Generalized Fractal Dimensions and Intermittency on Coupled Map Lattices

Authors: Ayşe GORBON, Ayşe ERZAN

Abstract: 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 generalized. 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 non--linear 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.

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