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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.
We review the occurrence of the patterns of the onset of chaos in low-dimensional nonlinear dissipative systems in leading topics of condensed matter physics and complex systems of various disciplines. We consider the dynamics associated with the att
We numerically study the two-dimensional, area preserving, web map. When the map is governed by ergodic behavior, it is, as expected, correctly described by Boltzmann-Gibbs statistics, based on the additive entropic functional $S_{BG}[p(x)] = -kint d
Universal scaling laws form one of the central issues in physics. A non-standard scaling law or a breakdown of a standard scaling law, on the other hand, can often lead to the finding of a new universality class in physical systems. Recently, we foun
Linking thermodynamic variables like temperature $T$ and the measure of chaos, the Lyapunov exponents $lambda$, is a question of fundamental importance in many-body systems. By using nonlinear fluid equations in one and three dimensions, we prove tha
One-dimensional maps exhibiting transient chaos and defined on two preimages of the unit interval [0,1] are investigated. It is shown that such maps have continuously many conditionally invariant measures $mu_{sigma}$ scaling at the fixed point at x=