Lyapunov exponents explain disorder-induced polarization and soliton teleportation in a mechanical Markov system

Materials with structural disorder often exhibit mechanical properties that are difficult to predict due to the absence of periodicity. Here we develop a new framework using Lyapunov exponents to explain zero mode (ZM) localization in disordered one-dimensional (1D) mechanical systems constructed via the spatial analog of a Markov process, which we call "mechanical Markov systems". We show that, despite their strong disorder, these mechanical Markov systems exhibit robust zero mode (ZM) polarization predicted by their Lyapunov exponents. These ZMs become mobile solitons in the nonlinear regime, and display a set of new nonlinear dynamics features including reflectionless chirality-dependent teleportation, which can also be explained using Lyapunov exponents. Our results establish Markov formalism as a powerful tool to explain and design localization and dynamics in disordered mechanical systems, opening opportunities for programmable metamaterials with novel linear and nonlinear responses.