An 8D Hyperchaotic System of Fractional-Order Systems Using the Memory Effect of Grünwald–Letnikov Derivatives

Abstract

We utilize Lyapunov exponents to quantitatively assess the hyperchaos and categorize the limit sets of complex dynamical systems. While there are numerous methods for computing Lyapunov exponents in integer-order systems, these methods are not suitable for fractional-order systems because of the nonlocal characteristics of fractional-order derivatives. This paper introduces innovative eight-dimensional chaotic systems that investigate fractional-order dynamics. These systems exploit the memory effect inherent in the Grünwald–Letnikov (G-L) derivative. This approach enhances the system’s applicability and compatibility with traditional integer-order systems. An 8D Chen’s fractional-order system is utilized to showcase the effectiveness of the presented methodology for hyperchaotic systems. The simulation results demonstrate that the proposed algorithm outperforms existing algorithms in both accuracy and precision. Moreover, the study utilizes the 0–1 Test for Chaos, Kolmogorov–Sinai (KS) entropy, the Kaplan–Yorke dimension, and the Perron Effect to analyze the proposed eight-dimensional fractional-order system. These additional metrics offer a thorough insight into the system’s chaotic behavior and stability characteristics.

Keywords

• Lyapunov exponents
• Grünwald–Letnikov derivative
• fractional-order systems
• Kaplan–Yorke dimension
• hyperchaotic system