A Comparative Numerical and Semi-Analytical Analysis of a Fractional-Order Symmetric Chaotic System Using SCFD and LRPSM

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.