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

TL;DR

This paper investigates a nonlinear non-Hermitian model revealing complex bifurcation scenarios and basin structures that determine the coexistence of skin and extended states, emphasizing the importance of basin geometry over spectral analysis.

Contribution

It introduces a global bifurcation framework and basin analysis to understand stationary states in nonlinear non-Hermitian systems, extending beyond traditional spectral concepts.

Findings

Identification of a subcritical Hopf bifurcation and saddle-node of limit cycles controlling state coexistence.

Development of a basin-fraction order parameter showing a first-order-like transition.

Discovery of long-lived transients and hysteresis phenomena within the coexistence window.

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.