Nonlinear normal modes and localization in two bubble oscillators.

We investigated a bifurcation structure of coupled nonlinear oscillation of two spherical gas bubbles subject to a stationary sound field by means of nonlinear modal analysis. The goal of this paper is to describe an energy localization phenomenon of coupled two-bubble oscillators, resulting from symmetry-breaking bifurcation of the steady-state oscillation. Approximate asymptotic solutions of nonlinear normal modes (NNMs) and steady state oscillation are obtained based on the method of multiple scales. It is found that localized oscillation arises in a neighborhood of the localized normal modes. The analytical solutions of the amplitude and the phase shift of the steady-state oscillation are compared to numerical results and found to be in good agreement within the limit of small-amplitude oscillation. For larger amplitude oscillation, a bifurcation diagram of the localized solution as a function of the driving frequency and the separation distance between the bubbles is provided in the presence of the thermal damping. The numerical results show that the localized oscillation can occur for a fairly typical parameter range used in practical experiments and simulations in the early literatures.