Topological Mechanics and Nonlinearity

Due to the recent discovery of topological insulators in condensed matter physics, a new notion of topology has emerged in association with the intrinsic wave dispersion of a structure. It has led to a plethora of mechanical designs with nontrivial and robust localization of energy -- potentially offering novel applications in energy harvesting, vibration isolation, and structure health monitoring.
The framework of topology fundamentally rests on the linear dynamics of the system. In this presentation, we will talk about our recent efforts to understand the interplay of nonlinearity and topology in mechanical systems. Our system obeys the dynamics that is governed by a second-order differential equation akin to electronic circuits. In particular, we study one-dimensional nonlinear lattices of both Fermi–Pasta–Ulam–Tsingou and Klein-Gordon types and discuss the amplitude-dependent topological transition, soliton formation, linear stability, and nonlinear Dirac physics. The findings highlight the effect of nonlinearity on the characteristics of topologically-robust edge states and the role of topology in interpreting purely nonlinear states.