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

1. Problem Statement

The paper investigates the interplay between non-Hermiticity and nonlinearity in the context of the Non-Hermitian Skin Effect (NHSE). While the linear NHSE is well-understood (where bulk eigenstates accumulate at boundaries due to non-reciprocity), the behavior of nonlinear extensions remains an open frontier.

Specifically, the authors address three critical questions regarding a nonlinear Hatano–Nelson model with saturating nonreciprocity:

1. How does the system behave if the amplitude-dependent nonreciprocity is non-monotonic and saturates (i.e., gain is suppressed at large amplitudes)?
2. What global dynamical bifurcations (beyond simple supercritical Hopf bifurcations) occur, and what stationary localization phenomena do they imply?
3. Can these models host bistability, where multiple stable stationary states (skin modes and extended states) coexist under identical parameters, leading to history-dependent outcomes?

2. Methodology

The authors employ a dynamical systems approach to analyze the stationary states of a continuum nonlinear Schrödinger equation.

• Model Formulation: They study a stationary nonlinear Schrödinger equation with a Hamiltonian $\hat{H}(\psi) = \frac{\hat{k}^2}{2} + i F(|\psi|^2)\hat{k}$, where the non-Hermitian term $F(|\psi|^2) = \gamma + a|\psi|^2 - b|\psi|^4$ includes a saturating nonlinearity ($a>0, b>0$).
• Phase-Space Flow: The stationary equation is reformulated as a planar dynamical system where the spatial coordinate $x$ acts as the evolution time. By defining $v = \partial_x \psi$, the system becomes:
  $$\partial_x \psi = v, \quad \partial_x v = 2F(\psi^2)v - 2E\psi$$
  Stationary states correspond to trajectories in the $(\psi, v)$ phase space.
• Attractor-Localization Correspondence:
  • Skin Modes: Trajectories attracted to the origin $(0,0)$ correspond to exponentially decaying wavefunctions.
  • Extended States: Trajectories attracted to a stable limit cycle correspond to bounded, oscillatory wavefunctions with non-vanishing asymptotic amplitude.
• Analytical Tools:
  • Numerical Continuation: Used to trace bifurcation diagrams and limit cycle branches.
  • Averaging Method: An analytical derivation of a one-dimensional amplitude equation to predict limit cycle amplitudes and bifurcation thresholds.
  • Basin Geometry Analysis: Introduction of a "basin-fraction order parameter" to quantify the probability of a system settling into a skin vs. extended state based on initial boundary conditions.

3. Key Contributions

The paper moves beyond local bifurcation analysis to establish a global bifurcation scenario governed by two distinct thresholds.

1. Discovery of a Subcritical Hopf Bifurcation: Unlike previous studies that found supercritical Hopf bifurcations, this model exhibits a subcritical Hopf bifurcation at $\gamma = 0$. Here, the origin loses stability, but an unstable small-amplitude limit cycle exists for $\gamma < 0$.
2. Identification of Saddle-Node of Limit Cycles (SNLC): A SNLC bifurcation occurs at a critical value $\gamma_c < 0$. At this point, a pair of limit cycles (one stable, one unstable) is created/annihilated.
3. Bistability Window: These two bifurcations create a finite coexistence window ($\gamma_c < \gamma < 0$) where both the stable origin (skin mode) and a stable outer limit cycle (extended state) are attractors.
4. Basin Separatrix Mechanism: The coexistence is separated not by a spectral mobility edge (as in linear disordered systems) but by a nonlinear basin separatrix organized by the unstable inner limit cycle.
5. Analytical Predictions: The authors derived a closed-form averaged amplitude equation that accurately predicts the SNLC threshold ($\gamma_c = -a^2/8b$) and the amplitudes of the limit cycles.

4. Key Results

• Global Bifurcation Diagram:
  • $\gamma < \gamma_c$: Only the origin is stable (Skin-only regime).
  • $\gamma_c < \gamma < 0$: Bistability. The origin and the outer limit cycle are both stable. The system's final state depends on the initial boundary slope $s = \partial_x \psi(0)$.
  • $\gamma > 0$: The origin is unstable; only the stable limit cycle exists (Extended-only regime).
• Basin-Fraction Order Parameter: The authors define $p_{skin}(\gamma)$ as the fraction of initial slopes leading to the skin state.
  • At the SNLC point ($\gamma_c$), $p_{skin}$ exhibits a first-order-like jump because a finite measure of initial conditions is immediately captured by the newly created extended attractor.
  • As $\gamma \to 0^-$, the basin of the skin state shrinks continuously until the origin becomes unstable.
• Physical Phenomena in the Coexistence Window:
  • Long-lived Spatial Transients: Trajectories initialized near the separatrix (unstable inner cycle) exhibit long-lived oscillatory transients that mimic an extended state before eventually decaying to the skin state (or vice versa). This is distinct from standard exponential skin decay.
  • Hysteresis: Under adiabatic parameter sweeps, the system exhibits memory.
    • Decreasing $\gamma$ from positive values: The system stays extended until $\gamma_c$, then jumps to skin.
    • Increasing $\gamma$ from negative values: The system stays skin until $\gamma = 0$, then jumps to extended.
    • This creates a hysteresis loop between $\gamma_c$ and $0$.

5. Significance

• Beyond Linear Spectral Theory: The work demonstrates that in nonlinear non-Hermitian systems, stationary states cannot be fully understood via linear spectral concepts (like mobility edges). Instead, global attractor-basin geometry is the governing principle.
• New Mechanism for Coexistence: It establishes a mechanism for the coexistence of localized and extended states that is purely dynamical and nonlinear, distinct from Anderson localization or linear spectral transitions.
• Control and Memory: The identification of hysteresis and separatrix-induced transients suggests new ways to control non-Hermitian systems, potentially enabling memory devices or switches based on the history of parameter tuning.
• Theoretical Framework: The derivation of the averaged amplitude equation provides a powerful, simplified analytical tool for predicting complex bifurcation structures in nonlinear non-Hermitian models, bridging the gap between numerical simulation and analytical theory.

In summary, this paper fundamentally shifts the perspective on nonlinear non-Hermitian skin effects from a spectral problem to a global dynamical systems problem, revealing rich phenomena like bistability, hysteresis, and complex transient behaviors driven by the geometry of phase-space basins.