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

TL;DR

This work studies a continuum nonlinear Hatano–Nelson model with saturating amplitude-dependent nonreciprocity and analyzes stationary states through a planar phase-space flow. It reveals a global bifurcation scenario governed by a subcritical Hopf bifurcation at $\gamma=0$ and a saddle-node of limit cycles (SNLC) at $\gamma_c<0$, creating a finite coexistence window $\gamma_c<\gamma<0$ where skin and extended states coexist at fixed energy $E$, separated by a nonlinear basin separatrix. An averaged amplitude equation $\partial_x r = h(r)$ yields closed-form predictions for inner/outer limit-cycle amplitudes and the SNLC threshold $\gamma_c^{\rm(th)} = -\frac{a^2}{8b}$, and its predictions agree with numerical bifurcation data. Building on the basin geometry, a basin-fraction order parameter $p_{\rm skin}(\gamma;\mu)$ shows a first-order-like jump at SNLC, and the coexistence window hosts separatrix-induced long-lived transients and hysteresis under adiabatic ramps, highlighting the central role of global attractor–basin geometry beyond linear spectral concepts in nonlinear non-Hermitian systems.

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.

Figures (4)

• Figure 1: Global bifurcation diagram of the phase-space flow Eq. \ref{['eq:phase_flow']}, plotted in terms of the limit-cycle amplitude $A$ versus $\gamma$. Solid (dashed) curves denote stable (unstable) invariant sets. The shaded interval $\gamma_c<\gamma<0$ indicates bistability, where the fixed point (skin) and the limit cycle (extended) coexist. Parameters are $a=1/2$, $b=1/32$, and $E=8$.
• Figure 2: Representative phase-space flow portraits and wave function profiles across the three regimes in Fig. \ref{['fig:bif_diag']}. Phase-space flows are shown in $(\psi,\partial_x\psi)$ with the vertical axis normalized by the natural frequency $\sqrt{2E}$. Skin regime$\;(\gamma<\gamma_c)$: (a) flow at $\gamma=-1.2$; (b) a typical skin mode with $\partial_x \psi(0)=6$. Coexistence regime$\;(\gamma_c<\gamma<0)$: (c) flow at $\gamma=-0.5$, showing a stable outer limit cycle (red) and an unstable inner limit cycle (green); (d-g) representative profiles with $\partial_x \psi(0)=7$ (skin), $\partial_x \psi(0)=8.69755$ (near-separatrix trajectory returning to the skin attractor), $\partial_x \psi(0)=8.69756$ (near-separatrix trajectory captured by the extended attractor), and $\partial_x \psi(0)=10$ (extended), respectively. Extended regime$\;(\gamma>0)$: (h) flow at $\gamma=0.2$, featuring a stable limit cycle (red); (i) a representative extended state with $\partial_x\psi(0)=2$. For (e-g) and (i), guide lines (green/red) indicate the inner/outer cycle amplitudes predicted in Sec. \ref{['sec:ave_eqn']}. For all subfigures, parameters are $a=1/2$, $b=1/32$, $E=8$.
• Figure 3: Comparison between the limit-cycle amplitudes predicted by the averaging theory and those obtained from numerical continuation of the phase-space flow dynamics. Inset: relative deviation $(A^{\rm (num)}-A^{\rm (th)})/A^{\rm (th)}$ for the outer and inner branches. Parameters are $a=1/2$, $b=1/32$, $E=8$, and the theoretical prediction of $\gamma_c$ is $\gamma_c^{\rm (th)} = -a^2/8b = -1$.
• Figure 4: Basin-geometry characterization of the phase diagram. (a) Threshold $s_{*}(\gamma)$ separating boundary slope ($s=\partial_x\psi(0)$) that flow to the origin from those attracted to the outer limit cycle. In the coexistence window $\gamma_c<\gamma<0$ (shaded), the separatrix is the unstable inner limit cycle, whose projection onto the boundary-slope axis yields $s_{*}(\gamma)$. Outside this window the attractor is unique and no separatrix exists. (b) Basin-fraction order parameter $p_{\mathrm{skin}}(\gamma)$. A first-order-like jump occurs at SNLC. Parameters are $a=1/2$, $b=1/32$ and $E=8$.

Theorems & Definitions (14)

• Lemma B.1: Lyapunov function kuznetsov1998elements
• proof
• Lemma B.2: Forward boundedness
• proof
• Proposition B.3: Only stable origin for sufficiently negative $\gamma$
• proof
• Proposition B.4: Hopf bifurcation at $\gamma=0$
• proof
• Proposition B.5: Subcritical Hopf for $a>0$
• proof