In this paper, we propose a novel numerical method for solving fractal--fractional differential equations with variable-order and exponential kernel. The method integrates the variable-order fractal derivative with the memory-preserving properties of the exponential kernel to better model complex and chaotic dynamical systems. A new predictor--corrector algorithm is developed to accommodate the nonlocal and nonlinear structure of fractal--fractional operators. The proposed scheme is shown to be accurate, stable, and computationally efficient through a detailed stability analysis. To validate the method, we apply it to several nonlinear systems, including classical and generalized chaotic models such as the Rucklidge, Chua--Hartley, Arnedo, and Wang--Sun systems. Numerical simulations demonstrate that the scheme captures intricate dynamic behavior and chaotic attractors under both constant and variable-order settings. The graphical results confirm that the approach is robust and adaptable for a wide range of applications involving memory-dependent and scale-invariant phenomena. This work contributes a reliable and flexible numerical tool for advancing the study of complex systems governed by variable-order fractional dynamics.
Nagpurkar et al. (Thu,) studied this question.
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