This research article presents a comprehensive theoretical and methodological synthesis of stability preserving numerical frameworks designed for stiff, nonlinear, and oscillatory dynamical systems that emerge across applied mathematics, physics, and biological modeling. Drawing exclusively upon the corpus of references provided, the study integrates the conceptual foundations of differential transformation methods, multistage numerical algorithms, Runge Kutta schemes, Padé approximation, and stability preserving solvers to establish a unified interpretive and computational framework. The increasing complexity of modern mathematical models in plasma physics, chemical reaction kinetics, stem cell therapy dynamics, nonlinear oscillators, and infectious disease modeling has produced a critical need for numerical solvers that maintain accuracy, stability, and long time reliability. Classical numerical techniques often encounter severe limitations when applied to stiff systems, strongly nonlinear oscillators, or chaotic regimes, particularly in the presence of sharp transients or sensitive dependence on initial conditions. In response to this challenge, contemporary numerical research has developed advanced strategies that blend analytical transformations with multistage discretization techniques, ensuring robust convergence even in the most demanding mathematical environments.

The differential transformation method has emerged as a powerful semi analytical framework for constructing approximate solutions to differential equations by transforming them into recurrence relations that can be computed iteratively. However, as shown by Bervillier in his extensive theoretical assessment of the method, its naive application may suffer from divergence, slow convergence, or sensitivity to truncation in stiff regimes. These limitations have stimulated the development of multistage differential transformation approaches, as proposed by Do and Jang and further refined by Aljahdaly and Ashi, in which the solution interval is subdivided into multiple segments, each of which applies the differential transformation locally to preserve accuracy and stability. When combined with Padé approximants, as demonstrated by Domairry and Hatami, the resulting hybrid frameworks significantly improve convergence behavior and enable the resolution of nonlinear boundary value problems and nanofluid flow systems with strong coupling effects.