Constructing Higher-Dimensional Digital Chaotic Systems via Loop-State Contraction Algorithm

TL;DR

This paper introduces a new method for constructing higher-dimensional digital chaotic systems using a loop-state contraction algorithm, proves their topological mixing property, and demonstrates improved time series prediction performance.

Contribution

It provides a novel general design method for HDDCS based on loop-state contraction, expanding theoretical understanding and practical applications.

Findings

Proved topological mixing for HDDCS, confirming chaos in the system.

Constructed chaotic echo state networks with improved Mackey-Glass prediction accuracy.

Higher-dimensional systems outperform lower-dimensional ones in prediction tasks.

Abstract

In recent years, the generation of rigorously provable chaos in finite precision digital domain has made a lot of progress in theory and practice, this article is a part of it. It aims to improve and expand the theoretical and application framework of higher-dimensional digital chaotic system (HDDCS). In this study, topological mixing for HDDCS is strictly proved theoretically at first. Topological mixing implies Devaney's definition of chaos in compact space, but not vise versa. Therefore, the proof of topological mixing improves and expands the theoretical framework of HDDCS. Then, a general design method for constructing HDDCS via loop-state contraction algorithm is given. The construction of the iterative function uncontrolled by random sequences (hereafter called iterative function) is the starting point of this research. On this basis, this article put forward a general design method to solve the problem of HDDCS construction, and a number of examples illustrate the effectiveness and feasibility of this method. The adjacency matrix corresponding to the designed HDDCS is used to construct the chaotic echo state network (ESN) for the Mackey-Glass time series prediction. The prediction accuracy is improved with the increase of the size of the reservoir and the dimension of HDDCS, indicating that higher-dimensional systems have better prediction performance than lower-dimensional systems.