Singular basins in multiscale systems: tunneling between stable states

TL;DR

This paper investigates how singular funnel-shaped basins of attraction in multiscale systems influence stability and challenge traditional reduction methods, with implications for understanding resilience in complex systems.

Contribution

It introduces the concept of singular basins with geometric properties affecting system resilience and demonstrates their universality across various multiscale models.

Findings

Singular funnels can extend across phase space, affecting stability.

Singular basins are robust and occur in multiple multiscale systems.

Traditional reduction methods may fail due to singular basin geometry.

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 quasi-static 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.