Topological Protection in a Strongly Nonlinear Interface Lattice.

We study a one-dimensional mechanical analog of the Shu-Schrieffer-Heeger interface model with strong nonlinearity of the cubic form. Two nonlinear half-lattices make a topological interface system with a nonlinear coupling added to the stiff spring, while linear grounding springs are added on all oscillators. The frequency-energy dependence of the nonlinear bulk modes and topologically insulated mode is explored using Numerical continuation of the system's nonlinear normal modes (NNMs), i.e., of standing waves. Moreover, the linear stability of the NNMs are investigated using Floquet Multipliers (FMs) and Krein signature analysis. We find that the nonlinear topological lattice supports a family of topologically insulated NNMs that are parameterized by the total energy of the system and are stable within a range of frequencies. Using numerical simulations, we empirically recover the geometric Zak phase to determine at which energies the nontrivial phase conditions cease to exists. We compare the predictions from FM analysis and the numerical phase analysis by numerically investigating the interface system for harmonic excitation applied to the interface site. It is shown that empirical calculations of the geometric Zak Phase lead to reliable measures for predicting the existence of the topological mode in the nonlinear system, and that while the FMs of the NNMs are dependable predictors of the NNMs stability, they are inferior to the empirical phase calculations for predicting at which energies the lattice supports waves localized at the interface.