Non-orbital characterizations of strange attractors: Effective intervals and multifractality measures.

Abstract

Numerical simulations reveal statistical distributions given by power laws resulting from movements of large quantities of phase points captured by strange attractors immersed in one-dimensional or two-dimensional phase spaces, attractors linked to ten specific dynamic systems. Unlike the characterization given by classical approaches as generalized dimensions and spectrum of singularities, the aforementioned distributions do not have their origin in observations of successive orbits, as consequence properties that would otherwise remain hidden are revealed. Specifically, occupancy times and occupancy numbers associated with small hypercubes that cover attractors obey well-defined statistical distributions given by power laws. One application concerns the determination of the intervals in which the most likely values of those numbers and times are located (effective intervals). The use of the effective interval with occupancy numbers to quantify the multifractalities (multifractality measures) is another application. The statistical approaches underlying the results consist of new paradigms that join the well-known classic paradigms to expand knowledge about strange attractors. The possibility that other attractors immersed in spaces with the same dimensions as those considered here exhibit analogous distributions is not ruled out due to the arbitrariness of the set taken.