  Mechanical topological insulators are well understood for linear and weakly
nonlinear systems, however traditional analysis methods break down for strongly
nonlinear systems since linear methods can not be applied in that case. We
study one such system in the form of a one-dimensional mechanical analog of the
Su-Schrieffer-Heeger interface model with strong nonlinearity of the cubic
form. 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), and 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. Next, it is shown that
empirical calculations of the geometric Zak Phase can define an energy
threshold to predict the excitability of the nonlinear topological mode, and
that this threshold coincides with the energy that the topological NNM
intersects the linear bulk-spectrum. These predictions are validated with
numerical simulations of the nonlinear topological system. These results are
also tested for parametric perturbations which preserve and break chirality in
the system. Thus, we provide a new method for analyzing and predicting the
existence of topologically insulated modes in a strongly nonlinear lattice
based on the physical observable of band topology.
