Topological edge states and amplitude-dependent delocalization in quasiperiodic elliptically geared lattices

Abstract: We present a class of mechanical lattices based on elliptical gears with quasiperiodic modulation and geometric nonlinearity, capable of exhibiting topologically protected modes and amplitude-driven transitions. Starting from a one-dimensional chain of modulated elliptical gears, we demonstrate the emergence of localized edge states arising from quasiperiodic variation in the gears' moments of inertia, analogous to the topological edge modes of the Aubry-Andre-Harper model. Under increasing excitation amplitude, the system undergoes a nonlinear transition, where edge localization breaks down and energy delocalizes into the bulk. By coupling multiple such chains with varying modulation phase, we construct a two-dimensional lattice in which the phase acts as a synthetic dimension. This structure supports topological wave propagation along the synthetic dimension. Nonlinearity again induces a breakdown of topological states, leading to complex, amplitude-dependent wave propagation. We further propose a numerical continuation approach to analyzing the periodic orbits and their linear stability, effectively discovering the boundary of the basin of bounded motion and detecting the occurrence of delocalization under certain excitation amplitudes. Our results reveal that elliptical geared systems offer a passive, amplitude-dependent platform for exploring topological phenomena and synthetic dimensionality in mechanical metamaterials.