Comparative Analysis of Correlation and Kaplan–Yorke Dimensions for Discrete-Time Fractional Systems

Abstract

The aim of this paper is to investigate the discrete-time fractional systems from the following aspects. First, the discrete-time fractional unified system in Caputo sense is established with the help of Euler’s discretization method. Furthermore, the dynamic behaviors of the discrete-time fractional Lü system (DFLS) which is deemed as a representative for unified system are observed. Then, the correlation dimension ([Formula: see text]) and Kaplan–Yorke dimension ([Formula: see text]) of the DFLS are evaluated by the aid of Grassberger–Procaccia algorithm and the Lyapunov exponent spectrum, respectively. Finally, the intrinsic connections between [Formula: see text] and [Formula: see text] are analyzed by the statistical modeling idea when the DFLS is in chaotic vibrations. The main results show that [Formula: see text] shares a positive correlation with [Formula: see text] for the chaotic DFLS, while the differences between [Formula: see text] and [Formula: see text] are not only related to the ratio of the largest and smallest Lyapunov exponents, but also closely tied up with the fractional order [Formula: see text] itself.