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

Using a mapping between spatial disorder and temporal stochasticity, we develop a new framework using Lyapunov exponents to explain exotic wave localization and mobility phenomena in disordered one-dimensional (1D) mechanical systems that can be constructed via a spatial analog of a Markov process, which we call "mechanical Markov systems." We show that disorder induces robust polarization of zero modes (ZMs) in these mechanical Markov systems, and this phenomenon is explained using Lyapunov exponents. Remarkably, these ZMs become mobile solitons in the nonlinear regime despite the disorder-controlled localization of all other modes, and display a set of new nonlinear dynamics features including reflectionless chirality-dependent jumping, which can also be explained using Lyapunov exponents. Our results establish the 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.