Intrinsic localized modes of 1/2-order subharmonic oscillations in nonlinear oscillator arrays

Abstract1/2-order subharmonic oscillations in an array with $$N$$N nonlinear, identical oscillators are theoretically investigated. Each oscillator is connected by weak, linear springs and subjected to sinusoidal excitation. Van der Pol’s method is employed to determine the frequency response curves for 1/2-order subharmonic oscillations when $$N=2$$N=2 and 3. Frequency response curves for hard- and soft-type nonlinearities of the oscillators are examined and compared with each other. All patterns of oscillations are classified depending on the results of the response curves, and it is determined in which pattern intrinsic localized modes (ILMs) appear. When disturbances are input in order to observe ILMs, basins of attraction demonstrate which pattern of ILMs may occur. Bifurcation sets are also calculated to examine the influence of the connecting spring constants on the response curves and the occurrence of Hopf bifurcations and amplitude-modulated motions (AMMs), including chaotic vibrations. Furthermore, the influence of oscillator imperfections on the response curves is investigated by slightly changing the value of the spring constant. The numerical simulations for $$N=2$$N=2 and 3 confirm the validity of the frequency response curves and the occurrence of AMMs. ILMs were also observed in the numerical simulations conducted for $$N=10$$N=10.

[1]  鈴木 増雄.  A. H. Nayfeh and D. T. Mook: Nonlinear Oscillations, John Wiley, New York and Chichester, 1979, xiv+704ページ, 23.5×16.5cm, 10,150円. , 1980 .

[2]  B. Hubbard,et al.  Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array. , 2003, Physical review letters.

[3]  Balakumar Balachandran,et al.  Intrinsic localized modes in microresonator arrays and their relationship to nonlinear vibration modes , 2008 .

[4]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[5]  Cetin Cetinkaya,et al.  Mode Localization in a Class of Multidegree-of-Freedom Nonlinear Systems with Cyclic Symmetry , 1993, SIAM J. Appl. Math..

[6]  Alexander F. Vakakis,et al.  A Multiple-Scales Analysis of Nonlinear, Localized Modes in a Cyclic Periodic System , 1993 .

[7]  Yuri S. Kivshar,et al.  Multiple states of intrinsic localized modes , 1998 .

[8]  T. Ikeda,et al.  Intrinsic Localized Modes of Harmonic Oscillations in Nonlinear Oscillator Arrays , 2013 .

[9]  C. Hayashi,et al.  Nonlinear oscillations in physical systems , 1987 .

[10]  Takashi Ikeda,et al.  Nonlinear Parametric Vibrations of an Elastic Structure with a Rectangular Liquid Tank , 2003 .

[11]  Takashi Hikihara,et al.  Stability change of intrinsic localized mode in finite nonlinear coupled oscillators , 2008 .

[12]  R. Brent.  Table errata: Algorithms for minimization without derivatives (Prentice-Hall, Englewood Cliffs, N. J., 1973) , 1975 .

[13]  Alexander F. Vakakis,et al.  A very complicated structure of resonances in a nonlinear system with cyclic symmetry: Nonlinear forced localization , 1995, Nonlinear Dynamics.

[14]  A. Sievers,et al.  Intrinsic localized modes in anharmonic crystals. , 1988, Physical review letters.

[15]  C. R. Willis,et al.  Discrete Breathers , 1997 .