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Intrinsic Localized P-Mode in Forced Nonlinear Oscillator Array

  • Edmon PerkinsEmail author
  • Timothy Fitzgerald
Conference paper
Part of the Understanding Complex Systems book series (UCS)

Abstract

Intrinsic localized modes (ILMs) are energy localizations that may occur in arrays of discrete, nonlinear oscillators. When present in physical systems, these energy localizations may cause undesirable dynamics or damaging effects. If properly understood, ILMs may be used to increase the sensing capacity of inertial sensors, store information, or move energy through an array. Depending on the system parameters, ILMs may have a variety of profiles (e.g., the symmetric ST-mode or the antisymmetric P-mode). Using the method of restricted normal modes, a displacement profile is calculated for the P-mode. After performing numerical simulations using the P-mode profile as initial conditions, the P-mode is found to be persistent when forced at 3 times the linear natural frequency. Although persistent, this P-mode ILM is found to have chaotic properties. This ILM may have been previously overlooked because of its positive Lyapunov exponent, meaning that there might be larger ranges of parameters capable of supporting these energy localizations.

References

  1. 1.
    P.W. Anderson, Absence of diffusion in certain random lattices. Phys. Rev. 109(5), 1492 (1958)CrossRefGoogle Scholar
  2. 2.
    S. Flach, A.V. Gorbach, Discrete breathers–advances in theory and applications. Phys. Rep. 467(1–3), 1–116 (2008)CrossRefGoogle Scholar
  3. 3.
    M. Sato, A. Sievers, Direct observation of the discrete character of intrinsic localized modes in an antiferromagnet. Nature 432(7016), 486 (2004)CrossRefGoogle Scholar
  4. 4.
    A. Ustinov, Solitons in Josephson junctions. Phys. D: Nonlinear Phenom. 123(1–4), 315–329 (1998)CrossRefGoogle Scholar
  5. 5.
    A. Ustinov, Imaging of discrete breathers. Chaos: Interdiscip. J. Nonlinear Sci. 13(2), 716–724 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    P. Binder, D. Abraimov, A. Ustinov, S. Flach, Y. Zolotaryuk, Observation of breathers in Josephson ladders. Phys. Rev. Lett. 84(4), 745 (2000)CrossRefGoogle Scholar
  7. 7.
    J.W. Fleischer, M. Segev, N.K. Efremidis, D.N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature 422(6928), 147 (2003)CrossRefGoogle Scholar
  8. 8.
    S.F. Mingaleev, Y.S. Kivshar, R.A. Sammut, Long-range interaction and nonlinear localized modes in photonic crystal waveguides. Phys. Rev. E 62(4), 5777 (2000)CrossRefGoogle Scholar
  9. 9.
    S.F. Mingaleev, Y.B. Gaididei, P.L. Christiansen, Y.S. Kivshar, Nonlinearity-induced conformational instability and dynamics of biopolymers. EPL (Europhys. Lett.) 59(3), 403 (2002)CrossRefGoogle Scholar
  10. 10.
    M. Sato, B. Hubbard, L.Q. English, A. Sievers, B. Ilic, D. Czaplewski, H. Craighead, Study of intrinsic localized vibrational modes in micromechanical oscillator arrays. Chaos: Interdiscip. J. Nonlinear Sci. 13(2), 702–715 (2003)CrossRefGoogle Scholar
  11. 11.
    E. Perkins, M. Kimura, T. Hikihara, B. Balachandran, Effects of noise on symmetric intrinsic localized modes. Nonlinear Dyn. 85(1), 333–341 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    M. Kimura, T. Hikihara, Coupled cantilever array with tunable on-site nonlinearity and observation of localized oscillations. Phys. Lett. A 373(14), 1257–1260 (2009)CrossRefGoogle Scholar
  13. 13.
    A. Sievers, S. Takeno, Intrinsic localized modes in anharmonic crystals. Phys. Rev. Lett. 61(8), 970 (1988)CrossRefGoogle Scholar
  14. 14.
    J. Page, Asymptotic solutions for localized vibrational modes in strongly anharmonic periodic systems. Phys. Rev. B 41(11), 7835 (1990)CrossRefGoogle Scholar
  15. 15.
    A. Dick, B. Balachandran, C. Mote, Intrinsic localized modes in microresonator arrays and their relationship to nonlinear vibration modes. Nonlinear Dyn. 54(1–2), 13–29 (2008)CrossRefGoogle Scholar
  16. 16.
    S. Bickham, S. Kiselev, A. Sievers, Stationary and moving intrinsic localized modes in one-dimensional monatomic lattices with cubic and quartic anharmonicity. Phys. Rev. B 47(21), 14206 (1993)CrossRefGoogle Scholar
  17. 17.
    K. Sandusky, J. Page, Interrelation between the stability of extended normal modes and the existence of intrinsic localized modes in nonlinear lattices with realistic potentials. Phys. Rev. B 50(2), 866 (1994)CrossRefGoogle Scholar
  18. 18.
    M. Kimura, T. Hikihara, Stability change of intrinsic localized mode in finite nonlinear coupled oscillators. Phys. Lett. A 372(25), 4592–4595 (2008)CrossRefGoogle Scholar
  19. 19.
    B. Balachandran, E. Perkins, T. Fitzgerald, Response localization in micro-scale oscillator arrays: influence of cubic coupling nonlinearities. Int. J. Dyn. Control 3(2), 183–188 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    M.T. Rosenstein, J.J. Collins, C.J. De Luca, A practical method for calculating largest lyapunov exponents from small data sets. Phys. D: Nonlinear Phenom. 65(1–2), 117–134 (1993)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Auburn UniversityAuburnUSA
  2. 2.Gonzaga UniversitySpokaneUSA

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