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Singular Basins in Multiscale Systems: Tunneling between Stable States

S. Yanchuk1,2,*, S. Wieczorek1, H. Jardón-Kojakhmetov3, and H. Alkhayuon1,*

  • *These authors contributed equally to this work.
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Phys. Rev. Lett. 137, 147202 – Published 28 September, 2026

DOI: https://doi.org/10.1103/jtkh-9lz5

Abstract

Real-world systems often evolve on different timescales and possess multiple coexisting stable states. Whether or not a system returns to a given stable state after being perturbed away from it depends on the shape and extent of its basin of attraction. We show that basins of attraction in multiscale systems can exhibit special geometric properties in the form of singular funnels. Although singular funnels are narrow, they can extend to different regions of the phase space and, unexpectedly, impact the system’s resilience to perturbations. Consequently, singular funnels may prevent common dimensionality reductions in the limit of large timescale separation, such as the quasistatic approximation, adiabatic elimination, and time averaging of the fast variables. We refer to basins of attraction with singular funnels as singular basins. We show that singular basins are universal and occur robustly in a range of multiscale systems: the normal form of a pitchfork bifurcation with a slowly adapting parameter, an adaptive active rotator, and an adaptive network of phase rotators.

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Beware the Funnel: When Simplifying Dynamics Can Mislead

Published 28 September, 2026

Simplified models of systems with fast and slow dynamics can miss a narrow region of phase space—a “singular funnel”—leading to misleading predictions of the system’s behavior.

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Phys. Rev. Lett. 137, 147202– Published 28 September, 2026
  • Received 30 January 2026
  • Accepted 24 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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