A Progressive Homotopy Expansion Scheme for Analysis of Fractional Dynamical Systems: Applications on Brusselator Oscillator Networks with Memory Effects

This work develops a progressive homotopy expansion scheme (PHES) for investigating Erdélyi–Kober (EK) fractional dynamical systems with memory effects, nonlinear interactions, and oscillatory behavior. The proposed framework is formulated in suitable Banach spaces with compact–open structures, enabling an operator-theoretic formulation for analytical and computational analysis. Existence and uniqueness of solutions are established using fixed-point arguments under suitable contraction conditions. The PHES is then constructed through recursive homotopy procedures and fractional expansion representations, and its convergence and Hyers–Ulam stability are rigorously analyzed. As an application, the proposed framework is employed to study EK fractional Brusselator oscillator networks describing memory-dependent chemical reaction dynamics. Numerical simulations for different fractional orders illustrate the influence of memory effects on oscillatory behavior, wave propagation, and collective network dynamics. A comparison with the Homotopy Perturbation Method (HPM) demonstrates excellent agreement, while the proposed PHES achieves smaller residual errors and higher numerical accuracy. These results confirm that PHES is an accurate and reliable approach for analyzing nonlinear EK fractional dynamical systems and memory-dependent oscillator networks.