Global bifurcations and basin geometry of the nonlinear non-Hermitian skin effect

Abstract: We study a continuum Hatano--Nelson model with a saturating nonlinear nonreciprocity and analyze its stationary states via the associated phase-space flow. We uncover a global scenario controlled by a subcritical Hopf bifurcation and a saddle-node of limit cycles, which together generate a finite coexistence window. In this window, skin modes and extended states are both stable at a fixed energy $E$, separated by a nonlinear basin separatrix in phase space rather than a spectral (mobility-edge) mechanism in a linear system. An averaged amplitude equation yields closed-form predictions for the limit-cycle branches and the SNLC threshold. Building on the basin geometry, we introduce a basin-fraction order parameter that exhibits a first-order-like jump at SNLC. Intriguing physical phenomena in the coexistence window are also revealed, such as separatrix-induced long-lived spatial transients and hysteresis. Overall, our findings highlight that, beyond linear spectral concepts, global attractor-basin geometry provides a powerful and complementary lens for understanding stationary states in nonlinear non-Hermitian systems.