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Generalized exceptional points in nonlinear and stochastic dynamics

Cheyne Weis1, Michel Fruchart1,2, Ryo Hanai3, Kyle Kawagoe1,4,5, Peter B. Littlewood1,6, and Vincenzo Vitelli1,4

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Phys. Rev. Research 7, 043157 – Published 12 November, 2025

DOI: https://doi.org/10.1103/mnn4-b298

Abstract

We study a class of bifurcations generically occurring in dynamical systems with nonmutual couplings ranging from models of coupled neurons to predator-prey systems and nonlinear oscillators. In these bifurcations, extended attractors such as limit cycles, limit tori, and strange attractors merge and split in a similar way as fixed points in a pitchfork bifurcation. We show that this merging and splitting coincide with the coalescence of covariant Lyapunov vectors with vanishing Lyapunov exponents, a feature that generalizes the exceptional points that can exist in families of non-Hermitian matrices or operators. We distinguish two classes of bifurcations associated with generalized exceptional points, corresponding respectively to continuous and discontinuous behaviors of the covariant Lyapunov vectors at the transition depending on the presence of a Z2 symmetry. We outline some physical consequences of this class of theories exhibiting generalized exceptional points, including nonreciprocal responses, the destruction of isochrons, and anomalous noise effects. In particular, we show that the effective diffusion coefficient on the attractor can stay finite or even diverge when the noise strength vanishes. We illustrate our results with concrete examples from neuroscience, ecology, and physics.

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Phys. Rev. Research 7, 043157– Published 12 November, 2025
  • Received 18 November 2022
  • Revised 2 August 2023
  • Accepted 28 May 2025
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