Strong nonlinear mixing evolutions within phononic frequency combs

Abstract

Phononic frequency combs (PFCs) represent an emerging attractive nonlinear vibrational phenomenon characterized by equidistant spectral lines. Despite the extensive experimental studies, the complex nonlinear mixing nature of PFCs continues to present significant challenges in solving and investigating their complete dynamics, which is difficult to achieve by existing computational approaches. In this paper, the entire solution space within a representative PFC induced by a 1:2 internal resonance is elucidated by conducting continuation computations and numerical long-time integrations. The proposed continuation approach is achieved by integrating our developed semi-analytical residue-regulating homotopy method (RRHM) with a pseudo arc-length continuation technique. In this solution space, we unearth wide-range nonlinear evolutions including overlapping intervals between the periodic and quasi-periodic branches, abundant multivalued sub-intervals, cyclic-fold (CF) bifurcations, and torus-doubling (TD) routes to chaos. In addition, multiple coexistences of a chaotic attractor and a periodic orbit, a chaotic attractor and a quasi-periodic orbit, as well as a periodic orbit and three quasi-periodic orbits are identified. Furthermore, we meticulously dissect and distinguish non-smooth variations in PFC morphology, which manifest as multiple jumps in comb spacing as the excitation frequency is swept across. This study could serve as a general guide for a comprehensive exploration of PFC dynamics and can offer insights to inform and inspire related experimental studies.