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

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.

💡 Research Summary

In this work the authors investigate a continuum Hatano‑Nelson model in which the non‑Hermitian (non‑reciprocal) term is modulated by the local wave amplitude through a saturating cubic‑quintic function
(F(|\psi|^{2})=\gamma + a|\psi|^{2} - b|\psi|^{4}) (with (a,b>0)). The stationary nonlinear Schrödinger equation (\hat H(\psi)\psi = E\psi) is rewritten as a two‑dimensional dynamical system by treating the spatial coordinate (x) as a time‑like evolution parameter:
(\partial_{x}\psi = v,\qquad \partial_{x}v = 2F(\psi^{2})v - 2E\psi.)
The origin ((\psi,v)=(0,0)) is the only fixed point; its linear stability is governed by the eigenvalues (\lambda = \gamma \pm \sqrt{\gamma^{2}-2E}). Consequently the origin is attracting for (\gamma<0) (skin‑localized states) and repelling for (\gamma>0) (extended states).

The central result is a global bifurcation scenario involving two distinct codimension‑one bifurcations as the control parameter (\gamma) is varied. First, at (\gamma=0) the origin undergoes a subcritical Hopf bifurcation: an unstable small‑amplitude limit cycle exists for (\gamma\to0^{-}) and collapses into the origin at (\gamma=0). Second, at a negative value (\gamma=\gamma_{c}<0) a saddle‑node of limit cycles (SNLC) occurs, where a pair of periodic orbits (one stable, one unstable) are created and annihilate each other as (\gamma) passes through (\gamma_{c}). The combination of these two bifurcations yields three dynamical regimes:

1. Skin‑only regime ((\gamma<\gamma_{c})) – the origin is the unique attractor, all trajectories spiral into it, and the physical wave decays exponentially, reproducing the linear Hatano‑Nelson skin effect.
2. Extended‑only regime ((\gamma>0)) – the origin is unstable, bounded trajectories are forced onto a stable limit cycle, and the wave approaches a non‑vanishing oscillatory profile (growth with saturation).
3. Coexistence regime ((\gamma_{c}<\gamma<0)) – both the origin (skin) and the outer stable limit cycle (extended) are simultaneously stable. An inner unstable limit cycle forms a separatrix that partitions the phase‑space basins of attraction. Initial slopes (s=\partial_{x}\psi(0)) on opposite sides of this separatrix evolve to completely different long‑range states, despite identical system parameters and energy (E).

To obtain quantitative predictions, the authors apply a averaging method. By scaling the velocity variable with the natural frequency (\omega=\sqrt{2E}) and separating fast oscillations from slow amplitude dynamics, they derive a one‑dimensional amplitude equation for the envelope (A(x)). This reduced equation yields closed‑form expressions for the amplitudes of the inner (unstable) and outer (stable) limit cycles and predicts the SNLC threshold (\gamma_{c}). Comparison with numerical continuation shows excellent agreement, confirming that the saturating nonlinearity shifts the SNLC to a negative (\gamma) and determines the size of the coexistence window.

The paper further introduces a basin‑fraction order parameter
(\phi(\gamma)=\frac{V_{\text{ext}}}{V_{\text{ext}}+V_{\text{skin}}}),
where (V_{\text{ext}}) and (V_{\text{skin}}) are the measures (in the space of initial slopes) of the basins of the extended and skin attractors, respectively. As (\gamma) crosses (\gamma_{c}), (\phi) exhibits a discontinuous jump from 0 to 1, analogous to a first‑order phase transition. This provides a global, geometry‑based characterization of the bistability that is independent of any spectral mobility‑edge picture.

Physical consequences of the basin geometry are explored. Near the separatrix, trajectories linger for long distances before committing to either attractor, leading to long‑lived spatial transients. When (\gamma) is cycled across the coexistence window, the system displays hysteresis: the final state depends on the direction of parameter change because the basin boundaries shift only at the SNLC. These phenomena have no counterpart in linear non‑Hermitian systems, where coexistence of localized and extended modes is tied to spectral features such as mobility edges.

Overall, the study demonstrates that (i) a saturating nonlinear non‑reciprocity can generate a subcritical Hopf together with an SNLC, producing a finite bistable window; (ii) the global attractor‑basin structure in phase space, rather than the spectrum, governs the coexistence of skin and extended states; and (iii) analytical averaging provides accurate predictions for limit‑cycle branches and bifurcation thresholds. By highlighting the importance of global basin geometry, the work offers a complementary perspective to the prevailing spectral approaches in non‑Hermitian physics and opens avenues for experimental observation of hysteresis and long‑lived transients in photonic, mechanical, circuit, or cold‑atom platforms that realize nonlinear non‑reciprocal couplings.