Nonlinear Exceptional Points

• Nonlinear Exceptional Points are spectral singularities where multiple nonlinear eigenstates coalesce, organizing bifurcation and stability transitions.
• They are mathematically framed via nonlinear eigenproblems and catastrophe theory, revealing universal topological structures in parameter space.
• NEPs enable practical applications such as non-reciprocal dynamics, enhanced sensing, and robust topological invariants for improved system performance.

Nonlinear Exceptional Points (NEPs) are spectral singularities in nonlinear dynamical or wave systems where multiple nonlinear eigenvalues and eigenstates coalesce, generalizing the linear non-Hermitian concept of exceptional points (EPs) to nonlinear, stochastic, and even Hermitian nonlinear settings. NEPs organize bifurcation and stability transitions, encode universal topologies in parameter space, underpin critical phenomena in nonlinear wave and matter systems, and enable unique dynamical and topological responses including enhanced responsivity, non-reciprocal behavior, and noise resilience.

1. Mathematical Formulation and Universal Topology

In a nonlinear eigenproblem, the central mathematical object is an operator (or matrix) $H$ that depends nonlinearly on the state (or on the eigenvalue), e.g.,

$$H(x) x = E x,\quad \text{with} \quad H \text{ nonlinear in } x$$

or more generally,

$F(\omega, k) | \psi(\omega, k) \rangle = 0$

where $F$ is a matrix-valued function of eigenvalue $\omega$ and parameter $k$, and may be nonlinear in $\omega$. In contrast to linear systems where EPs are isolated degeneracies (e.g., $\det[H(k) - \omega I] = 0$ with a repeated root and defective eigenvectors), nonlinearities unfold these singularities into structurally richer sets.

The universal topology of NEPs in the parameter space is described by catastrophe theory. When a small nonlinear perturbation ($\alpha$) is added to a linear $2 \times 2$ EP, the EP becomes the organizing center of the elliptic umbilic (D$_{-4}$) singularity (Kwong et al., 26 Feb 2025). Near this singularity, the Lyapunov potential has the canonical form

$$V_{EU}(r, \phi, \frac{\alpha}{\gamma}, \frac{\beta'}{\gamma}, \frac{\delta}{\gamma}) = -\left[ \frac{\phi^3}{3} - \phi r^2 + \frac{\alpha}{\gamma} r^2 - 2 \frac{\beta'}{\gamma} \phi - 2 \frac{\delta}{\gamma} r \right]$$

Here, as parameters are varied, regions with multiple (e.g., four) nonlinear eigenstates emerge, and the locus of NEPs forms cones with three-cusped deltoid cross-sections. At cusps, three eigenstates coalesce (NEP3), and at the apex, a fourth-order coalescence occurs.

Key Features:

Regime	Number of Nonlinear Eigenstates	Singularity Type
Inside cone (deltoid section)	4	Fold/cusp (NEP3)
Outside cone	2	Standard (unfolded) region
Apex of cone	4	Higher-order NEP

This topology is universal for systems supporting second-order linear EPs and generalizes to higher dimension via higher-codimension catastrophes.

2. NEPs in Nonlinear Wave, Quantum, and Lattice Systems

NEPs arise naturally in nonlinear generalizations of canonical non-Hermitian models, including light-matter systems, condensates, and coupled resonators:

• Non-Hermitian Dimers with Saturable Nonlinearities (Gu et al., 24 Feb 2024): Inclusion of mode-dependent saturable nonlinearities leads to additional nonlinear eigenvalues and makes the EP location tunable. For a two-mode system, the EP detuning shifts according to

$\delta_{\text{EP}} = -\frac{g_1 - g_2}{3}$

where $g_1, g_2$ are nonlinear coefficients in the two resonators. The NEP is identified when both eigenvalues and eigenvectors (found from a high-order polynomial in the population imbalance) coalesce.

• Dissipative Bose-Hubbard and Cavity Arrays (Lyubarov et al., 2018): Nonlinear dissipative photon-photon interactions induce formation and mixing of two-photon bound states (doublons) and scattering states. The NEP marks the momentum threshold $k_c$ where

$\cos k = U/(2J)$

and pair (doublon) and continuum states coalesce.

• Optical Waveguides with Kerr Nonlinearity (Dey et al., 2019): In three-mode non-Hermitian waveguides with local Kerr nonlinearity, encirclement of multiple EPs leads to anomalous mode collapsing: all inputs collapse to a single output mode, and the outcome (which branch) can be selected by the sign of the nonlinearity (focusing or defocusing). Non-chiral output emerges in contrast to the linear (chiral) EP regime.
• Nonlinear Lasing Systems (Benzaouia et al., 2022): In coupled lasers above threshold, NEPs are the points where distinct nonlinear lasing solutions (PT-symmetric and PT-broken) merge. At these points, lasing output and frequency display kinks, and scattering experiments reveal quartic divergences in the response spectrum. Such NEP lasers are stabilized if the inversion relaxation rate is large enough.

