Generalized fractal dimensions and intermittency on coupled map lattices

Özet

The fractal dimension of the wrinkledness of a graph [1] of a function is a measure of the smoothness or the seeming randomness of the function considered. In this study, we introduce the generalized dimensions of the graph, $\beta(q)$, which are the scaling exponents of the moments of the averaged graph length. These will have a nonlinear dependence on the moments, q, if the wrinkledness is not equally distributed. Moreover, the relation between the graph dimensions and the scaling exponent of the $1^{st}$ order structure function [2] can be eneralized. To understand how the non-uniformity of the wrinkledness of the graph is distributed, the generalized dimensions of the support,D(q), are introduced. These dimensions are related with the generalized graph dimensions and the $q^{th}$ order structure functions. D(q) are related to $\beta(q)$, and the scaling exponents of the $q^{th}$ order structure functions, $\zeta_ q$. We have computed $\beta(q)$, $\zeta_ q$, D(q) and the $f(\alpha)$ spectrum for a number of coupled map lattices [3,4], which may be thought as simple replacements for nonlinear partial differential equations [5]. We find that the graph of these CML display multiscaling properties, with $\beta(q)$ and D(q) depending weakly on q.