Calculation of the Statistical Properties in Intermittency Using the Natural Invariant Density

Abstract

We use the natural invariant density of the map and the Perron–Frobenius operator to analytically evaluate the statistical properties for chaotic intermittency. This study can be understood as an improvement of the previous ones because it does not introduce assumptions about the reinjection probability density function in the laminar interval or the map density at pre-reinjection points. To validate the new theoretical equations, we study a symmetric map and a non-symmetric one. The cusp map has symmetry about x=0, but the Manneville map has no symmetry. We carry out several comparisons between the theoretical equations here presented, the M function methodology, the classical theory of intermittency, and numerical data. The new theoretical equations show more accuracy than those calculated with other techniques.

Review of Chaotic Intermittency

“…A new methodology to obtain analytical expressions for statistical functions describing chaotic intermittency is introduced here. This methodology implements the Perron-Frobenius operator, called the continuity technique [39,[65][66][67]. This methodology has been successfully applied to type-II and V intermittencies [65,66].…”

“…It utilizes the Perron-Frobenius operator to compute the reinjection probability density function. In the same way as the methodology of the M function, the continuity technique has been shown to be accurate for several maps displaying different types of intermittency [65][66][67].…”