This paper investigates the dynamical properties of the fractional-order Arneodo system using a Grünwald–Letnikov-based numerical discretization. Fractional-order operators introduce memory and hereditary effects, enabling a more realistic description than classical integer-order models. The local stability of equilibrium points is examined through eigenvalue analysis of the Jacobian matrix, along with dissipativity conditions and the emergence of complex attractors. A comprehensive dynamical investigation is presented through phase portraits, time series, Lyapunov exponents, and bifurcation diagrams for varying fractional orders. Numerical findings demonstrate the emergence of new chaotic and hyperchaotic attractors. The results confirm that the fractional order strongly influences the system’s stability, sensitivity, and complexity. Our results confirm the relevance of fractional-order modeling in applications, such as secure communication, random number generation, and complex system analysis.

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.

The main aim of this paper is to investigate the dynamics of the fractional-order Lorenz system within a unified computational framework using both semi-analytical and numerical approaches. The Laplace Residual Power Series Method (LRPSM) is employed to obtain computationally efficient semi-analytical solutions, while a numerical scheme based on the Caputo fractional derivative (SCFD) is developed for the considered system. The resulting dynamics are systematically analyzed through Lyapunov exponents, bifurcation diagrams, phase-space portraits, and time series to characterize the transition to fractional-order chaos and the geometric structures of the resulting attractors. For validation, the results obtained by LRPSM and SCFD are compared with those of the Adams–Bashforth–Moulton predictor–corrector method. The critical fractional order associated with the onset of chaotic behavior is also determined, and the corresponding fractional chaotic attractors are successfully obtained. The results demonstrate that LRPSM and SCFD provide consistent and computationally efficient approaches for studying nonlinear fractional-order dynamics.