(untitled) (European Journal of Pure and Applied Mathematics, 2025)
This work applies two efficient techniques to investigate fractional-order systems: the Residual Power Series Method (RPSM) and the Caputo fractional derivative (CFD). Both are utilized, especially to evaluate chaotic behavior and investigate the complex dynamics associated with a four-dimensional fractional-order chaotic system. Reliable and efficient numerical simula- tions with the complexity of chaotic behavior are obtained using the CFD approach. Furthermore, derived analytical solutions of the fractional-order 4D system using the RPSM are obtained. This approach is highly preferred as it can handle several beginning conditions, is computationally efficient, and is numerically stable, hence producing rather precise results. Combining RPSM’s analytical capacity with CFD’s strong numerical accuracy offers a comprehensive understanding of the system dynamics. Although RPSM is best for stability and simplicity of use, the CFD technique is especially helpful because of its great accuracy in predicting chaotic behavior. The suggested methods precisely define their dynamics, provide exact solutions, and clearly identify chaotic attractors. For instance, representative parameter values used in our analysis are (ν = 0.95, 0.99, 1). These results show how well CFD- and RPSM-based methods represent and solve challenging problems in engineering and scientific research.