Multi-scale analysis of highly nonlinear regimes emerging in the inertially coupled locally resonant structure

In this talk I'll present the asymptotical study of resonant, planar dynamics of the unit-cell model comprising the outer mass element subject to a two-dimensional, nonlinear local potential and incorporating internal rotator. Current study focuses on the analysis of the two asymptotic limits, namely the limit of high and low amplitude motion of the external element. I'll show the two distinct singular and regular multi-scale approaches unveiling the rather intriguing bifurcations undergone by the system regimes for both considered limits, respectively. The basic question of possible coexistence of various stationary and non-stationary regimes as well as their local and global bifurcations is addressed via the reduction of the global flow on the slow invariant manifold in the vicinity of the fundamental resonance. In the limit of low amplitude oscillations we prove the complete integrability of the reduced averaged flow enabling the analytical description of the intriguing nonlinear phenomenon of unidirectional energy transport.