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

作者：Ma, Li^*; Huang, Changbao

来源：International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2022, 32(15): 2250222.

DOI：10.1142/S0218127422502224

• Discrete-time fractional system
• correlation dimension
• Kaplan-Yorke dimension
• Lyapunov exponent spectrum
• Grassberger-Procaccia algorithm

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 Lu system (DFLS) which is deemed as a representative for unified system are observed. Then, the correlation dimension (D2) and Kaplan-Yorke dimension (DKY) of the DFLS are evaluated by the aid of Grassberger-Procaccia algorithm and the Lyapunov exponent spectrum, respectively. Finally, the intrinsic connections between D2 and DKY are analyzed by the statistical modeling idea when the DFLS is in chaotic vibrations. The main results show that D2 shares a positive correlation with DKY for the chaotic DFLS, while the differences between D2 and DKY 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.