Life expectancy calculation of transient chaos in the 2D micro-chaos map

Abstract

We analyse a simple model of a digitally controlled mechanical system, which may perform chaotic vibrations. As a consequence of the digital effects, i.e., the sampling and the round-off error, the behaviour of this system can be described by the so-called micro-chaos map. If dry friction is present in the system, it can stop the motion. In such cases the resulting behaviour is referred to as transient chaos, the duration of which can be closely related to the control time. We developed a method for the exact calculation of the mean lifetime N_m of transient chaos in case of the 1D micro-chaos map, and showed that in certain cases N_m characterizes the duration of chaotic transients better than the so-called escape rate. In the present paper, we try to extend these results to a 2D version of the micro-chaos map.