Chaotic dynamics of a Fractal-Fractional order Energy-Saving and Emission-Reduction system: stability analysis and machine learning validation

This paper investigates the dynamics of a fractal-fractional order energy-saving and emission-reduction chaotic system, extending the original integer-order model using the Caputo fractal-fractional derivative. The existence and uniqueness of the solution are proved using the Banach fixed point theorem, while Ulam-Hyers stability is demonstrated both analytically and numerically. A fourth-order Runge-Kutta numerical scheme is developed for fractal-fractional systems to simulate the transformed system for various fractal and fractional order values. The numerical results show that the considered fractional-ordered system also exhibits chaotic properties across parameter variations, with trajectories showing sensitivity to both fractional parameters α and β. Furthermore, a neural network approach validates the numerical solutions, achieving exceptional performance with mean square error analysis of 1.1282×10 -5 at epoch 6. The close alignment between integer-order and fractal-fractional-order systems confirms the consistency and physical relevance of the proposed scheme.