Fractional order dynamical systems have emerged as one of the most powerful mathematical frameworks for understanding complex physical, biological, and engineering phenomena that exhibit memory, hereditary behavior, and long term dependency. Classical integer order models often fail to capture the full richness of such processes, particularly when chaos, stiffness, or multiscale dynamics are involved. The fractional order Rossler system, in particular, represents one of the most widely studied prototypes for fractional chaotic dynamics because it combines nonlinear feedback, sensitive dependence on initial conditions, and nonlocal temporal effects. However, solving such systems remains a fundamental challenge because fractional derivatives introduce nonlocality, while chaos amplifies numerical errors, and stiffness creates extreme sensitivity to step size. This article develops a comprehensive analytic and numerical investigation of fractional order chaotic and stiff systems using the differential transform method and its multi step and adaptive extensions. Building strictly upon the theoretical and computational foundations established by Zhang and Zhou, Freihat and Momani, Odibat and collaborators, Alomari, Yildirim, Gokdogan, and other contributors, the study examines how generalized and multi step differential transform techniques overcome the core limitations of classical solvers such as Runge Kutta and standard MATLAB routines when applied to chaotic fractional dynamics. The methodology section presents a detailed conceptual explanation of how the differential transform method converts differential equations into recursive algebraic structures that preserve nonlinearity and fractional memory while maintaining computational efficiency. The results demonstrate that multi step and adaptive differential transform strategies achieve superior stability, accuracy, and long time convergence when compared to conventional numerical approaches, particularly in stiff and chaotic regimes. The discussion elaborates on the theoretical implications of these findings for chaos theory, fractional calculus, and numerical analysis, and highlights how the interaction between memory effects and adaptive step control leads to qualitatively different dynamical representations. Limitations of the approach are critically analyzed, including convergence radius, truncation sensitivity, and computational cost. Finally, the article outlines future research directions for hybrid fractional chaos solvers and real world applications in physics, engineering, and biological modeling.