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

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 ( D 2 ) and Kaplan–Yorke dimension ( D K Y ) of the DFLS are evaluated by the aid of Grassberger–Procaccia algorithm and the Lyapunov exponent spectrum, respectively. Finally, the intrinsic connections between D 2 and D K Y are analyzed by the statistical modeling idea when the DFLS is in chaotic vibrations. The main results show that D 2 shares a positive correlation with D K Y for the chaotic DFLS, while the differences between D 2 and D K Y are not only related to the ratio of the largest and smallest Lyapunov exponents, but also closely tied up with the fractional order v itself. [ABSTRACT FROM AUTHOR]