure the normal heat conductivity in these systems. The character of the energy transfer is determined by the spectrum of nonlinear excitations peculiar for every given model and therefore depends on the concrete potential shape and a temperature of the lattice. The reason is that the peculiarities of the nonlinear excitations and their interactions prescribe the energy scattering mechanism in each model. For sine-Gordon and φ4 models, linear waves (phonons) are scattered at a dynamical lattice of topological solitons; for sinh-Gordon and for φ4 in a different parameter regime the phonons are scattered at localized high-frequency breathers (in the case of φ4 the scattering mechanism switches with the growth of the temperature).

Key words

Nonlinear normal modes breathers heat transfer 


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  1. [1]
    E. Fermi, J. Pasta, and S. Ulam, Los Alamos Report No. LA-1940, 1955.Google Scholar
  2. [2]
    R.E. Peierls, Quantum theory of solids, Oxford University Press, London 1955.Google Scholar
  3. [3]
    J.G. Kirkwood, J Chem. Phys. 7(7) 506, 1939.MathSciNetCrossRefGoogle Scholar
  4. [4]
    K. Pitzer, J Chem. Phys. 8(8) 711, 1940.CrossRefGoogle Scholar
  5. [5]
    L.I. Manevitch, and A.V. Savin, Phys. Rev. E 55 4713, 1997.CrossRefGoogle Scholar
  6. [6]
    A.V. Savin, and L.I. Manevitch, Phys. Rev. B 58(17) 11386, 1998.CrossRefGoogle Scholar
  7. [7]
    L.I. Manevitch, Nonlinear Dynamics 25 95–109, 2001.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    L.I. Manevitch, Polymer Science C 4(2) 117–181, 2001.Google Scholar
  9. [9]
    O.V. Gendelman, and A.V. Savin, Phys. Rev. Lett. 84 2381, 2000.CrossRefGoogle Scholar
  10. [10]
    A.V. Savin, and O.V. Gendelman, Phys. Rev. E 67 041205, 2003.Google Scholar
  11. [11]
    B. Hu, B. Li, and H. Zhao, Phys. Rev. E 57 2992, 1998.CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Leonid I. Manevitch
    • 1
  • Oleg V. Gendelman
    • 1
  • Alexander V. Savin
    • 1
  1. 1.N. N. Semenov Institute of Chemical PhysicsRussian Academy of SciencesMoscowRussia

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