A numerical estimate of the regularity of a family of Strange Non--Chaotic Attractors

TL;DR

This paper numerically estimates the regularity of a family of Strange Non--Chaotic Attractors using wavelet analysis, overcoming challenges posed by their discontinuous and implicit nature.

Contribution

It introduces a novel algorithm based on the Fast Wavelet Transform to estimate regularity of complex attractors without explicit formulas.

Findings

Successful estimation of attractor regularity despite discontinuities

Development of ad-hoc techniques for wavelet analysis in complex systems

Validation of the method through quality checks of wavelet coefficients

Abstract

We estimate numerically the regularities of a family of Strange Non--Chaotic Attractors related with one of the models studied in C. Grebogy et al. (1984) (see also G. Keller (1996)). To estimate these regularities we use wavelet analysis in the spirit of R. de la Llave et al. (2002) together with some ad-hoc techniques that we develop to overcome the theoretical difficulties that arise in the application of the method to the particular family that we consider. These difficulties are mainly due to the facts that we do not have an explicit formula for the attractor and it is discontinuous almost everywhere for some values of the parameters. Concretely we propose an algorithm based on the Fast Wavelet Transform. Also a quality check of the wavelet coefficients and regularity estimates is done.