Cross-frequency couplings in non-sinusoidal dynamics of interacting oscillators: Acoustic estimation of the radial position and spatial stability of nonlinear oscillating bubbles.

In this work, the analysis of cross-frequency couplings (CFC) is introduced in the context of nonlinear acoustics related to the dynamics of bubble(s)-resonator systems. The results obtained from experiments specifically designed to untangle the causal connection between the CFC patterns observed at the signal level and the underlying physical processes, are discussed. It was found that "causal" amplitude-to-amplitude (AAC) and amplitude-to-phase (APC) couplings emerge in the system dynamics as a consequence of the bubble(s)-resonator mechanistic interaction in the oscillatory steady-state. In these CFC patterns, the amplitude of the fundamental frequency component (f) effectively modulates the amplitude and relative phase of the harmonic components (Nf). Moreover, these AAC and APC couplings give rise to "epiphenomenal" phase-to-amplitude (PAC) and phase-to-phase (PPC) couplings, in which the link between modulating and modulated parameters represents a correlation rather than a causal connection. It is shown that these CFC patterns can be exploited to determine the presence, spatial stability and radial position of nonlinear oscillating bubble(s) trapped within the acoustic chamber. Potential applications of the proposed techniques are also discussed. Substantial evidence is presented showing that CFC patterns emerging from quasi-periodic non-sinusoidal waveforms are informative on the interaction between underlying oscillators.