3D phase space geometry of wave-defect interaction in bistable mechanical lattices in the presence of radiation damping

The fate of a solitary wave propagating in a bistable mechanical lattice can be tailored using a local inhomogeneity (`defect'). The defect gives rise to a localized oscillatory mode (`breather'). The resulting wave-breather dynamics were studied in our prior work, where we derived a 2 degrees-of-freedom reduced-order Hamiltonian model, and analyzed the system using the methods of lobe dynamics. The lobe dynamics analysis showed that depending upon its initial speed, an incoming solitary wave can get transmitted, reflected, or temporarily captured upon interaction with the defect. In the current work, we modify the reduced-order model to include the loss of energy in the two localized degrees-of-freedom to the dispersive modes (`phonons'), a phenomenon known as `radiation damping'. We employ the framework of (partial) Lagrangian Coherent Structures (LCS) in the resulting 3D non-Hamiltonian system to compute 2D transport barriers in the phase space. We demonstrate that these 2D LCS can be used to perform lobe dynamics computations. These computations give a more accurate delineation of initial conditions that lead to the different fates for the incoming wave, and provide an insight into wave control strategies. This research represents a step forward in establishing a systematic approach to defect engineering for manipulating nonlinear waves in mechanical metamaterials.