Determine the scaling coefficient α governing crisis-induced escape probabilities

Determine the coefficient α in the asymptotic scaling law Q∞(δ) ≈ exp[−α (ln δ)² + β ln δ] for the escape probability near crisis transitions in non-autonomous (time-dependent) dynamical systems, beyond the simple stochastic one-dimensional model where α is linked to the instability of the fixed point whose stable manifold forms the basin boundary; clarify how the attractor’s fractal dimension and temporal correlations of the modulation affect α.

Background

The paper establishes that crises occur in non-autonomous systems and that the escape probability near the transition follows a nonstandard scaling of the form exp[−α (ln δ)² + β ln δ], where δ is the distance from criticality and α is model-dependent. In the simple linear stochastic grey-zone model, α is related to the instability of a fluctuating saddle that defines the basin boundary.

However, for more complex systems (e.g., stochastically modulated logistic and Hénon maps and the Kuramoto model with inertia), the precise determination of α is left open, with indications that properties such as attractor fractal dimension and temporal correlations likely influence α. The authors explicitly state this as future work.