Universal topology of exceptional points in nonlinear non-Hermitian systems

Exceptional points are non-Hermitian degeneracies where eigenvalues and eigenvectors coalesce, giving rise to unusual physical effects across scientific disciplines. The concept of exceptional points 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 second-order exceptional points in the linear regime. Knowledge of this topology (called elliptic umbilic singularity in bifurcation theory) deepens our understanding of second-order linear exceptional points, which here emerge as coalescence of four nonlinear eigenvectors. This helps guide future experimental discovery of nonlinear exceptional points and their classification, establish rigorous bounds of sensitivity enhancement of exceptional points in nonlinear systems, and helps envision and optimize technological applications of nonlinear exceptional points. Our theoretical approach is general and can be extended to nonlinear perturbations of third-order and higher-order exceptional points.