Local characterization of transient chaos on finite times in opensystems

To characterize local finite-time properties associated with transient chaos in open dynamicalsystems, we introduce an escape rate and fractal dimensions suitable for this purpose in acoarse-grained description. We numerically illustrate that these quantifiers have a considerablespread across the domain of the dynamics, but their spatial variation, especially on long butnon-asymptotic integration times, is approximately consistent with the relationship that wasrecognized by Kantz and Grassberger for temporally asymptotic quantifiers. In particular,deviations from this relationship are smaller than differences between various locations, whichconfirms the existence of such a dynamical law and the suitability of our quantifiers to representunderlying dynamical properties in the non-asymptotic regime.