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

为了表征开放动力系统中与瞬态混沌相关的局部有限时间特性，我们在粗粒度描述中引入了适合此目的的逃逸率和分形维数。我们用数值说明这些量词在动力学域中具有相当大的分布，但它们的空间变化，尤其是在长但非渐近积分时间上，与 Kantz 和 Grassberger 对时间渐近量词所认识的关系大致一致。特别是，与这种关系的偏差小于不同位置之间的差异，这证实了这种动力学定律的存在以及我们的量词在非渐近状态下表示潜在动力学特性的适用性。