3. NEPs, Bifurcation Phenomena, and Generalized Dynamics

In nonlinear dynamical systems—including population models, limit-cycle oscillators, and neural/chaotic networks—NEPs generalize to generalized exceptional points (GEPs):

• Coalescence of Covariant Lyapunov Vectors (CLVs) (Weis et al., 2022):
  • Continuous: CLVs approach each other smoothly at the bifurcation.
  • Discontinuous: CLVs “jump” into alignment, causing abrupt system responses.
• Physical outcomes:
  • Non-reciprocal transient dynamics.
  • Enhanced noise sensitivity (large phase diffusion, stochastic switching).
  • Destruction of isochrons (loss of unique phase assignment across the system's basin of attraction).
• Applications:
  • Neuroscience: Onset of bistable phase relationships in coupled Wilson-Cowan neurons.
  • Ecology: Phase-locked bifurcations in predator-prey dynamics (Rosenzweig–MacArthur model) (Felski et al., 19 Nov 2024).

4. Topological and Sensing Applications

• Hybrid and Higher-Order Topological NEPs (Bai et al., 2022): Nonlinearity enables realization of higher-dimensional topological invariants (e.g., hybrid topological invariants—HTI) not accessible in linear systems. An "exceptional nexus" (EX) emerges as an order-3 NEP, supporting anisotropic (direction-dependent) critical exponents for phase rigidity and response scaling. These are directly observable in circuit experiments with two coupled resonators employing saturable gain.
• Spectral Topology Control and Enhanced Sensing (Wingenbach et al., 2023): In exciton-polariton condensates described by nonlinear Gross–Pitaevskii equations, the inclusion of Kerr-type nonlinearities and saturable gain can shift and rotate the position of the EPs, tilt the Riemann surfaces, and even make higher-order spectral branches intersect. This controllable topology enables nonlinearity-enhanced sensing capabilities, offering sensitivity increases that can exceed 150% over the linear case.
• NEP-Based Sensors: Noise, SNR, and Feedback
  • Enhanced SNR: In neuromorphic and optical platforms, sensors operating near an NEP demonstrate orders-of-magnitude SNR improvement relative to linear EP-based sensors, due to sublinear response laws (e.g., frequency splitting ∝ $\sqrt{\delta}$ or $\delta^{1/3}$) and built-in nonlinear feedback.
  • Noise Regulation Mechanisms: Nonlinearity (especially saturable gain) provides a feedback mechanism that limits the frequency uncertainty, even in the presence of diverging responsivity. This resolves the fundamental concern that NEPs may not offer viable SNR enhancement, even as Petermann factors diverge in the linearized theory.
  • Fluctuation Dynamics: Fluctuations are governed by nonlinear or Bogoliubov–de Gennes (BdG) operators. Correct treatment predicts that SNR enhancement persists, with frequency uncertainty saturating at a finite value due to nonlinear feedback rather than diverging as in the linear Petermann analysis (Zheng et al., 11 Oct 2024).

5. Nonlinear Topological Invariants and Robustness

Topological characterization of NEPs in nonlinear systems generalizes formulas from linear non-Hermitian physics:

• Winding number for nonlinear eigenproblems (Yoshida et al., 30 Jul 2024):

$$W(\omega_{\text{ref}}) = \oint \frac{dk}{2\pi i} \nabla_k \arg[\det F(\omega_{\text{ref}}, k)]$$

ensuring that NEPs are robust against parameter perturbations.

• Nonlinear pseudo-Hermiticity and Chern numbers:

When additional nonlinear pseudo-Hermiticity exists, a zeroth Chern number is defined via the Hermitianized matrix and its spectrum, correctly quantizing topological changes at NEPs.

• NEPs in single-component systems:

Nonlinear eigenvalue dependence allows for EPs even in 1×1 (single site) systems, as shown for models like the Kapitza pendulum.

6. NEPs in Hermitian Nonlinear Systems and Dynamical Encircling

• EP-like points (ELPs) in Hermitian Kerr Systems (Fang et al., 7 Aug 2024): In coupled resonator systems with nonlinear (Kerr) interactions but no explicit gain/loss, EP-like points arise where the spectrum shows turning (fold) points with square-root (for order-2) or cube-root (for order-3) scaling—critical signatures identical to EPs. Through isospectral mapping, these ELPs correspond uniquely to EPs in linear non-Hermitian systems.
• Chiral State Transfer:

Encircling an ELP in parameter space induces chiral state transfer, mirroring the nonadiabatic state evolution familiar from non-Hermitian EP encircling. The direction of encirclement determines the final state, which can result in a robust mode conversion or comb spectra, relevant for unidirectional transport and nonreciprocal device engineering.

7. NEPs, Non-Hermitian Band Topology, and Nonlinear Hall Effects

• Exceptional Rings and Nonlinear Hall Responses (Qin et al., 10 Nov 2024): In non-Hermitian band structures, such as dissipative Dirac models with exceptional rings, NEPs organize singularities in Berry curvature and higher-order (nonlinear) electronic responses. Broken inversion enables a Berry curvature dipole (BCD), resulting in a nonlinear Hall response that scales with non-Hermiticity as $D_{yxx} \propto \sqrt{\gamma}$, while Berry connection polarizability (BCP) scales as $\propto 1/\sqrt{\gamma}$. These mechanisms offer tunable harmonic generation in optoelectronic devices.

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In summary, Nonlinear Exceptional Points unify bifurcation theory, spectral topology, and non-Hermitian physics in nonlinear dynamical and wave systems. They organize transitions via universal catastrophe structures (e.g., elliptic umbilic), regulate topological responses, and enable robust, tunable, and high-performance sensing and signal processing, with special resilience against noise due to intrinsic nonlinear feedback. Their formulation and technological exploitation are currently a frontier of nonlinear dynamics, open quantum systems, photonics, and condensed matter research.