Singular basins in multiscale systems: tunneling between stable states

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.

💡 Research Summary

The paper investigates a previously unnoticed geometric feature of basins of attraction in slow–fast (multiscale) dynamical systems, termed a “singular funnel” (SF). While multistability—the coexistence of several stable attractors—is well studied, the authors show that the shape of the basins can acquire narrow, tube‑like extensions that stretch across large regions of phase space. These SFs become exponentially thin as the timescale separation parameter ε (ratio of slow to fast time) tends to zero, following a volume scaling V(ε) ~ exp(–C/ε). Despite their vanishing measure in the singular limit, they remain present in the full system and enable trajectories to move from one attractor’s basin to another through pathways that are completely absent in reduced models obtained by quasi‑static approximation, adiabatic elimination, or time‑averaging of the fast variables.

The authors first illustrate the phenomenon with a two‑dimensional slow–fast version of the supercritical pitchfork normal form: ˙x = x(µ – x²), ˙µ = ε(–µ + a x – b). For ε≪1, adiabatic elimination yields a one‑dimensional slow equation dµ/dτ = f(µ) whose basins are separated by a single critical value µ = µ_b. In the full two‑dimensional system, however, a narrow SF emanates from the vicinity of (x,µ) = (0,0) and allows trajectories starting on either side of µ_b to converge to the same stable equilibrium e₁. This “tunneling” effect persists for arbitrarily small ε, demonstrating that the reduced model misses a genuine transition pathway. A modified system without the non‑hyperbolic degeneracy (using a tanh nonlinearity) exhibits the same behavior, confirming that the effect is structural rather than a degeneracy artifact.

Next, the paper studies an adaptive phase oscillator: ˙φ = ω + µ – sin φ, ˙µ = ε(–µ + η