Restricted normal mode analysis and chaotic response of p-mode intrinsic localized mode

摘要

Nonlinearities in arrays of discrete oscillators can cause energy localizations called intrinsic localized modes (ILMs). Two spatial physical manifestations of localization within the array are the ST-mode ILM (a symmetric mode) and the P-mode ILM (an asymmetric mode). The free response and forced response of the asymmetric vibratory P-mode are discussed within this paper, with particular attention given to the linear coupling between the oscillators. The free, undamped response is studied by means of the restricted normal mode approach, which gives the amplitude profile of the P-mode ILM. The initial P-mode ILM amplitude profile, as calculated from the restricted normal mode approach, is capable of creating both forced and unforced persistent ILMs. For the unforced and undamped free response, the P-mode ILM is stable with and without linear coupling, and its response frequency is incommensurate with the response of the surrounding oscillators. It was found that the frequency of oscillation of this mode has a strong relationship to the amplitude of vibration. Moreover, the amplitude profiles generated by the restricted normal mode approach are used to generate initial conditions for the forced, damped case. The existence of the damped and forced P-mode ILM is strongly dependent upon the linear coupling term. It is shown that if linear coupling does not exist, then this mode is persistent but chaotic. To the author's knowledge, this is the first pinned and persistent chaotic ILM to be observed, which is not predicted from local analysis.