Universal Topology of Exceptional Points in Nonlinear Non-Hermitian Systems

Exceptional points (EPs) are non-Hermitian degeneracies where eigenvalues and eigenvectors coalesce, giving rise to unusual physical effects across scientific disciplines. The concept of EPs has recently been extended to nonlinear physical systems. We theoretically demonstrate a universal topology in the nonlinear parameter space for a large class of physical systems that support $2^\mathrm{nd}$ order EPs in the linear regime. Knowledge of this topology (called elliptic umbilic singularity in bifurcation theory) deepens our understanding of $2^\mathrm{nd}$ order linear EPs, which here emerge as coalescence of 4 nonlinear eigenvectors. This helps guide future experimental discovery of nonlinear EPs and their classification, and helps envision and optimize technological applications of nonlinear EPs. Our theoretical approach is general and can be extended to nonlinear perturbations of $3^\mathrm{rd}$ and higher-order EPs.