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

Abstract

1/2-order subharmonic oscillations in an array with \(N\) 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 \(N=2\) 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 \(N=2\) 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 \(N=10\).