Dynamics and Chaos Analysis of a Novel 4D Chaotic System Using Constant- and Variable-Order Fractional Calculus

In this work, a new 4D chaotic system is presented with Liouville–Caputo fractional derivatives, including constant-order (C) and variable-order (V). In this regard, the paper examines the equilibrium of the system, local stability, and dissipative property. The use of variable order through incorporating time-dependent fractional order in the system, which includes periodic and exponential functions, is an appropriate way of simulating the adaptive memory effect of the fractional-order systems. The analytical and numerical studies reveal that the proposed system is able to show the chaotic behaviour and sensitive dependence on parameters. Specifically, in terms of maximum Lyapunov exponent and Kaplan–Yorke dimension, the performance of the variable-order system is superior to the constant-order one, reaching the values of λmax≈0.275 and DKY≈2.223. The approach presented here offers a more realistic framework to model memory-dependent chaotic systems and finds applications in the design of nonlinear circuits, secure communications, neuromorphic computing, and advanced control systems.