Fractional calculus has emerged as a powerful mathematical framework for modeling real-world systems characterized by memory, nonlocality, and complex dynamical behaviors. Unlike classical integer-order models, fractional-order formulations provide a more flexible description of processes with long-term memory and hereditary effects, which are commonly observed in physics, engineering, biology, and information sciences. In particular, fractional-order chaotic and hyperchaotic systems have received considerable attention for their ability to capture richer dynamical features, making them highly relevant in areas such as secure communications, signal encryption, random number generation, and nonlinear control. In this work, we explore a four-dimensional fractional-order chaotic model by integrating the Residual Power Series Method (RPSM) and the Caputo fractional derivative (CFD). The CFD is employed to obtain highly accurate numerical simulations that reveal the sensitivity of chaotic attractors to changes in the fractional order, while the RPSM provides analytical approximations that are computationally efficient, stable, and capable of handling diverse initial conditions. This hybrid framework bridges the gap between purely numerical and purely analytical methods, ensuring both accuracy and mathematical insight. Our findings show that fractional order is a key parameter for tuning system behavior between chaotic and stable regimes. This study contributes new insights into the modeling of hyperchaotic systems, with potential relevance for engineering, cryptography, and nonlinear sciences.