Determine the zero-coupling escape rate and lifetime

Determine the escape rate and mean lifetime of trajectories in the alternating-feedback dynamical system in the limit as the coupling parameter a approaches zero, where the escape rate is expected to vanish and the transient lifetime is expected to diverge.

Background

The paper studies an alternating-feedback system in which chaotic dynamics of the variable x induce diffusive motion in the feedback variable y, while periodic regimes produce long-lived transient attractors. The coupling parameter a controls both the diffusive scale and the cycle-to-cycle escape step from periodic regions. As a decreases, trajectories remain near transient-attractor basin boundaries for longer periods, producing increasingly long transients.

Numerical results indicate that the escape rate tends toward zero and the mean lifetime tends toward infinity as a approaches zero. However, the authors report that they cannot numerically determine either quantity in this limiting regime. Establishing their asymptotic behavior would clarify the precise scaling of super-long transients beyond the finite-a numerical regime.

References

Here, we are unable to obtain the escape rate or lifetime numerically in the limit of $a\to 0$, but we can see that the lifetime grows extremely large as $a \to 0$.

— A general feedback mechanism unifying stochasticity, long transients, and resilience  (2610.21834 - Chai et al., 15 Oct 2026) in Section 3, Numerical Results, discussion following Eq. (NLM_cutoff) and Fig. NLM_Sur_a