Title: Topological Boundary Modes in Nonlinear Dynamics with Chiral Symmetry

Abstract: Particle-hole symmetry and chiral symmetry play a pivotal role in multiple areas of physics, yet they remain unstudied in systems with nonlinear interactions whose nonlinear normal modes do not exhibit $\textbf{U}(1)$-gauge symmetry. In this work, we establish particle-hole symmetry and chiral symmetry in such systems. Chiral symmetry ensures the quantization of the Berry phase of nonlinear normal modes and categorizes the topological phases of nonlinear dynamics. We show topologically protected static boundary modes in chiral-symmetric nonlinear systems. Our theoretical framework extends particle-hole and chiral symmetries to nonlinear dynamics, whose nonlinear modes do not necessarily yield $\textbf{U}(1)$-gauge symmetry.