Symmetric Chaotic Behavior in 4D Variable-Order Fractional Systems Under Liouville–Caputo Operators

Abstract

In this paper, a novel four-dimensional variable-order fractional chaotic system is presented through the use of the Liouville–Caputo variable-order fractional derivative to represent the time-varying memory properties of nonlinear dynamical behaviors. Special attention will be paid to the chaotic behavior of the proposed system and their chaotic attractors depending on different memory effects. For calculation of the solutions of such systems, a numerical approach for the variable-order fractional systems is used, and three memory functions of the variable order are considered to investigate their effect on the behavior of the system. The obtained behavior will be analyzed by means of time responses, phase portraits, bifurcation diagrams, the spectrum of the Lyapunov exponents, and the Kaplan–Yorke fractal dimension. From the numerical simulations, it is shown that different fractional orders lead to completely different dynamical regimes and the appearance of symmetric chaotic attractors with different geometrical and topological structures while the physical parameters of the system are kept the same. These results highlight the important role of time-varying memory in controlling and generating chaotic structures in fractional-order nonlinear systems.