Criticality of the distributed-memory Hopf bifurcation

Determine whether the oscillatory instability produced by the distributed gamma-memory kernel is a supercritical or subcritical Hopf bifurcation by computing the first Lyapunov coefficient of the finite-dimensional linear-chain system.

Background

The analysis derives linear Hopf thresholds for the delayed five-box AMOC formulation and shows that oscillatory instability can occur before the saddle-node fold. However, linear stability analysis does not determine the nonlinear character of the bifurcation. A supercritical Hopf bifurcation would produce small-amplitude oscillations after instability, whereas a subcritical Hopf bifurcation could permit finite-amplitude loss of the active state and alter basin geometry. The authors identify computation of the first Lyapunov coefficient for the finite-dimensional chain representation as the relevant unresolved analysis.

References

For the largest target the Floquet calculation returned a neutral-mode error of $1.9\times10{-3}$ at both 512 and 1024 steps per period, above the $10{-3}$ tolerance of the variational integrator, and was rejected; the stability of that orbit is not established.

— Designing collective behaviour within a fixed coarse-grained description  (2609.29834 - Wen et al., 24 Sep 2026) in Supplementary Note 9, subsection “Towards the amplitude boundary”

We have computed linear thresholds only and have not determined the criticality, so no claim about the outcome at onset can be made here.

— Conservation Constraints and Distributed Advective Memory in a Reduced Model of Atlantic Overturning Hysteresis  (2609.24000 - Herho et al., 21 Sep 2026) in Section Discussion, paragraph beginning “The erosion of the steady window”