Asymptotic expansions for the escape rate of stochastically perturbed unimodal maps

TL;DR

This paper investigates the asymptotic expansion of the escape rate in stochastically perturbed unimodal maps, revealing complex oscillatory behavior of the noise dimension parameter related to the system's fractal structure.

Contribution

It provides new data and insights into the behavior of the noise dimension parameter, challenging previous simpler interpretations and showing oscillations with respect to the map parameter.

Findings

The noise dimension parameter exhibits oscillations as a function of the map parameter.

The relation between the noise dimension and fractal dimensions is more complex than previously thought.

Previous models did not account for the oscillatory behavior observed in new data.

Abstract

The escape rate of a stochastic dynamical system can be found as an expansion in powers of the noise strength. In previous work the coefficients of such an expansion for a one-dimensional map were fitted to a general form containing a few parameters. These parameters were found to be related to the fractal structure of the repeller of the system. The parameter alpha, the "noise dimension", remains to be interpreted. This report presents new data for alpha showing that the relation to the dimensions is more complicated than predicted in earlier work and oscillates as a function of the map parameter, in contrast to other dimension-like quantities.